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Purchased from American Mathematical Society for the exclusive use of Dipesh Bhattarai (bhdpxc) Copyright 1997 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.

Selected Title s i n Thi s Serie s

36 J o s e p h G . R o s e n s t e i n , D e b o r a h S . P r a n z b l a u , a n d Fre d S . R o b e r t s , E d i t o r s , Discrete Mathematic s i n th e School s

35 D i n g z h u D u , J u n G u , a n d P a n o s M . P a r d a l o s , E d i t o r s , Satisfiabilit y Problem :

Theory an d Application s

34 N a t h a n i e l D e a n , E d i t o r , Africa n American s i n Mathematic s

33 R a v i B . B o p p a n a a n d J a m e s F . L y n c h , E d i t o r s , Logi c an d rando m structure s

32 J e a n - C h a r l e s G r ^ g o i r e , G e r a r d J . H o l z m a n n , a n d D o r o n A . P e l e d , E d i t o r s , T h e

S P I N verificatio n syste m

31 N e i l I m m e r m a n a n d P h o k i o n G . K o l a i t i s , E d i t o r s , Descriptiv e complexit y an d finite models

30 S a n d e e p N . B h a t t , E d i t o r , Paralle l Algorithms : Thir d DIMAC S Implementatio n Challenge

29 D o r o n A . P e l e d , V a u g h a n R . P r a t t , a n d G e r a r d J . H o l z m a n n , E d i t o r s , P a r t i a l Order Method s i n Verificatio n

28 L a r r y F i n k e l s t e i n a n d W i l l i a m M . K a n t o r , E d i t o r s , Group s an d C o m p u t a t i o n I I

27 R i c h a r d J . L i p t o n a n d Eri c B . B a u m , E d i t o r s , DN A Base d C o m p u t e r s

26 D a v i d S . J o h n s o n a n d M i c h a e l A . Trick , E d i t o r s , Cliques , Coloring , an d Satisfiability: Secon d DIMAC S Implementatio n Challeng e

25 G i l b e r t B a u m s l a g , D a v i d E p s t e i n , R o b e r t G i l m a n , H a m i s h S h o r t , a n d C h a r l e s S i m s , E d i t o r s , Geometri c an d Computationa l Perspective s o n Infinit e Group s

24 L o u i s J . B i l l e r a , C u r t i s G r e e n e , R o d i c a S i m i o n , a n d R i c h a r d P . S t a n l e y , E d i t o r s , Formal Powe r Serie s an d Algebrai c Combinatorics/Serie s formelle s e t combinatoir e algebrique, 199 4

23 P a n o s M . P a r d a l o s , D a v i d I . S h a l l o w a y , a n d G u o l i a n g X u e , E d i t o r s , Globa l Minimization o f Nonconve x Energ y Functions : Molecula r Conformatio n an d Protei n Foldin g

22 P a n o s M . P a r d a l o s , M a u r i c i o G . C . R e s e n d e , a n d K . G . R a m a k r i s h n a n , E d i t o r s , Parallel Processin g o f Discret e Optimizatio n Problem s

21 D . Fran k H s u , A r n o l d L . R o s e n b e r g , a n d D o m i n i q u e S o t t e a u , E d i t o r s , Interconnection Network s an d Mappin g an d Schedulin g Paralle l C o m p u t a t i o n s

20 W i l l i a m C o o k , L a s z l o L o v a s z , a n d P a u l S e y m o u r , E d i t o r s , Combinatoria l Optimization

19 I n g e m a r J . C o x , P i e r r e H a n s e n , a n d B e l a J u l e s z , E d i t o r s , Partitionin g D a t a Set s

18 G u y E . B l e l l o c h , K . M a n i C h a n d y , a n d S u r e s h J a g a n n a t h a n , E d i t o r s , Specification o f Paralle l Algorithm s

17 Eri c S v e n R i s t a d , E d i t o r , Languag e Computation s

16 P a n o s M . P a r d a l o s a n d H e n r y W o l k o w i c z , E d i t o r s , Quadrati c Assignmen t an d Related Problem s

15 N a t h a n i e l D e a n a n d G r e g o r y E . S h a n n o n , E d i t o r s , C o m p u t a t i o n a l Suppor t fo r Discrete Mathematic s

14 R o b e r t C a l d e r b a n k , G . D a v i d Forney , J r . , a n d N a d e r M o a y e r i , E d i t o r s , Codin g and Quantization : D I M A C S / I E E E Worksho p

13 J i n - Y i C a i , E d i t o r , Advance s i n Computationa l Complexit y Theor y

12 D a v i d S . J o h n s o n a n d C a t h e r i n e C . M c G e o c h , E d i t o r s , Networ k Flow s an d Matching: Firs t DIMAC S Implementatio n Challeng e

11 Larr y F i n k e l s t e i n a n d W i l l i a m M . K a n t o r , E d i t o r s , Group s an d C o m p u t a t i o n

10 J o e l F r i e d m a n , E d i t o r , Expandin g Graph s

9 W i l l i a m T . T r o t t e r , E d i t o r , Plana r Graph s

8 S i m o n G i n d i k i n , E d i t o r , Mathematica l Method s o f Analysi s o f Biopolyme r Sequence s

7 L y l e A . M c G e o c h a n d D a n i e l D . S l e a t o r , E d i t o r s , On-Lin e Algorithm s

6 J a c o b E . G o o d m a n , R i c h a r d P o l l a c k , a n d W i l l i a m S t e i g e r , E d i t o r s , Discret e and Computationa l Geometry : Paper s fro m t h e DIMAC S Specia l Yea r

(Continued in the back of this publication) Purchased from American Mathematical Society for the exclusive use of Dipesh Bhattarai (bhdpxc) Copyright 1997 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.

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DIMACS Series i n Discret e Mathematic s

a n d Theoretica l Compute r Scienc e

V o l u m e 3 6

Discrete Mathematic s in th e School s

J o s e p h G . Rosenstei n Deborah S . Franzbla u

Fred S . Robert s Editors

NSF Scienc e an d Technolog y Cente r in Discret e Mathematic s an d Theoretica l Compute r Scienc e A consortiu m o f Rutger s University , Princeto n University ,

AT&T Labs , Bel l Labs , an d Bellcor e

American Mathematica l Societ y National Counci l o f Teacher s o f Mathematic s

I ^ I C T I V I

https://doi.org/10.1090/dimacs/036

T h i s D I M A C S v o l u m e i s a collectio n o f a r t i c l e s b y e x p e r i e n c e d e d u c a t o r s e x p l a i n i n g w h y a n d h o w d i s c r e t e m a t h e m a t i c s c a n a n d s h o u l d b e t a u g h t i n K - 1 2 c l a s s r o o m s . I t als o d i s c u s s e s h o w d i s c r e t e m a t h e m a t i c s c a n b e u s e d a s a vehicl e fo r a c h i e v i n g t h e b r o a d e r goals o f t h e m a j o r effor t n o w u n d e r w a y t o i m p r o v e m a t h e m a t i c s e d u c a t i o n . T h i s v o l u m e d e v e l o p e d fro m a conferenc e t h a t t o o k p l a c e a t R u t g e r s U n i v e r s i t y o n O c t o b e r 2 - 4 , 1992 .

1991 Mathematics Subject Classification. P r i m a r y 0 0 A 0 5 , 0 0 A 3 5 .

L i b r a r y o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a

Discrete mathematic s i n th e school s / Josep h G . Rosenstein , Debora h S . Franzblau , Fre d S . Roberts, editors .

p. cm . — (DIMAC S serie s i n discret e mathematic s an d theoretica l compute r science , ISS N 1052-1798 ; v . 36 )

P a p e r s fro m a conferenc e hel d a t DIMAC S a t Rutger s Universit y i n Oct . 1992 . "NSF Scienc e an d Technolog y Cente r i n Discret e Mathematic s an d Theoretica l Compute r

Science. A consortiu m o f Rutger s University , Princeto n University , AT& T Labs , Bel l Labs , an d Bellcore."

Includes bibliographica l references . ISBN 0-8218-0448- 0 (hardcove r : alk . paper ) 1. Mathematics—Stud y an d teaching—Congresses . I . Rosenstein , Josep h G . II . Franzblau ,

Deborah S. , 1957 - . III . Roberts , Fre d S . IV . NS F Scienc e an d Technolog y Cente r i n Discret e Mathematics an d Theoretica l Compute r Science . V . Series . QA11.A1D57 199 7 511'.07'1—dc21 97-2327 7

C I P

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10 9 8 7 6 5 4 3 2 0 3 0 2 0 1 0 0

C o n t e n t s

Foreword i x

Preface x i

Vision Statemen t fro m 199 2 Conferenc e xii i

Overview an d Abstract s x v

Introduction Discrete Mathematic s i n th e Schools : A n Opportunit y t o Revitalize Schoo l Mathematic s J O S E P H G . R O S E N S T E I N xxii i

Section 1 . T h e Valu e o f Discret e M a t h e m a t i c s : V i e w s fro m t h e Classroo m

The Impac t o f Discret e Mathematic s i n M y Classroo m B R O . P A T R I C K C A R N E Y 3

Three fo r th e Money : A n Hou r i n th e Classroo m N A N C Y C A S E Y 9

Fibonacci Reflections—It' s Elementary ! J A N I C E C . KOWALCZY K 2 5

Using Discret e Mathematic s t o Giv e Remedia l Student s a Secon d Chanc e SUSAN H . P I C K E R 3 5

What We'v e Go t Her e I s a Failur e t o Cooperat e R E U B E N J . S E T T E R G R E N 4 3

Section 2 . T h e Valu e o f Discret e M a t h e m a t i c s : Achieving Broade r Goal s

Implementing th e Standards : Let' s Focu s o n th e Firs t Fou r N A N C Y C A S E Y AN D M I C H A E L R . F E L L O W S 5 1

Discrete Mathematics : A Vehicl e fo r Proble m Solvin g an d Excitemen t M A R G A R E T B . COZZEN S 6 7

Logic an d Discret e Mathematic s i n th e School s SUSANNA S . E P P 7 5

vi C O N T E N T S

Writing Discrete(ly ) R O C H E L L E L E I B O W I T Z 8 5

Discrete Mathematic s an d Publi c Perception s o f Mathematic s J O S E P H M A L K E V I T C H 8 9

Mathematical Modelin g an d Discret e Mathematic s H E N R Y O . P O L L A K 9 9

The Rol e o f Application s i n Teachin g Discret e Mathematic s

F R E D S . R O B E R T S 10 5

Section 3 . W h a t I s Discret e M a t h e m a t i c s : T w o P e r s p e c t i v e s

What I s Discret e Mathematics ? Th e Man y Answer s S T E P H E N B . M A U R E R 12 1

A Comprehensiv e Vie w o f Discret e Mathematics : Chapte r 1 4 o f th e Ne w Jersey Mathematic s Curriculu m Framewor k J O S E P H G . R O S E N S T E I N 13 3

Section 4 . Integratin g Discret e M a t h e m a t i c s int o Existin g M a t h e m a t i c s Curricula , Grade s K - 8

Discrete Mathematic s i n K- 2 Classroom s VALERIE A . D E B E L L I S 18 7

Rhythm an d Pattern : Discret e Mathematic s wit h a n Artisti c Connectio n fo r Elementary Schoo l Teacher s R O B E R T E . JAMISO N 20 3

Discrete Mathematic s Activitie s fo r Middl e Schoo l EVAN M A L E T S K Y 22 3

Section 5 . Integratin g D i s c r e t e M a t h e m a t i c s int o Existin g M a t h e m a t i c s Curricula , Grade s 9—1 2

Putting Chao s int o Calculu s Course s R O B E R T L . DEVANE Y 23 9

Making a Differenc e wit h Differenc e Equation s J O H N A . D O S S E Y 25 5

Discrete Mathematica l Modelin g i n th e Secondar y Curriculum : Rational e and Example s fro m Th e Core-Plu s Mathematic s Projec t E R I C W . H A R T 26 5

A Discret e Mathematic s Experienc e wit h Genera l Mathematic s Student s B R E T H O Y E R 28 1

Algorithms, Algebra , an d th e Compute r La b P H I L I P G . L E W I S 289

CONTENTS vi i

Discrete Mathematic s I s Alread y i n th e Classroo m - Bu t It' s Hidin g J O A N R E I N T H A L E R 29 5

Integrating Discret e Mathematic s int o th e Curriculum : A n Exampl e

J A M E S T . SANDEFU R 30 1

Section 6 . Hig h Schoo l Course s o n D i s c r e t e M a t h e m a t i c s

The Statu s o f Discret e Mathematic s i n th e Hig h School s H A R O L D F . BAILE Y 31 1

Discrete Mathematics : A Fres h Star t fo r Secondar y Student s L. C H A R L E S B I E H L 31 7

A Discret e Mathematic s Textboo k fo r Hig h School s

N A N C Y C R I S L E R , P A T I E N C E F I S H E R , AN D G A R Y F R O E L I C H 32 3

Section 7 . Discret e M a t h e m a t i c s an d Compute r Scienc e

Computer Science , Proble m Solving , an d Discret e Mathematic s P E T E R B . H E N D E R S O N 33 3

The Rol e o f Compute r Scienc e an d Discret e Mathematic s i n th e Hig h Schoo l Curriculum V I E R A K . P R O U L X 34 3

Section 8 . R e s o u r c e s fo r Teacher s

Discrete Mathematic s Softwar e fo r K-1 2 Educatio n NATHANIEL D E A N AN D YANX I L I U 35 7

Recommended Resource s fo r Teachin g Discret e Mathematic s D E B O R A H S . FRANZBLA U AN D J A N I C E C . KOWALCZY K 37 3

The Leadershi p Progra m i n Discret e Mathematic s J O S E P H G . ROSENSTEI N AN D V A L E R I E A . D E B E L L I S 41 5

Computer Softwar e fo r th e Teachin g o f Discret e Mathematic s i n th e School s M A R I O VASSALL O AN D A N T H O N Y R A L S T O N 43 3

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Foreword

This DIMAC S volum e o n "Discret e Mathematic s i n th e Schools " con - tains referee d article s o n th e them e o f a conferenc e hel d a t DIMAC S a t Rutgers Universit y i n Octobe r 1992 . Th e conferenc e wa s sponsore d b y DI - MACS wit h fundin g fro m th e Nationa l Scienc e Foundation .

We woul d especiall y lik e t o than k Josep h G. Rosenstei n fo r organizin g the conference , an d hi m an d Debora h S . Franzbla u wh o togethe r wit h Fre d S. Robert s serve d a s editor s o f thi s volume .

Fred S . Roberts , Directo r Bernard Chazelle , co-Directo r Stephen R . Mahaney , Associat e Directo r

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Preface

Discrete mathematics ca n and should be taught i n K-12 classrooms. Thi s volume, a collectio n o f article s b y experience d educators , explain s wh y an d how, includin g evidenc e fo r "why " an d practica l guidanc e o n "how" . I t als o discusses ho w discret e mathematic s ca n b e use d a s a vehicl e fo r achievin g the broade r goal s of the majo r effor t no w underway t o improv e mathematic s education.

This volum e i s intende d fo r severa l differen t audiences . Teacher s a t all grad e level s wil l fin d her e a grea t dea l o f valuabl e materia l tha t wil l help the m introduc e discret e mathematic s i n thei r classrooms , a s wel l a s examples o f innovativ e teachin g techniques . Schoo l an d distric t curriculu m leaders wil l fin d article s tha t addres s thei r question s o f whethe r an d ho w discrete mathematic s ca n b e introduce d int o thei r curricula . Colleg e facult y will fin d idea s an d topic s tha t ca n b e incorporate d int o a variet y o f courses , including mathematic s course s fo r prospectiv e teachers . A descriptio n o f the organizatio n o f thi s volum e an d a n annotate d summar y o f th e article s it contain s ca n b e foun d i n th e O v e r v i e w a n d A b s t r a c t s .

This volum e develope d fro m a conferenc e tha t too k plac e a t Rutger s University o n Octobe r 2-4 , 1992 . Th e conference , entitle d "Discret e Mathe - matics i n th e Schools : Ho w D o W e Mak e a n Impact? " wa s attende d b y 3 3 people, fro m hig h school s an d colleges , wh o ha d playe d leadershi p role s i n introducing discret e mathematic s a t precolleg e levels. 1 Th e conferenc e wa s sponsored b y th e Cente r fo r Discret e Mathematic s an d Theoretica l Com - puter Scienc e (DIMACS) 2 an d funde d b y th e Nationa l Scienc e Foundatio n (NSF).

The invitatio n t o th e conferenc e note d tha t "Althoug h primaril y a re -

lA lis t o f conferenc e participant s an d a n abbreviate d conferenc e progra m appea r a s appendices t o t h e I n t r o d u c t i o n .

2 DIMACS i s a n NSF-funde d Scienc e an d Technolog y Cente r whic h wa s founde d i n 1989 a s a consortiu m o f Rutger s an d Princeto n Universities , AT& T Bel l Laboratories , and Bellcor e (Bel l Communication s Research) . Wit h th e reorganizatio n o f AT& T Bel l Laboratories i n 1996 , i t wa s replace d i n th e DIMAC S consortiu m b y AT& T Lab s an d Bell Lab s (par t o f Lucen t Technologies) . DIMAC S i s als o funde d b y th e Ne w Jerse y Commission o n Scienc e an d Technology , it s partne r organizations , an d numerou s othe r agencies.

X l l PREFACE

search center , DIMAC S i s committe d t o educationa l program s involvin g discrete mathematic s . . . a s discret e mathematic s activitie s a t K-1 2 level s increase, i t i s appropriat e fo r a nationa l cente r i n discret e mathematic s t o bring togethe r thos e associate d wit h suc h activitie s fo r a n opportunit y t o reflect o n ho w al l o f ou r activitie s ca n mak e a n impac t o n mathematic s ed - ucation nationally. " Th e rational e fo r th e conferenc e i s further describe d i n the Introduction , an d th e Visio n Statemen t concernin g discret e math - ematics i n th e school s tha t emerge d fro m th e conferenc e appear s directl y after thi s Preface .

This volum e wa s originall y conceive d a s th e proceeding s o f th e confer - ence. However , a s w e bega n receivin g an d reviewin g articles , w e realize d that a n expande d an d mor e comprehensiv e boo k woul d hav e greate r valu e and impact . Accordingly , w e solicite d additiona l article s fro m appropriat e authors; approximatel y two-third s o f th e article s ar e base d o n conferenc e presentations, an d th e remainde r wer e writte n independently . Al l o f th e authors receive d comment s an d suggestion s fro m bot h anonymou s referee s and th e editors , an d revise d thei r article s accordingly ; thi s lengthene d con - siderably th e tim e t o produc e th e volume , bu t greatl y enhance d it s quality .

The editor s wis h t o than k th e author s fo r thei r cooperatio n an d pa - tience, a s wel l as for thei r contributions . W e also thank th e referee s fo r thei r assistance, Reube n Settergre n fo r man y hour s spen t i n editoria l work , type - setting, an d creatin g figures, Pa t Pravat o fo r he r abl e secretaria l help , an d NSF fo r a supplementar y gran t tha t enable d u s t o complet e th e volume .

Compiling a volume like this, involvin g 34 articles from differen t authors , is no t a n eas y task , an d w e ar e quit e please d tha t thi s tas k ha s no w bee n completed.

Joseph G. Rosenstei n Deborah S . Franzbla u Fred S . Robert s

Vision S t a t e m e n t fro m 199 2 Conference 1

A majo r refor m effor t i s no w underwa y i n mathematic s education . Th e goals o f thi s refor m ar e t o enabl e u s t o educat e informe d citizen s wh o ar e better abl e t o functio n i n ou r increasingl y technologica l society ; hav e bette r reasoning powe r an d problem-solvin g skills ; ar e awar e o f th e importanc e o f mathematics i n ou r society ; an d ar e prepare d fo r futur e career s whic h wil l require ne w an d mor e sophisticate d analytica l an d technica l tools .

We fee l tha t discrete mathematics is an exciting and appropriate vehi- cle for working toward and achieving these goals. I t i s a n excellen t too l fo r improving reasonin g an d problem-solvin g skills . I t lend s itsel f wel l t o th e evolving consensus o n effective instructiona l strategie s expresse d i n the Cur- riculum and Evaluation Standards for School Mathematics o f th e Nationa l Council o f Teacher s o f Mathematic s (NCTM) . Discret e mathematic s ha s many practica l application s tha t ar e usefu l fo r solvin g som e o f the problem s of ou r societ y an d tha t ar e meaningfu l t o ou r students . It s problem s mak e mathematics com e aliv e fo r students , an d hel p the m se e th e relevanc e o f mathematics t o th e rea l world . Discret e mathematic s doe s no t hav e exten - sive prerequisites , ye t pose s challenge s t o al l students . I t i s fu n t o do , i s often geometricall y based , an d stimulate s a n interes t i n mathematic s o n th e part o f student s a t al l level s an d o f al l abilities .

At th e sam e time , w e fee l tha t discrete mathematics needs to be intro- duced into the K-12 curriculum for its own sake. Durin g th e pas t 3 0 years , discrete mathematic s ha s grow n rapidl y an d ha s becom e a significan t are a of mathematics . Increasingly , discret e mathematic s i s the mathematic s tha t is bein g use d b y decision-maker s i n busines s an d government ; b y worker s in field s suc h a s telecommunication s an d computin g tha t depen d upo n in - formation transmission ; an d b y thos e i n man y rapidl y changin g profession s involving healt h care , biology , chemistry , automate d manufacturing , trans - portation, etc . Increasingly , discret e mathematic s i s the languag e o f a larg e body o f scienc e an d underlie s decision s tha t individual s wil l hav e t o mak e

*An initia l draf t o f thi s "visio n statement " wa s develope d durin g th e Octobe r 199 2 conference, reflectin g th e goal s of the conferenc e an d th e consensu s o f it s participants. Th e statement wa s revise d a t a meetin g o f a designate d committe e o f conferenc e participant s the followin g January .

xiii

XIV VISION STATEMEN T

in thei r ow n lives , i n thei r professions , an d a s citizens . It shoul d b e stressed , however , tha t we are not advocating any specific

set of topics in discrete mathematics that should be taught; discret e math - ematics include s man y differen t areas , eac h o f whic h i s valuable . Rather , we fee l i t i s importan t tha t student s b e abl e t o spea k th e languag e o f dis - crete mathematic s an d b e expose d t o th e way s o f thinkin g an d reasonin g that ar e inheren t i n moder n discret e mathematics ; al l student s shoul d kno w and b e abl e t o appl y discret e mathematic s concept s an d skill s i n a variet y of contexts . An d i t i s especially importan t fo r teacher s t o becom e excite d about thei r ow n experience s wit h discret e mathematic s an d t o shar e tha t excitement wit h thei r students .

Overview an d A b s t r a c t s

As noted i n the Preface , thi s volume makes the cas e that discret e math - ematics shoul d b e include d i n K-1 2 classroom s an d curricula , an d provide s practical assistanc e an d guidanc e o n ho w this ca n b e accomplished . Th e or - ganization o f this volum e parallel s thes e tw o goals. Afte r th e Introductio n the article s ar e arrange d i n th e followin g eigh t clusters :

Section 1 . Th e Valu e o f Discret e Mathematics : View s fro m th e Classroom

Section 2 . Th e Value of Discrete Mathematics: Achievin g Broade r Goals

Section 3 . Wha t i s Discret e Mathematics : Tw o Perspective s

Section 4 . Integratin g Discrete Mathematics int o Existing Math - ematics Curricula , Grade s K- 8

Section 5 . Integratin g Discret e Mathematics int o Existing Math - ematics Curricula , Grade s 9-1 2

Section 6 . Hig h Schoo l Course s o n Discret e Mathematic s Section 7 . Discret e Mathematic s an d Compute r Scienc e Section 8 . Resource s fo r Teacher s

Everyone's firs t questio n i s o f course , "Wha t i s discret e mathematics? " Everyone's secon d questio n is , "Wh y shoul d I us e discret e mathematics? " Explicit discussio n o f th e firs t questio n i s delaye d unti l Sectio n 3 , an d th e focus o f th e Introductio n an d Section s 1- 2 i s th e secon d question . Thes e sections mak e th e cas e fo r discret e mathematic s — fro m th e perspectiv e o f teachers i n th e classroom , an d fro m th e perspectiv e o f researcher s involve d in improvin g mathematic s education . Thes e article s encompas s a variet y of agenda s — implementin g th e fou r NCT M proces s standard s (problem - solving, reasoning , communicatin g mathematica l ideas , an d makin g con - nections), improvin g th e public' s perceptio n o f mathematics , conveyin g th e usefulness o f mathematics , an d providin g a ne w star t fo r students , teachers , and curricula .

Everyone's thir d questio n is , "Ho w can I use discrete mathematic s i n m y classroom?" Thi s questio n i s addressed i n Sections 4-7. On e set o f response s

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involves incorporatin g discret e mathematic s int o existin g curricula ; thes e responses appea r i n Section s 4 an d 5 , arrange d b y grad e level . Anothe r se t of response s involve s introducin g ne w courses , typicall y a t th e hig h schoo l level, an d thes e ar e addresse d i n Sectio n 6 . Sectio n 7 addresse s th e rol e o f computer scienc e in the hig h school curriculum, a s well as the rol e of discret e mathematics i n th e teachin g o f compute r science .

Section 8 describe s resource s availabl e t o teacher s wh o decid e t o enric h their classroom s wit h discret e mathematics .

Following ar e abstract s o f th e article s i n thi s volume , prepare d b y th e editors. Th e abstract s ar e arrange d b y section , an d withi n eac h sectio n ar e presented alphabetically , a s ar e th e article s i n th e volume .

Introduction

Joseph G. Rosenstein's article Discrete M a t h e m a t i c s i n t he Schools : A n Opportunit y t o Revitaliz e Schoo l M a t h e m a t i c s serve s a s a n in - troduction t o thi s volum e an d describe s wh y discret e mathematic s ca n b e a useful vehicl e fo r improvin g mathematic s educatio n an d revitalizin g schoo l mathematics. H e provide s rationale s fo r introducin g discret e mathematic s in th e schools , notin g tha t discret e mathematic s i s applicable , accessible , attractive, an d appropriate , an d argue s tha t discret e mathematic s offer s a "new start " i n mathematic s fo r students . Thi s articl e i s base d o n a concep t document distribute d t o participant s prio r t o th e Octobe r 199 2 conference , and o n th e openin g presentatio n o f th e conference .

Section 1 . T h e Valu e o f Discret e M a t h e m a t i c s : V i e w s fro m t h e Classroo m

Bro. Patric k Carney' s articl e T h e Impac t o f Discret e M a t h e m a t i c s in M y Classroo m describe s anecdotall y ho w th e autho r arouse d i n hi s students a n interes t i n mathematics , an d develope d i n hi s student s a mor e "positive attitud e towar d mathematic s an d thei r abilit y t o d o it" .

Nancy Casey' s articl e Three fo r t h e M o n e y : A n Hou r i n t h e Class - r o o m describe s th e excitemen t generate d i n a clas s o f hig h schoo l students , participating i n a special summer program , whe n the y ar e presente d wit h a n unsolved mathematica l problem , an d th e mathematica l journey s tha t the y take t o lear n wha t th e proble m i s an d t o tr y t o solv e it . I t als o provide s a vivid descriptio n o f ho w th e teacher' s rol e i n th e classroo m change s whe n the clas s embark s o n a n uncharte d adventur e o f mathematica l discovery .

Janice C . Kowalczyk' s articl e Fibonacc i Reflections : It' s Elemen - tary! i s a n accoun t o f he r experience s givin g a worksho p o n th e Fibonacc i sequence (1 , 1,2 , 3 , 5 , 8 , . . . ) t o a fourth-grad e class . Sh e give s a detaile d description o f the workshop activities , includin g student investigation s o f the classical rabbi t populatio n proble m tha t lead s t o th e sequence , an d spiral - counting i n pinecones , sunflowers , shells , an d othe r object s whos e growt h patterns exhibi t th e sequence . Th e articl e illustrate s ho w usin g a topi c wit h

OVERVIEW AN D ABSTRACT S x v i i

a stron g visua l appeal , alon g wit h a focu s o n studen t exploration , ca n brin g out th e strength s i n man y student s wh o hav e ha d difficultie s i n th e tradi - tional elementar y mathematic s curriculum .

Susan H . Picker' s articl e U s i n g Discret e M a t h e m a t i c s t o Giv e R e - medial Student s a Secon d Chanc e i s a n accoun t o f he r experience s in - troducing discret e mathematic s t o a clas s o f remedia l tenth-grad e student s in Manhattan, an d thei r succes s in solving complex graph-coloring problems . More tha n that , i t i s a n accoun t o f th e impac t tha t thi s cours e ha d o n th e students' perception s o f mathematic s an d thei r ow n abilities , a s wel l a s o n their subsequen t schoo l careers . Th e autho r learne d fro m thi s experienc e the exten t t o whic h students ' dislik e o f arithmeti c serve s a s a n obstacl e t o their progres s an d succes s i n mathematics .

Reuben J . Settergren' s articl e "Wha t We'v e Go t Her e i s a Failur e t o Cooperate " describe s a cooperativ e game , base d o n th e classica l Pris - oner's Dilemma , tha t th e autho r playe d wit h twelve-year-ol d student s i n a summer program . Th e gam e gav e student s insigh t int o wh y individual s ar e sometimes motivate d t o behav e i n a wa y tha t harm s th e large r community , providing a n opportunit y t o discuss moral an d socia l issues in a mathematic s class.

Section 2 . T h e Valu e o f Discret e M a t h e m a t i c s : Achieving Broade r Goal s

Nancy Case y an d Michae l R . Fellows ' article Implementin g t h e Stan - dards: Let' s Focu s o n t h e Firs t Fou r argue s tha t i n orde r t o properl y address th e NCT M proces s standard s — reasoning , problem-solving , com - munications, an d connection s — i n th e elementar y schoo l classroom , ne w content mus t b e introduce d int o th e K- 4 mathematic s curriculum . Th e authors sho w b y exampl e ho w elementar y version s o f proble m situation s that aris e i n compute r scienc e an d discret e mathematic s mak e i t possibl e to realiz e th e goal s o f th e proces s standards . The y describ e thei r approac h to teachin g mathematic s a s paralle l t o th e "whol e language " approac h t o teaching reading .

Margaret B . Cozzens ' articl e Discret e M a t h e m a t i c s : A Vehicl e fo r P r o b l e m Solvin g an d E x c i t e m e n t provide s example s o f discret e math - ematics activitie s fro m severa l curriculu m developmen t project s funde d b y the NS F divisio n tha t th e autho r heads . Th e autho r argue s tha t discret e mathematics ca n motivat e student s t o thin k mathematically , t o becom e bet - ter proble m solvers , an d t o increas e thei r interes t i n mathematics .

Susanna S . Epp' s articl e Logi c an d Discret e M a t h e m a t i c s i n t h e Schools argue s tha t logica l reasonin g shoul d b e a componen t o f th e dis - crete mathematic s tha t i s discusse d a t al l grad e levels . Student s shoul d no t have to wai t unti l the y ar e colleg e students t o explor e the reasonin g involve d in "and" , "or" , an d "if-then " statements , o r t o understan d ho w quantifier s are used . Thi s nee d no t b e don e formall y (e.g. , throug h trut h tables ) bu t

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through concret e activitie s whic h ultimately wil l support th e students ' tran - sition t o abstrac t mathematica l thinking . Th e autho r illustrate s th e valu e of explici t discussio n o f logi c wit h experience s fro m a discret e mathematic s course sh e ha s taugh t a t DePau l University .

Rochelle Leibowitz ' articl e Writin g Discrete(ly ) argue s tha t discret e mathematics serve s a s a n excellen t vehicl e fo r teachin g student s t o com - municate mathematically . Throug h describin g carefull y simpl e proof s an d algorithms (e.g. , instruction s fo r buildin g a Leg o model) , student s acquir e technical writin g skill s tha t wil l b e usefu l i n a variet y o f caree r an d lif e situations.

Joseph Malkevitch' s articl e Discret e M a t h e m a t i c s an d Publi c Per - ceptions o f M a t h e m a t i c s contrast s th e kind s o f problem s typicall y dis - cussed i n hig h schoo l mathematic s classes , usuall y involvin g extensiv e ma - nipulation o f symbols, wit h th e kind s o f problem s tha t manifes t th e way s i n which mathematic s influence s dail y life . Malkevitc h argue s tha t th e nega - tive perception s tha t th e genera l publi c ha s abou t mathematic s aris e in par t from a n unbalance d mathematica l die t — too muc h o f th e former , to o littl e of the latte r — and note s tha t problem s fro m discret e mathematic s ca n pla y an importan t rol e i n changin g thes e perceptions .

Henry O . Pollak' s articl e Mathematica l M o d e l i n g an d Discret e M a t h e m a t i c s discusse s mathematical modelin g in general, noting that "ap - plied mathematics" , "proble m solving" , an d "wor d problems " al l star t wit h an idealize d versio n o f a real world problem , an d s o normally omi t th e initia l and final part s o f th e modelin g process . Th e autho r note s tha t i n discret e mathematics situations , however , i t i s often possibl e t o introduc e th e entir e mathematical modelin g proces s int o th e classroom ; h e provide s five exam - ples o f modelin g situation s whic h lea d t o discret e mathematic s an d whic h can b e mad e accessibl e t o hig h schoo l students .

Fred S . Roberts' articl e T h e Rol e o f Application s i n Teachin g Dis - crete M a t h e m a t i c s note s tha t "on e o f th e majo r reason s fo r th e grea t increase i n interest i n discrete mathematic s i s its importanc e i n solving prac- tical problems. " Th e autho r introduce s severa l "rule s o f thumb " abou t th e role o f application s i n teachin g discret e mathematics , an d illustrate s thos e by providin g man y application s o f th e Travelin g Salesma n Problem , grap h coloring, an d Eule r paths .

Section 3 . W h a t i s Discret e M a t h e m a t i c s : T w o Perspective s

Stephen B . Maurer's articl e "Wha t i s Discret e M a t h e m a t i c s ? " T h e M a n y Answer s provide s an d discusse s a variet y o f propose d definition s and description s o f discret e mathematics , alon g wit h severa l propose d goal s and benefit s fo r includin g discret e mathematic s i n th e schools . Th e articl e concludes wit h a se t o f goal s an d topic s fo r discret e mathematic s i n th e schools o n whic h th e autho r think s ther e migh t b e genera l agreement .

OVERVIEW AN D ABSTRACT S x i x

Joseph G . Rosenstein' s articl e A Comprehensiv e V i e w o f D i s c r e t e M a t h e m a t i c s : Chapte r 1 4 o f t h e N e w Jerse y M a t h e m a t i c s Cur - riculum Framewor k contain s a comprehensiv e discussio n o f topic s o f dis - crete mathematic s appropriat e fo r eac h o f th e K-2 , 3-4 , 5-6 , 7-8 , an d 9-1 2 grade levels . Th e autho r spearheade d th e developmen t o f the Framewor k i n his rol e a s Directo r o f th e Ne w Jerse y Mathematic s Coalition . Grade-leve l overviews ar e accompanie d b y severa l hundre d activitie s appropriat e fo r th e various grad e levels . Th e materia l reflect s th e experience s o f teacher s i n th e Leadership Progra m i n Discrete Mathematics , discusse d i n a separate articl e in Sectio n 8 .

Section 4 . Integratin g Discret e M a t h e m a t i c s int o Existin g M a t h e m a t i c s Curricula , Grade s K - 8

Valerie A . DeBellis ' articl e D i s c r e t e M a t h e m a t i c s i n K - 2 Class - rooms describe s th e author' s visit s t o severa l classroom s an d wha t sh e learned abou t th e reasonin g an d problem-solvin g skill s exhibite d b y youn g children wh o ar e introduce d t o situation s involvin g discret e mathematics . It als o describe s ho w topics i n discret e mathematic s ca n b e reformulate d fo r children a t earl y elementar y levels .

Robert E . Jamison' s articl e R h y t h m an d P a t t e r n : D i s c r e t e M a t h - ematics w i t h a n Artisti c C o n n e c t i o n fo r Elementar y Schoo l Teach - ers describe s th e materia l tha t th e autho r ha s use d i n program s fo r bot h inservice an d preservic e elementar y schoo l teachers . I t focuse s o n ho w el - ementary schoo l teacher s ca n us e geometri c activitie s involvin g drawin g polygons an d plana r representation s o f polyhedra, movin g i n geometri c pat - terns, an d usin g modula r arithmeti c i n movemen t an d musi c — t o provid e their student s wit h foundationa l experience s fo r futur e stud y o f mathemat - ics.

Evan Maletsky' s articl e Discret e M a t h e m a t i c s A c t i v i t i e s i n M i d - dle Schoo l provide s a wealt h o f activitie s tha t ar e appropriat e a t th e mid - dle schoo l level ; thes e involv e countin g (e.g. , finding th e triangula r number s when yo u coun t rectangle s o n a folde d piec e o f paper) , graphs , an d itera - tion (e.g. , generatin g Sierpinsk i triangles) . Th e autho r discusse s ho w thes e can b e incorporate d int o th e activitie s tha t ar e alread y takin g plac e i n th e classroom.

Section 5 . Integratin g D i s c r e t e M a t h e m a t i c s int o E x i s t i n g M a t h e m a t i c s Curricula , Grade s 9 - 1 2

Robert L . Devaney' s articl e P u t t i n g Chao s int o Calculu s Course s describes ho w fundamenta l idea s o f dynamica l systems , includin g iteration , attracting an d repellin g points , an d chaos , ca n b e introduce d i n a beginnin g calculus class, through a n in-dept h investigatio n o f the behavio r o f Newton' s Method, usin g a compute r o r graphin g calculator . Th e author' s approac h

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integrates discret e wit h continuou s mathematic s an d provide s a connectio n from calculu s t o th e fascinatin g worl d o f fractal s an d chaos .

John A . Dossey's article Making a Differenc e w i t h Differenc e Equa - tions show s ho w differenc e equation s ca n b e use d t o mode l chang e i n a number o f real-world settings . Th e autho r recommend s th e us e o f differenc e equations t o provid e a unified developmen t o f standard sequence s studie d i n mathematics, suc h a s arithmetic , geometric , an d Fibonacc i sequences .

Eric W. Hart' s articl e Discrete M a t h e m a t i c a l M o d e l i n g i n t h e Sec - ondary Curriculum : Rational e an d E x a m p l e s fro m t h e Core-Plu s M a t h e m a t i c s Projec t ( C P M P ) discusse s th e question s o f wha t discret e mathematics belong s i n th e secondar y curriculum , an d ho w i t shoul d b e incorporated, fro m th e perspectiv e o f th e curriculu m developer . Th e ar - ticle present s example s adapte d fro m CPM P material s whic h illustrat e th e CPMP approac h — that discret e mathematics shoul d b e woven into an over - all integrate d mathematic s curriculum , an d tha t th e emphasi s shoul d b e o n discrete mathematica l modeling .

Bret Hoyer' s articl e A Discret e M a t h e m a t i c s Experienc e w i t h General M a t h e m a t i c s Student s describe s ho w th e autho r introduce d topics i n discret e mathematic s firs t int o intermediat e algebr a an d geometr y classes, an d then , a s a result o f the students ' positiv e experiences , int o othe r classes a s well — including genera l mathematic s an d consume r mathematic s courses. Th e articl e focuse s o n th e "Stree t Networks " uni t o n Eule r path s and circuit s tha t wa s wove n int o thes e courses .

Philip G . Lewis ' articl e Algorithms , Algebra , an d t h e Compute r Lab describe s ho w th e author' s hig h schoo l student s use d th e LOG O com - puter environmen t t o explor e an d develo p concept s i n linea r algebra . Thes e explorations, whic h too k plac e i n a compute r lab , enable d student s t o vie w linear algebr a algorithmicall y an d t o lear n ho w t o construc t an d analyz e algorithms.

Joan Reinthaler' s articl e Discret e M a t h e m a t i c s i s Alread y i n t h e Classroom — B u t It' s Hidin g argue s tha t man y problem s i n hig h schoo l courses ar e discusse d a s problem s wit h continuou s domain s whe n a discret e perspective woul d b e mor e realistic , an d woul d lea d t o differen t investiga - tions an d solutions . Severa l example s ar e give n involvin g standar d textboo k problems i n algebra .

James T . Sandefur' s articl e Integratin g Discret e M a t h e m a t i c s int o t h e Curriculum : A n E x a m p l e describe s ho w h e use s th e handshak e problem t o revie w wit h hi s precalculus clas s the notion s o f function, domai n and range , an d graphin g quadrati c functions . Th e autho r argue s tha t "thi s approach integrate s discret e mathematic s int o th e existin g curriculum , re - sults i n deepe r studen t understanding , an d ca n b e accomplishe d i n abou t the sam e amoun t o f tim e a s i s presentl y devote d t o thes e topics. "

OVERVIEW AN D A B S T R A C T S x x i

Section 6 . Hig h Schoo l Course s o n Discret e M a t h e m a t i c s

Harold F . Bailey' s articl e T h e S t a t u s o f Discret e M a t h e m a t i c s i n t h e Hig h School s report s o n a survey tha t th e autho r di d t o ascertai n ho w many hig h school s offe r course s i n discret e mathematics , wha t thos e course s contain, an d th e goal s o f th e school s i n offerin g suc h courses .

L. Charle s Biehl' s articl e Discret e M a t h e m a t i c s : A Fres h Star t for Secondar y Student s describe s a project-base d discret e mathematic s course developed b y the autho r fo r juniors an d senior s of average ability. Th e students explore d a variet y o f mathematica l topic s i n real-worl d settings ; moreover, sinc e man y topic s i n discret e mathematic s hav e fe w prerequisites , these students wer e able to become successful proble m solver s and t o develo p more positiv e attitude s t o mathematics . Th e articl e include s a n outlin e o f the course .

Nancy Crisler , Patienc e Fisher , an d Gar y Froelich' s articl e A Discret e M a t h e m a t i c s Textboo k fo r H i g h School s describe s th e textboo k the y have co-authored, providin g a discussion o f its origins and development . Th e organization an d conten t o f the boo k i s based o n the NCT M report , Discrete Mathematics and the Secondary Mathematics Curriculum] i t addresse s fiv e broad area s (socia l decisio n making , grap h theory , countin g techniques , ma - trix models , an d th e mathematic s o f iteration ) an d interweave s si x unifyin g themes (modeling , us e o f technology , algorithmi c thinking , recursiv e think - ing, decisio n making , an d mathematica l induction) . Th e articl e include s summaries o f an d example s draw n fro m eac h chapte r o f th e book .

Section 7 : Discret e M a t h e m a t i c s an d Compute r Scienc e

Peter B . Henderson' s articl e C o m p u t e r Science , P r o b l e m Solving , and Discret e M a t h e m a t i c s addresse s th e rol e o f discret e mathematic s in a firs t cours e i n compute r science , base d o n th e author' s experienc e i n developing a "Fundamental s o f Compute r Science " cours e a t SUN Y Ston y Brook. Althoug h th e cours e describe d wa s develope d originall y fo r student s planning a caree r i n compute r science , i t ha s draw n student s wit h a wid e variety o f goals . Th e autho r note s tha t "Wit h it s emphasi s o n logica l rea - soning an d proble m analysi s an d solution , discret e mathematic s provide s a catalyst fo r genera l thinkin g an d problem-solvin g skill s . . . , " makin g suc h a cours e valuabl e fo r teachin g compute r scienc e t o hig h schoo l student s a s well.

Viera K . Proulx ' articl e T h e Rol e o f C o m p u t e r Scienc e an d D i s - crete M a t h e m a t i c s i n t h e Hig h Schoo l Curriculu m identifie s si x ke y themes i n compute r scienc e tha t th e autho r argue s shoul d b e taugh t t o al l high schoo l students , an d sketche s activitie s fo r student s t o explor e thes e themes. Th e idea s i n th e articl e gre w ou t o f th e author' s participatio n i n the Associatio n fo r Computin g Machiner y (ACM ) Tas k Forc e o n th e Hig h School Curriculum , whic h produce d a "Mode l Hig h Schoo l Compute r Sci - ence Curriculum " i n 1993 .

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Section 8 . Resource s fo r Teacher s

Nathaniel Dea n an d Yanx i Liu' s articl e Discret e M a t h e m a t i c s Soft - ware fo r K—1 2 Educatio n describe s two workshops involvin g teachers an d software developer s i n whic h teacher s solve d problem s usin g softwar e devel - oped fo r research , an d share d thei r reflection s o n th e feature s tha t woul d make suc h softwar e usefu l i n thei r classrooms . I n th e first workshop , teach - ers use d NETPAD , writte n b y Dea n whe n h e wa s a t Bellcore ; i n th e secon d workshop, teacher s use d Combinatorica , writte n b y Steve n Skien a o f SUN Y Stony Brook . Th e articl e als o provide s a n annotate d lis t o f othe r softwar e packages tha t ar e potentiall y usefu l t o teachers .

Deborah S . Pranzblau an d Janic e C . Kowalczyk's article R e c o m m e n d e d Resources fo r Teachin g Discret e M a t h e m a t i c s identifie s outstandin g resources, includin g books , modules , periodicals , literature , Interne t sites , software, an d video s fo r th e K-1 2 mathematic s teache r o r superviso r build - ing a core resource library fo r teachin g topics in discrete mathematics. Ther e are extensiv e review s o f fou r popula r textbooks ; othe r resource s ar e accom - panied b y briefe r descriptions . Th e lis t o f resources , whic h i s indexe d b y topic an d grad e level , an d whic h include s publishe r information , wa s devel - oped fro m recommendation s b y participants an d instructor s i n the DIMAC S Leadership Progra m i n Discret e Mathematics .

Joseph G . Rosenstei n an d Valeri e A . DeBellis ' articl e T h e Leadershi p Program i n Discret e M a t h e m a t i c s describe s th e DIMACS-sponsore d programs fo r K-1 2 teacher s tha t hav e take n plac e fo r th e pas t nin e year s a t Rutgers University , th e developmen t an d implementatio n o f th e program' s goals, an d ho w th e progra m i s servin g a s a continuou s resourc e fo r th e dissemination o f discret e mathematic s t o K-1 2 schools .

Mario Vassallo and Anthon y Ralston' s articl e C o m p u t e r Softwar e fo r t h e Teachin g o f Discrete M a t h e m a t i c s i n t h e School s provide s a num- ber o f criteria for judging the suitability of computer softwar e fo r educationa l use, an d the n describe s an d evaluate s thre e softwar e system s (Mathemat - ica/Combinatorica, GraphPack , an d SetPlayer ) agains t thes e criteria .

I n t r o d u c t i o n

Discrete M a t h e m a t i c s i n t h e Schools : An O p p o r t u n i t y t o Revitaliz e Schoo l M a t h e m a t i c s

Joseph G . Rosenstei n

This articl e serve s a s a n introductio n i n fou r differen t bu t overlappin g ways:

• A s a n introductio n t o a volum e advocatin g discret e mathematic s i n the schools , i t outline s th e cas e fo r thi s position .

• A s a n introductio n t o a collectio n o f thirty-fou r divers e articles , i t provides som e contex t fo r thos e articles .

• A s a n introductio n t o th e 199 2 conferenc e whic h le d t o thi s volume , it provide s informatio n abou t th e conferenc e an d it s themes .

• A s an introductio n t o m y perspective a s conference organizer , author , and editor , i t summarize s th e mai n reason s fo r m y involvement i n thi s enterprise.

T h e author' s perspectiv e

Starting a t th e end , whic h i s of cours e th e beginning , ther e ar e tw o ma - jor reason s fo r m y ongoin g effort s t o promot e discret e mathematic s i n th e schools — tha t i n tw o majo r ways , discret e mathematic s offer s a n opportu - nity t o revitaliz e schoo l mathematics .

• Discret e mathematic s offer s a ne w start fo r students . Fo r th e studen t who ha s bee n unsuccessfu l wit h mathematics , i t offer s th e possibilit y for success . Fo r th e talente d studen t wh o ha s los t interes t i n mathe - matics, i t offer s th e possibilit y o f challenge .

• Discret e mathematic s provide s a n opportunit y t o focu s o n how math - ematics i s taught , o n givin g teacher s ne w way s o f lookin g a t mathe - matics an d ne w way s o f makin g i t accessibl e t o thei r students . Fro m this perspective , teachin g discret e mathematic s i n th e school s i s no t an en d i n itself , bu t a too l fo r reformin g mathematic s education .

XXIV JOSEPH G . ROSENSTEI N

These tw o themes firs t appeare d i n a concept documen t tha t I develope d in Januar y 199 1 an d tha t gre w ou t o f th e first tw o year s o f m y experienc e directing th e Leadershi p Progra m i n Discret e Mathematics , a n NSF-funde d teacher enhancemen t progra m fo r hig h schoo l teachers , a t Rutger s Univer - sity.1 Participant s reporte d change s i n thei r classrooms , i n thei r students , and i n themselves . Thei r successe s taugh t u s tha t discret e mathematic s wa s not just anothe r piec e of the curriculum . Man y participants reporte d succes s with a variet y o f students a t a variet y o f levels, demonstrated a ne w enthusi - asm fo r teachin g i n ne w ways , an d proselytize d amon g thei r colleague s an d administrators.

These tw o theme s ar e discusse d furthe r i n thi s articl e i n section s enti - tled Discret e mathematics : A ne w star t fo r s t u d e n t s an d D i s c r e t e mathematics: A vehicl e fo r improvin g m a t h e m a t i c s e d u c a t i o n .

T h e Octobe r 199 2 Conferenc e

These tw o view s o f discret e mathematic s — a s a ne w star t fo r student s and a s a vehicl e fo r improvin g mathematic s educatio n — seeme d t o m e t o establish a n agend a fo r thos e intereste d i n bot h discret e mathematic s an d mathematics education . I f discret e mathematic s coul d hav e a significan t impact o n mathematics education , ho w can that impac t b e actualized ? Thi s question le d t o a conferenc e entitle d "Discret e Mathematic s i n th e Schools : How D o W e Mak e a n Impact? "

The Conferenc e too k plac e o n Octobe r 2-4 , 199 2 a t Rutger s Universit y and wa s sponsore d b y th e Cente r fo r Discret e Mathematic s an d Theoreti - cal Compute r Scienc e (DIMACS) , a n NSF-funde d Scienc e an d Technolog y Center. I t brough t togethe r thirty-thre e educator s wh o ha d bee n involve d in a variet y o f way s i n introducin g discret e mathematic s i n th e schools ; se e Appendix A fo r a lis t o f conferenc e participants . Th e concep t documen t containing th e tw o theme s describe d abov e wa s distribute d i n advanc e o f the conferenc e an d wa s reflecte d i n th e openin g presentatio n a t whic h I welcomed an d challenge d th e conferenc e participants .

The conferenc e progra m wa s designe d t o infor m th e participant s abou t various perspective s o f discret e mathematic s an d it s rol e i n K-1 2 education , and abou t al l o f th e variou s activitie s takin g plac e tha t promote d discret e mathematics i n the schools . A n abbreviate d versio n of the program, showin g presentations an d sessio n titles , appear s i n Appendi x B . Presentation s wer e followed b y extende d discussions .

lrThe NSF-funde d Leadershi p Progra m i n Discret e Mathematic s i s co-sponsore d b y the Cente r fo r Discret e Mathematic s an d Theoretica l Compute r Scienc e (DIMACS ) an d the Rutger s Cente r fo r Mathematics , Science , an d Compute r Scienc e Education (CMSCE) . Although originall y (i n 1989-1991 ) fo r hig h schoo l teachers , th e Leadershi p Progra m sub - sequently (beginnin g i n 1992 ) als o enrolle d middl e schoo l teachers , an d no w (sinc e 1995 ) focuses o n K - 8 teachers . Se e th e articl e b y Rosenstei n an d DeBelli s i n thi s volum e fo r further informatio n abou t th e Leadershi p Program .

D I S C R E T E MATHEMATIC S I N T H E SCHOOL S X X V

One outcome of the discussion s a t th e conferenc e wa s the Visio n State - ment whic h appear s a t th e beginnin g o f thi s volume . Tw o majo r point s o f the Visio n Statemen t wer e tha t "discret e mathematic s i s a n excitin g an d appropriate vehicl e fo r workin g towar d an d achievin g thes e goals " (refer - ring t o th e goal s o f thos e strivin g t o improv e mathematic s education) , an d that "discret e mathematic s need s t o b e introduce d int o th e curriculu m fo r its ow n sake " becaus e o f th e increasin g importanc e an d prevalenc e o f it s applications.

W h a t i s discret e m a t h e m a t i c s ?

It is , of course , natura l fo r K-1 2 teacher s an d administrators , a s wel l a s parents an d th e press , t o as k thi s question . Unfortunately , i t i s no t a n eas y question t o answer . Th e proble m i s tha t th e phras e "discret e mathemat - ics" doe s no t refe r t o a well-define d branc h o f mathematic s — lik e algebra , geometry, trigonometry , o r calculu s — bu t rathe r encompasse s a variet y o f loosely-connected concept s an d techniques . Moreover , i t i s no t a branc h o f mathematics whic h is generally familia r t o th e public . A t th e dedicatio n cer - emony o f DIMAC S a s a Cente r i n 1989 , then-Governo r Thoma s Kea n (NJ ) quipped that , befor e participatin g i n thi s ceremony , hi s impressio n wa s tha t discrete mathematic s wa s wha t accountant s di d behin d close d doors . Tha t may b e a commo n initia l impressio n o f discret e mathematics .

I have found tha t on e effective wa y of answering the questio n i s by givin g lots o f example s o f th e kind s o f situation s wher e th e mathematic s tha t i s used i s "discrete" . Thoug h no t actuall y definin g discret e mathematics , th e examples giv e a flavor o f wha t comprise s discret e mathematics , an d als o helps t o demystif y th e phrase . Her e i s the lis t tha t w e are currentl y usin g i n one o f th e brochure s o f th e Leadershi p Progra m i n Discret e Mathematics ; this lis t contain s example s tha t w e anticipate wil l make sens e to th e teacher s that w e hop e t o attrac t t o th e program .

• Wha t i s th e quickes t wa y t o sor t a lis t o f name s alphabetically ? • Whic h wa y o f connectin g a numbe r o f site s int o a telephon e networ k

requires th e leas t amoun t o f cable ? • Whic h versio n o f a lotter y give s th e bes t odds ? • I f each voter rank s th e candidate s fo r Presiden t i n order o f preference ,

how ca n a consensu s rankin g o f th e candidate s b e obtained ? • Wha t i s th e bes t wa y fo r a robo t t o pic k u p item s store d i n a n auto -

mated warehouse ? • Ho w doe s a C D playe r interpre t th e code s o n a C D correctl y eve n i f

the C D i s scratched ? • Ho w ca n a n estat e b e divide d fairly ? • Ho w ca n ic e crea m stand s b e place d a t variou s stree t corner s i n a

town s o tha t a t an y corne r ther e i s a stan d whic h i s a t mos t on e block away ?

• Ho w can representative s b e apportione d fairl y amon g th e state s usin g current censu s information ?

XXVI JOSEPH G . ROSENSTEI N

These problem s — an d man y other s fro m differen t area s withi n discret e mathematics — shar e severa l importan t characteristics . The y ar e easil y understood an d discussed , readil y see n a s dealing with real-worl d situations , and ca n b e explore d withou t extensiv e backgroun d i n schoo l mathematics . This i s discusse d i n mor e detai l i n th e followin g section .

Although I hav e use d thi s "definition-by-examples " o f discret e math - ematics fo r a numbe r o f years , i n th e sprin g o f 1996 , a s th e Ne w Jerse y Department o f Educatio n wa s preparin g t o presen t it s recommendation s fo r mathematics standard s t o th e Stat e Boar d o f Education , I wa s tol d tha t I ha d t o provid e a "rea l definition " fo r th e document . S o her e i s discret e mathematics a s i t appear s i n Ne w Jersey' s Core Curriculum Content Stan- dards:

Discrete mathematic s i s the branc h o f mathematic s tha t deal s with arrangement s o f discret e objects . I t include s a wid e va - riety o f topic s an d technique s tha t aris e i n everyda y life , suc h as ho w t o fin d th e bes t rout e fro m on e cit y t o another , wher e the object s ar e citie s arrange d o n a map . I t als o include s ho w to coun t th e numbe r o f differen t combination s o f topping s fo r pizzas, ho w bes t t o schedul e a lis t o f task s t o b e done , an d how computer s stor e an d retriev e arrangement s o f informatio n on a screen . Discret e mathematic s i s th e mathematic s use d by decision-maker s i n ou r society , fro m worker s i n governmen t to thos e i n healt h care , transportation , an d telecommunica - tions. It s variou s application s hel p student s se e th e relevanc e of mathematic s i n th e rea l world .

In Thi s Volume . Tw o article s i n Sectio n 3 of this volum e addres s directl y the question , "Wha t i s discret e mathematics? " Stephe n Maurer' s articl e explores a numbe r o f possible charactization s o f discrete mathematics , non e of which prove s t o b e full y satisfactory . Josep h Rosenstein' s articl e provide s an extende d elaboratio n o f th e descriptio n above , a s i t appear s i n th e New Jersey Mathematics Curriculum Framework.

W h y introduc e discret e m a t h e m a t i c s int o t h e curriculum ?

A numbe r o f differen t argument s hav e bee n presente d fo r includin g dis - crete mathematic s i n th e schoo l curriculum ; thes e argument s ca n eac h b e viewed agains t th e backdro p o f th e problem s pose d above . Discret e mathe - matics is :

Applicable: I n recen t years , topic s i n discret e mathematic s hav e be - come valuabl e tool s an d provid e powerfu l model s i n a numbe r o f dif - ferent areas .

Accessible: I n orde r t o understan d man y o f thes e applications , arith - metic i s ofte n sufficient , an d man y other s ar e accessibl e wit h onl y elementary algebra .

DISCRETE MATHEMATIC S I N TH E SCHOOL S XXVll

Attractive: Thoug h easil y stated , man y problem s ar e challenging , ca n interest an d attrac t students , an d len d themselve s t o exploratio n an d discovery.

Appropriate: Bot h fo r student s wh o ar e accustome d t o succes s an d are alread y contemplatin g scientifi c careers , an d fo r student s wh o ar e accustomed t o failur e an d perhap s nee d a fres h star t i n mathematics .

In Thi s Volume . A numbe r o f article s i n thi s volum e illustrat e an d elab - orate o n thes e reason s fo r incorporatin g discret e mathematic s int o th e cur - riculum. Severa l article s tha t particularl y addres s eac h o f th e abov e theme s are provide d below .

Applicable: Th e article s b y Henr y Pollak , Fre d Roberts , Joh n Dossey , and Eri c Har t addres s th e application s o f discret e mathematic s an d how i t provide s model s fo r real-worl d situations .

Accessible: Th e article s b y Janic e Kowalczyk , Susa n Picker , Nanc y Casey and Michae l Fellows, Joseph Rosenstein , Valeri e DeBellis, Rob- ert Jamison , an d Eva n Maletsk y show , fo r example , ho w discret e mathematics ca n b e use d i n elementar y an d middl e schoo l grades .

Attractive: Th e article s b y Patrick Carney , Nanc y Casey , Reube n Set - tergren, an d Margare t Cozzen s discus s ho w discret e mathematic s ex - cites studen t interest .

Appropriate: Th e article s b y Nanc y Casey , Susa n Picker , Bre t Hoyer, and L . Charle s Bieh l discuss ho w discrete mathematic s i s appropriat e for student s wh o nee d a fres h star t i n mathematics . Othe r article s i n this volum e discus s ho w discret e mathematic s ca n b e combine d wit h and enhanc e existin g topic s lik e algebr a (Bre t Hoyer , Phili p Lewis) , precalculus (Joh n Dossey , Joa n Reinthaler , Jame s Sandefur) , calcu - lus (Rober t Devaney) , an d compute r scienc e (Pete r Henderson , Ver a Proulx).

Discrete mathematics : A n e w star t fo r s t u d e n t s

The traditiona l topic s o f schoo l mathematic s — arithmetic , algebra , ge- ometry, etc . — ar e o f cours e important ; withou t a goo d groundin g i n thes e topics, student s wil l b e seriousl y disadvantage d i n caree r options . An d th e nation wil l continu e t o hav e a seriou s shortfal l i n technicall y skille d person - nel.

However, man y student s fin d schoo l mathematic s t o b e a seriou s stum - bling block , an d ultimatel y giv e up. Th e mos t frequentl y prescribe d remed y for student s wh o hav e faile d i n schoo l mathematic s appears , unfortunately , to b e mor e o f th e same . An d "mor e o f th e same " usuall y mean s no t onl y repetition o f content , bu t als o repetitio n o f method . Thus , man y student s come t o se e schoo l mathematic s onl y a s a se t o f unintelligibl e procedures , which is not surprisin g sinc e they wer e never given an opportunit y t o explor e concepts meaningfull y an d appl y the m i n ne w situations .

XXV111 JOSEPH G . ROSENSTEI N

At th e othe r en d o f th e spectrum , man y talente d student s als o fin d school mathematics t o be uninteresting an d irrelevant , an d thu s opt fo r othe r careers. Fo r thes e students , wh o ar e lookin g fo r a spar k o f lif e an d challeng e in mathematics , a frequen t respons e i s "wai t unti l yo u ge t t o calculus" ; bu t many hav e los t interes t b y th e tim e the y ge t t o calculus .

Discrete mathematic s offer s a ne w start . Fo r th e studen t wh o ha s bee n unsuccessful i n mathematics , discret e mathematic s offer s th e possibilit y o f success. Student s wh o hav e encountere d mathematic s whic h the y ca n d o successfully ar e encouraged t o take another loo k at th e mathematics a t whic h they hav e failed . Student s wh o hav e foun d tha t the y ca n solv e meaningfu l problems gai n a sense of empowerment. Teacher s in the Leadershi p Progra m have reported that , fo r students who have a history of failure i n mathematics , being abl e t o us e terminolog y an d solv e problem s i n area s wit h whic h othe r school personne l — teachers an d guidanc e counselors , a s well as student s — are unfamilia r i s a ver y head y experience .

The rank s o f student s wh o hav e bee n unsuccessfu l i n mathematic s con - tain a disproportionat e numbe r o f minoritie s an d women . Suc h students , who hav e give n u p hop e o f eve r learnin g schoo l mathematics , ca n becom e interested i n an d ca n lear n discret e mathematic s sinc e the y d o no t associat e it a t th e outse t wit h routin e schoo l mathematics. Teacher s i n the Leadershi p Program i n Discret e Mathematic s hav e use d discret e mathematic s success - fully wit h thes e student s i n al l type s o f schools , includin g thos e i n urba n areas.

For th e talente d studen t wh o ha s los t interes t i n mathematics , discret e mathematics offer s th e possibilit y o f challenge. Discret e mathematic s serve s as a natura l contex t fo r man y o f th e puzzle-lik e question s tha t intrigu e th e talented student , offer s open-ende d problem s whic h quickl y lea d t o th e fron - tiers o f knowledge , an d provide s eas y acces s t o application s whic h mathe - maticians ar e now making in a variety of real-life situations . On e can imagin e students engage d i n discret e mathematic s sayin g "Thi s i s ho w I woul d lik e to spen d m y professiona l life" , a s wel l a s "Thi s i s fun" .

In Thi s Volume . Se e th e article s cite d unde r "accessible" , "attractive" , and "appropriate " i n th e previou s section .

Discrete m a t h e m a t i c s : A vehicl e fo r improvin g m a t h e m a t i c s education.

The introductio n o f new material int o the curriculu m afford s a particula r opportunity t o infus e ne w instructiona l technique s a t th e sam e time . Whe n there i s no specific bod y o f material tha t district s an d teacher s fee l obligate d to "cover" , ther e i s clearl y "time " fo r experimentatio n — wit h computers , with grou p learning , wit h proble m solving . Whe n th e problem s ar e ne w t o the teachers , an d clos e t o th e cuttin g edg e o f knowledge , ther e i s greate r acceptance o f a classroo m ope n t o discussion , t o reasonin g together , an d t o the excitemen t o f discoverin g ne w solution s whic h ar e no t "i n th e book" .

DISCRETE MATHEMATIC S I N TH E SCHOOL S XXIX

Moreover, a s teacher s becom e familia r wit h thes e technique s an d se e that the y wor k wit h thei r student s i n thei r ow n classrooms , the y wil l adap t them fo r us e i n thei r othe r classes . Thos e teacher s wh o hav e take n th e tim e from traditiona l teacher-oriente d instructio n t o tr y thes e learner-oriente d techniques kno w tha t th e tim e i s wel l spent . Th e difficult y i s i n gettin g them t o try .

Discrete mathematic s offer s a wealt h o f ne w materia l and , mor e impor - tant i n this context, consist s of many topic s which lend themselves readil y t o approaches t o learnin g tha t ar e recommende d i n th e nationa l reports : dis - covery learning , experimentation , proble m solving , cooperativ e learning , us e of technology . Wit h discret e mathematics , student s ca n easil y becom e in - volved i n the doin g of mathematics, ca n see themselves a s "mathematicians " rather tha n a s follower s o f routin e instructions .

In Thi s Volume . Nanc y Case y an d Michae l Fellow s argu e i n thei r articl e that onl y i f the y us e discret e mathematic s wil l K- 4 teacher s hav e suffi - ciently ric h mathematica l conten t t o properl y addres s th e proces s standard s of "reasoning , problem-solving , communications , an d connections " stresse d in th e NCT M Standards. 2 Othe r article s focu s o n ho w discret e mathe - matics ca n hel p teacher s achiev e educationa l objective s suc h a s teachin g students mathematica l communicatio n (Rochell e Leibowitz) , reasonin g (Su - sanna Epp) , an d problem-solvin g (Margare t Cozzens , Peter Henderson) , an d change publi c perception s o f mathematic s (Josep h Malkevitch) . Th e articl e by Josep h Rosenstei n an d Valeri e DeBelli s discusse s th e impac t o f the Lead - ership Progra m i n Discret e Mathematic s o n the activitie s o f its participants .

Resources fo r introducin g discret e m a t h e m a t i c s i n t h e school s

At th e tim e o f th e conference , ther e wer e relativel y fe w resource s avail - able t o teacher s intereste d i n includin g discret e mathematic s i n thei r class - rooms an d curricula . Increasingl y i n recen t years , i n par t becaus e discret e mathematics i s addresse d i n th e NCT M Standards , mor e effor t ha s bee n placed bot h o n developin g material s relate d t o discret e mathematic s an d to incorporatin g discret e mathematic s activitie s i n textbooks . A s a resul t of th e effort s o f th e Leadershi p Progra m i n Discret e Mathematic s an d th e "Implementation o f th e NCT M Standar d i n Discret e Mathematic s Project " program directe d b y Margare t Kenne y a t Bosto n Colleg e an d othe r site s across th e country , ther e ar e no w nearl y 200 0 teacher s wh o hav e ha d ex - tensive exposur e t o discret e mathematics ; man y o f the m hav e bee n takin g leadership roles , developin g curriculu m material s an d makin g presentation s at conferences .

In Thi s Volume . Th e articl e b y Debora h Franzbla u an d Janic e Kowal - czyk, base d o n recommendation s o f teacher s i n th e Leadershi p Progra m i n Discrete Mathematics , provide s a n extensiv e revie w o f availabl e prin t an d

2 Curriculum and Evaluation Standards for School Mathematics, Nationa l Counci l o f Teachers o f Mathematics , 1989 , Reston , VA .

X X X JOSEPH G . ROSENSTEI N

video resources . Tw o articles , on e b y Eri c Har t an d th e othe r b y Nanc y Crisler, Patienc e Fisher , an d Gar y Proelich , discus s text s fo r hig h schoo l students whic h includ e discret e mathematics . Tw o articles , on e b y Nat e Dean an d Yanx i Liu , an d th e othe r b y Mari o Vassall o an d Anthon y Ral - ston, discus s discret e mathematic s software . Tw o articles , b y Harol d Baile y and L . Charle s Biehl , discus s hig h schoo l course s i n discret e mathematics . And th e articl e b y Josep h Rosenstei n an d Valeri e DeBelli s discusse s th e Leadership Progra m i n Discret e Mathematics .

Conclusion

Speaking fo r th e editors , th e conferenc e participants , an d th e authors , we hop e tha t thi s volum e wil l b e a majo r contributio n bot h t o facilitat - ing th e us e o f discret e mathematic s i n K-1 2 school s an d t o demonstratin g the potentia l o f discret e mathematic s a s a vehicl e t o improv e mathematic s education an d revitaliz e schoo l mathematics .

D E P A R T M E N T O F MATHEMATICS , R U T G E R S U N I V E R S I T Y

E-mail address: j oerQdimacs. r u t g e r s. ed u

DISCRETE MATHEMATIC S I N TH E SCHOOL S XXXI

A p p e n d i x A .

Discrete M a t h e m a t i c s i n t h e Schools : How D o W e Mak e a n Impact ?

October 2 - 4 , 199 2

Conference Participant s

NAME STATE AFFILIATIO N (a t tim e of conference )

Bailey, Harol d F . Biehl, L . Charle s Carrs, Marjori e Crisler, Nanc y Dance, Rosali e DeBellis, Valeri e Dean, Nathanie l Epp, Susann a Fellows, Michae l Froelich, Gar y Hart, Eri c Henderson, Pete r Hoover, Mar k Hoyer, Bre t Kenney, Margare t Kowalczyk, Janic e Lacampagne, Caro l B . Leibowitz, Rochell e Lewis, Phili p G . Malkevitch, Josep h Malt as, Jame s

Maurer, Stephe n McGraw, Su e An n Piccolino, Anthon y Picker, Susa n

Pollak, Henr y Proulx, Vier a Reinthaler, Joa n Roberts, Fre d Rosenstein, Josep h G . Saks, Michae l Vassallo, Mari o Yunker, Le e

NY DE

MO MD NJ NJ IL

ND IA NY NJ IA MA RI DC MA MA NY IA

PA OR NJ NY

NJ MA DC NJ NJ NJ NY IL

College o f Moun t Sain t Vincen t McKean HS , Wilmingto n University o f Queensland , Brisbane , Australi a Pattonville Schoo l Dist. , St . Loui s Count y Ballou Science/Mat h HS , Takom a Par k Rutgers Universit y Bellcore DePaul Universit y University o f Victoria , Britis h Columbia , Canad a Bismarck H S Maharishi Internationa l Universit y SUNY Ston y Broo k Educational Testin g Servic e John F . Kenned y HS , Ceda r Rapid s Boston Colleg e Teacher Educatio n an d Compute r Cente r U.S. Departmen t o f Educatio n Wheaton Colleg e Lincoln Sudbur y Regiona l H S York Colleg e (CUNY ) Malcolm Pric e Laborator y School ,

University o f Norther n Iow a Swarthmore Colleg e Lake Osweg o H S Montclair Stat e Colleg e Office o f th e Superintendent ,

Manhattan Publi c School s Columbia Universit y Northeastern Universit y The Sidwel l Friend s Schoo l Rutgers Universit y Rutgers Universit y Rutgers Universit y SUNY Fredoni a Community H S Dist . 94 , Wes t Chicag o

xxx n J O S E P H G . R O S E N S T E I N

A p p e n d i x B .

D i s c r e t e M a t h e m a t i c s i n t h e S c h o o l s : H o w D o W e M a k e a n I m p a c t ?

O c t o b e r 2 - 4 , 1 9 9 2

C o n f e r e n c e P r o g r a m ( A b b r e v i a t e d )

Friday Octobe r 2 Presentation: Josep h G . Rosenstei n

"Discrete m a t h e m a t i c s a s a n e w star t fo r s t u d e n t s an d teachers " Classroom Perspectives , Experiences , an d Model s — Sessio n 1

L. Charle s Bieh l — "Discret e m a t h e m a t i c s fo r s t u d e n t s o f averag e ability"

Susan Picke r — "Discret e m a t h e m a t i c s : Givin g remedia l s t u d e n t s a second chance "

Presentation: Stephe n Maure r "What i s discret e m a t h e m a t i c s : T h e m a n y answers "

Classroom Perspectives , Experiences , an d Model s — Sessio n 2 Gary Froelic h — "A semeste r discret e m a t h e m a t i c s cours e a t t h e

high schoo l level " James Malta s — "Implementin g a discret e m a t h e m a t i c s cours e fo r

n o n - m a t h students " Nancy Crisle r — "M y e x p e r i e n c e s a s a teache r an d m a t h

coordinator" Philip Lewi s — "Usin g a c o m p u t e r lab : A l g o r i t h m s , algebra , an d

axioms" Presentation: Josep h Malkevitc h

"Discrete m a t h e m a t i c s an d t h e public' s p e r c e p t i o n o f mathematics"

Classroom Perspectives , Experiences , an d Model s — Sessio n 3 Rosalie Danc e - "Integratin g discret e an d continuou s approache s i n

secondary math " Lee Yunke r - "Curren t an d futur e t r e n d s o n discret e m a t h e m a t i c s

in t h e curriculum " Presentation: Eri c Har t

"Curriculum material s fo r discret e m a t h e m a t i c s i n t h e schools " An overvie w an d a tast e o f .. .

For Al l Practica l Purpose s — Jo e Malkevitc h an d Ton y Piccolin o COMAP Projec t — Nanc y Crisle r an d Gar y Froelic h UCSMP material s — Susann a Ep p CORE-PLUS — Eri c Har t Several textbook s — Le e Yunke r

Saturday Octobe r 3 Programs fo r teacher s

Georgetown projec t — Rosali e Danc e an d Joa n Reinthale r NCTM projec t — Pe g Kenne y an d other s Iowa Projec t — Eri c Har t an d other s Rutgers Projec t — Jo e Rosenstei n an d other s

D I S C R E T E MATHEMATIC S I N T H E SCHOOL S xxxii i

Classroom Perspectives , Experiences , an d Model s — Sessio n 4 Joan Reinthale r — "Teachin g m o d e l i n g t o wea k m a t h students " Sue An n McGra w — "Integratin g discret e m a t h e m a t i c s int o

traditional m a t h courses " Bret Hoye r — " A discret e m a t h e m a t i c s cours e usin g Fo r Al l

Practical Purposes " Perspectives

Susanna Ep p — "Strengthenin g thinkin g skill s usin g discret e mathematics"

Roehelle Leibowit z — "Strengthenin g writin g skill s usin g discret e mathematics"

Anthony Piccolin o — "Discret e m a t h e m a t i c s : Makin g m a t h accessible t o all "

Presentation: Henr y Polla k "The rol e o f modelin g i n teachin g discret e mathematics "

Presentation: Fre d Robert s "The rol e o f application s i n teachin g discret e mathematics "

Presentation: Mari o Vassall o "Computer softwar e fo r teachin g discret e m a t h e m a t i c s i n t h e

schools" Presentation: Nat e Dea n

"What compute r softwar e i s currentl y bein g developed? " Presentation: Michae l Fellow s

"Discrete m a t h e m a t i c s an d c o m p u t e r scienc e i n t h e e l e m e n t a r y schools"

"How D o W e Mak e a n Impact? " Organizing ou r suggestion s Structuring Sunday' s discussion s

Sunday Octobe r 4 Perspectives

Viera Proul x — "Compute r scienc e i n hig h school " Peter Henderso n — "Compute r science , discret e m a t h e m a t i c s , an d

problem solving " Mark Hoove r — "Assessmen t an d discret e mathematics " Harold Baile y — "Assessin g curren t practic e i n discret e

mathematics" "How D o W e Mak e a n Impact? "

Work session s i n smalle r group s Reports fro m group s The nex t step s

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Section 1

T h e Valu e o f Discret e M a t h e m a t i c s : Views fro m t h e Classroo m

The Impac t o f Discret e Mathematic s i n M y Classroo m B R O . P A T R I C K C A R N E Y

Page 3

Three fo r th e Money : A n Hou r i n th e Classroo m N A N C Y C A S E Y

Page 9

Fibonacci Reflections—It' s Elementary ! J A N I C E C . KOWALCZY K

Page 2 5

Using Discret e Mathematic s t o Giv e Remedia l Student s a Secon d Chanc e

SUSAN H . P I C K E R

Page 3 5

What We'v e Go t Her e I s a Failur e t o Cooperat e R E U B E N J . S E T T E R G R E N

Page 4 3

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

T h e I m p a c t o f Discret e M a t h e m a t i c s i n M y Classroom

Bro. Patric k Carne y

Early i n the year , ou r schoo l has a "Bac k to Schoo l Night" whe n teacher s meet wit h parent s t o explai n th e course s an d answe r questions . On e paren t started th e evenin g sessio n b y askin g wha t I wa s doin g tha t resulte d i n he r daughter's lookin g at th e bar codes on envelopes and o n commercial product s at home . I t woul d b e unfai r t o describ e he r a s hostile , bu t sh e was certainl y questioning wha t w e wer e doin g i n th e mat h class . I briefl y explaine d th e various topics that coul d b e learne d throug h th e stud y o f codes, check digits , and th e like , an d showe d ho w the y le d student s t o revie w th e fou r basi c arithmetical operations , remainders , positio n value , etc . Sh e agree d tha t such revie w wa s probabl y necessary , an d tha t i t wa s no t a routin e tha t students woul d tak e t o wit h an y degre e o f enthusiasm . I the n pointe d ou t that th e mer e fac t tha t sh e wa s askin g m e abou t th e clas s indicate d tha t the youn g woma n i n questio n ha d enoug h interes t i n th e materia l o n code s to brin g i t hom e an d us e it . Th e student' s mothe r finally agree d tha t thi s gave a fres h approac h t o learnin g necessar y skills ; fro m m y vantag e poin t i n the fron t o f th e room , I coul d se e the look s o n th e face s o f the othe r parent s and nod s o f agreement . I n fact , on e o f th e mos t enthusiasti c seeme d t o b e a gentlema n wh o himsel f i s a mat h teache r i n anothe r school .

I begi n wit h thi s anecdot e t o mak e th e poin t tha t eve n thoug h discret e mathematics i s ofte n stresse d fo r it s practica l value , I believ e tha t a n eve n more importan t aspec t i s that i t capture s th e imaginatio n o f the student s i n a wa y tha t routin e dril l ca n neve r do . Man y thing s ar e practica l bu t d o no t create enthusias m fo r learning . I n m y opinion , th e rea l impac t o f discret e mathematics o n th e curriculu m i s its abilit y t o instil l a n interes t i n student s who migh t no t otherwis e find mathematic s a s excitin g a s w e teacher s do .

When I first studied i n the Leadership Program fo r Discret e Mathematic s at Rutger s Universit y [2 ] in th e summe r o f 1991 , I enjoyed wha t w e did ver y much an d though t abou t ho w bes t t o us e it . I teac h i n a smal l schoo l ( a little mor e tha n 30 0 students i n grades 6 - 1 2 ) an d ha d just finished on e yea r

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 .

https://doi.org/10.1090/dimacs/036/01

4 BRO. PATRIC K CARNE Y

of teachin g a Proble m Solvin g cours e t o 7t h an d 8t h graders . I t me t onc e a week, s o th e clas s ha d t o b e carefull y planne d t o b e self-contained . Whil e some thing s I ha d don e worke d well , I wa s not satisfie d wit h it . Som e o f th e discrete mat h activitie s seeme d perfectl y suite d t o th e cours e an d I decide d to wor k the m int o m y classe s a s replacement s fo r th e topic s tha t di d no t work well .

The mor e o f thes e item s I tried , th e mor e enthusiasti c I became . Eve n when on e wa s a disaste r suc h a s m y first tr y a t th e Parke r Brothers ' gam e Instant Insanity, I realize d tha t th e proble m wa s no t th e material , bu t m y approach. (I n thi s case , I ha d approache d i t to o abstractly . Whe n I revise d the activity , i t becam e ver y popular. ) A t th e en d o f eac h year , I aske d the student s wha t topic s wer e th e mos t interesting , useful , boring , difficult , etc. Tha t year , althoug h w e did man y differen t problems , th e discret e mat h problems clearl y wer e th e mos t popular .

In fact , ou r stud y o f fai r divisio n ranke d first. Tha t wa s a topi c I ha d barely hear d o f th e previou s yea r an d I certainl y ha d neve r though t o f in - cluding i t i n the program . I t i s interesting t o not e tha t initially , th e student s were very frustrate d wit h th e fai r divisio n problem . I had prepare d the m fo r it a fe w week s i n advanc e b y discussin g th e strateg y o f tw o peopl e sharin g a cand y ba r b y havin g on e chil d brea k i t i n "half " an d th e othe r selec t th e piece h e o r sh e wants . The y al l graspe d that . The n I said "wha t woul d b e a fair wa y t o divid e i t amon g thre e people? " A t th e en d o f each o f four weekl y classes, I pose d tha t questio n an d the y offere d solution s unti l th e bel l rang . I promise d the m tha t a lesso n wa s coming . I n th e beginning , the y wer e ver y creative (e.g. , successiv e dividin g b y 2 hoping t o ge t a multipl e o f 3 , havin g each perso n d o a differen t par t o f th e process , on e cut , th e othe r tw o selec t first, etc. ) bu t a s eac h o f their solution s wa s knocke d dow n becaus e o f som e flaw (usuall y foun d b y a peer) , the y becam e frustrated .

When on e bo y finally suggeste d "kil l one an d the n divid e i t betwee n th e two remaining, " I thought i t wa s abou t tim e t o hav e tha t lesson . W e looke d at th e ide a o f Person s A an d B breakin g th e whol e int o tw o equa l parts . Then the y eac h brok e thei r piec e int o wha t the y considere d thre e equa l parts. Perso n C the n selecte d on e o f th e "thirds " fro m eac h o f th e othe r two people . W e wen t throug h a coupl e o f othe r method s whic h reviewe d fractions (a t leas t halves , thirds , an d sixths ) an d the y seeme d t o agre e tha t that wa s a n answe r the y coul d understan d an d perhap s coul d hav e figured out. Bu t w e di d no t sto p there . On e o f m y colleague s ha d give n u s a computer progra m fo r th e "Movin g Knife." 1 Her e wa s a totall y differen t approach. A "knife " move s acros s th e "cand y bar " fro m lef t t o right . Whe n any perso n feel s th e amoun t t o th e lef t o f th e knif e i s a fai r share , h e o r sh e hits a key to sto p the knife an d th e compute r tell s what percen t wa s selected . The movin g knif e the n continue s unti l anothe r perso n i s satifie d wit h th e

lfThe moving knife softwar e I use d wa s writte n b y th e clas s o f Ji m Lorent z ( a fello w 1991 alumnu s o f th e Leadershi p Progra m i n Discret e Mathematics. ) Simila r softwar e i s distributed wit h [1] .

THE IMPAC T O F DISCRET E MATHEMATIC S I N M Y CLASSROO M 5

amount displayed . Al l o f their origina l approache s ha d bee n reall y modele d on dividing th e cand y ba r i n two pieces (excep t fo r th e totall y bizarr e ones) . Here wa s anothe r view .

We decided to use the software t o hold a tournament. W e broke the clas s up int o group s o f three . W e trie d th e simpl e rectangula r an d ova l shape d "candy bars " fo r practice , bu t th e contes t use d rando m shapes . Now , fro m my poin t o f view , w e were als o reinforcing hand-ey e coordinatio n and , mor e importantly, estimatio n skills . I hav e no w don e thi s fo r thre e year s wit h students o f differen t ages , an d I hav e observe d tha t mos t student s becom e much bette r a t estimatin g fraction s afte r playin g i t a fe w times .

We ra n th e contes t s o tha t th e winne r wa s th e individua l gettin g th e "largest" piece . Whe n I presente d thi s i n a follow-u p worksho p fo r teacher s at Rutgers , the y cam e u p wit h a grea t improvement . The y sai d wh y no t have th e student s compet e i n teams , an d le t th e winne r b e th e tea m whic h ends u p wit h th e closes t t o a fai r division . No w th e group s woul d hav e t o cooperate amon g themselve s t o compet e agains t others .

Competition ha s prove d t o b e a grea t motivato r i n anothe r way . Las t year, w e offered a Discret e Mathematic s cours e t o ou r hig h schoo l students . Generally, i t wa s take n b y student s wh o woul d probabl y no t elec t mathe - matics i n thei r senio r year . Certainl y som e ha d ha d grea t troubl e an d ha d even faile d previou s mat h classes . On e o f th e problem s w e studie d wa s th e the Travelin g Salesperso n Proble m (TSP) . Th e proble m i s t o fin d th e bes t route tha t a salesperso n coul d tak e i f h e o r sh e woul d begi n a t th e hom e base, visi t eac h customer , an d retur n t o th e hom e bas e ("best " wa s define d as minimizin g tota l distance) . I explaine d t o the m tha t nobod y ha d eve r come u p wit h a n algorith m t o solv e th e proble m whic h coul d ru n i n a rea - sonable amoun t o f time. Whe n w e had a n exam , I adde d a TS P proble m fo r extra credit . Whoeve r ha d th e first , second , an d thir d shortes t route s woul d be awarde d extr a points . I n addition , I woul d tr y th e proble m mysel f a t th e same tim e an d han g m y answe r outsid e o f the door . Ther e wer e extra point s to b e gaine d fo r beatin g m y answer . O f cours e ther e wer e enough citie s tha t they coul d neve r tr y al l case s i n th e allotte d time . Tw o girl s tie d fo r first , and whe n the y wen t outside , the y sa w tha t thei r answer s matche d mine . They cam e bac k int o th e classroo m al l excite d abou t it . I t ma y hav e bee n the firs t tim e they eve r foun d themselve s o n a par wit h th e teacher . I canno t say our stud y o f graph theor y wa s perfect, fo r ther e wer e area s whic h I neve r did communicat e t o m y satisfaction, bu t thi s on e aspec t reall y inspire d thei r interest. The y wer e ver y prou d o f themselves .

Later i n tha t course , w e wer e buildin g Sierpinski' s Pyramid , simila r t o the on e insid e whic h Valeri e DeBelli s (Associat e Directo r o f th e Leadershi p Program i n Discret e Mathematic s a t Rutger s University ) i s standing i n Fig - ure 1 .

I ha d hear d man y way s o f constructin g th e Pyramid , an d remembere d there wa s on e whic h hi t m e a s mos t suitabl e fo r a perso n wit h m y lac k of constructio n skills , bu t I ha d forgotte n th e details . I poste d a not e t o

6 BRO. PATRIC K C A R N E Y

F I G U R E 1 . Valeri e DeBelli s i n Sierpinski' s Pyrami d

discrete mat h teacher s vi a emai l an d go t bac k abou t 1 4 answers . On e wa s what I wanted , an d i t cam e fro m Eva n Maletsk y (Professo r o f Mathematic s at Montclai r Stat e Universit y an d a staf f membe r o f th e Rutger s program) . He close d wit h th e sentenc e "Sen d m e a pictur e whe n yo u finish - whoops , it's a fractal , you'l l neve r finish." I printe d i t an d brough t i t t o class . W e were using Evan' s boo k an d I showed the m hi s picture i n it. The y wer e mos t impressed tha t a famou s autho r woul d tak e tim e t o writ e t o ou r clas s an d also tha t h e woul d jok e wit h them .

We starte d th e projec t slowly , bu t eventuall y th e student s caugh t on . Only th e man y sno w holidays kep t u s fro m gettin g t o th e heigh t o f the class - room. Buildin g th e Pyrami d turne d ou t t o b e a grea t cooperativ e project . My student s wer e ver y prou d whe n othe r students , wh o wer e takin g th e more "advanced " mathematic s courses , woul d com e int o th e roo m an d as k what thi s wa s (eventuall y i t becam e har d t o miss ) an d I woul d hav e on e of m y student s explai n it . I thin k i t di d muc h fo r thei r self-confidenc e tha t there wer e area s o f mathematic s i n whic h the y wer e th e "experts. " Thi s may wel l hav e bee n th e first tim e i n thei r live s tha t thi s ha d occurred .

When w e studie d code s i n class , I assigne d student s t o researc h a cod e that w e di d no t stud y i n clas s an d giv e a n ora l presentatio n o n i t t o th e class. On e youn g ma n i n particula r (wh o ha d faile d m y clas s a s a freshma n and droppe d ou t o f ou r schoo l fo r a year ) di d a n outstandin g job . I wa s so impresse d tha t I offere d hi s handou t o n Vehicl e Identificatio n Number s

T H E IMPAC T O F D I S C R E T E MATHEMATIC S I N M Y CLASSROO M 7

to thos e i n our discrete mat h emai l group . Bu t I poste d i t s o that request s would b e sen t t o th e student' s address . Th e first reques t h e receive d cam e from a universit y professor . H e carried tha t lette r aroun d i n his book wit h the retur n addres s stickin g ou t ver y noticeabl y fo r abou t a week . H e late r got mor e involve d i n usin g th e Interne t an d no w I a m please d t o sa y he is enrolle d i n the loca l communit y college . N o doubt h e had mature d ove r time, but there was also a vast differenc e betwee n his approach to the discrete math cours e an d to the mor e traditiona l cours e i n whic h I ha d taugh t hi m previously.

I thin k tha t thes e anecdote s hel p illustrat e th e way that discret e math - ematics involve s individua l peopl e o f bot h sexes , o f al l age s (well , a t leas t grades 7-12) , an d of varying abilities . Th e impact migh t bes t b e summed u p by relatin g a discussio n I had with th e students whe n th e course ende d las t year. I aske d i f the course turne d ou t t o b e what the y expecte d whe n the y signed u p fo r it . Ther e wa s a unanimou s "No! " The n individual s adde d comments suc h a s " I didn' t thin k I wa s going t o lik e it, " or , " I though t i t would b e boring lik e the other mat h classe s I'v e had" ( I hate t o admi t tha t I wa s her teache r th e yea r before) , an d "Yo u should hav e tol d u s wha t i t was lik e an d more peopl e woul d hav e take n it. " I do not for a minute clai m that al l of these student s wil l becom e grea t mathematicians , bu t I do thin k they hav e a new and far more positiv e vie w of mathematics an d their abilit y to d o it. I f nothin g else , i t i s m y hop e tha t the y wil l no t pas s o n t o thei r children th e " I wa s neve r an y goo d a t mat h an d neve r foun d an y us e for it s o why should yo u try" syndrom e whic h haunt s s o many student s i n our schools today .

References

[1] Bennett , Sand i et . al. , Fair Divisions: Getting Your Fair Share, HiMA P Modul e # 9 , Consortium fo r Mathematic s an d Its Application s (COMAP) , 1987.

[2] Rosenstein , Josep h G. , an d Valerie A . DeBellis, "Th e Leadership Progra m i n Discret e Mathematics", thi s volume .

B I S H O P W A L S H M I D D L E / H I G H SCHOOL , CU MBERLA N D M D

E-mail address: p c a r n e y Q d i m a c s . r u t g e r s . edu

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

T h r e e fo r t h e Money : An Hou r i n t h e Classroo m

Nancy Case y

I a m th e teache r i n a crowded , windowless , cubicl e o f a classroom , wing - ing i t somewhat , a s always . Th e chair s ar e arrange d i n a circle , althoug h a t the ver y most , a quarte r o f th e student s ar e sittin g down . O n th e floor i n front o f u s i s th e larges t know n 3-regula r plana r grap h o f diamete r 3 . (Se e Figure 1 . I f thes e term s ar e unfamilia r t o th e reader , thei r meanin g i s ex - plained below. ) I t i s draw n wit h maskin g tap e an d it s 1 2 vertice s ar e larg e enough fo r a perso n t o stan d in . A jumbl e o f conversation s fills th e room . Students interrup t on e another ; occasionall y someon e ask s m e a question . Mostly I a m watching . Wha t I se e i s s o excitin g tha t i t i s har d fo r m e no t to deman d tha t the y al l si t dow n an d b e quie t s o tha t I ca n giv e a lectur e about wha t I see . Fo r th e first tim e sinc e th e clas s perio d began , I relax . Teaching, learning , th e classroom—i t i s al l goin g th e wa y i t i s suppose d to .

The 2 0 o r s o youn g peopl e i n th e classroo m ar e hig h schoo l student s participating i n th e Idah o Scienc e Camp. 1 Thi s i s ou r fourt h one-hou r ses - sion together ; w e ar e no t quit e th e stranger s tha t w e wer e t o on e anothe r a few day s earlier . I hav e bee n tryin g t o expan d thei r understandin g o f wha t it mean s t o d o mathematics , an d als o pu t int o practic e som e thing s tha t I understand theoretically—tha t student s ca n lear n muc h mor e i n a ric h an d stimulating environment , tha t hierarchica l expositio n isn' t alway s th e bes t way t o conve y information , an d tha t student s lear n a lo t b y talkin g t o eac h other abou t thei r ideas . I a m tryin g t o understan d wha t i t mean s t o hav e a learning-centered classroo m instea d o f a teacher-centere d one . I t i s th e un - certainty o f givin g u p tigh t contro l o f th e progra m tha t ha s mad e m e tense ; seeing wha t goo d thing s ca n happe n whe n I d o i s wha t make s m e relax .

Before th e student s arrived , I ha d draw n th e grap h o n th e floor wit h masking tape . I kno w a lo t abou t thi s particula r grap h an d abou t graph s i n general. Tryin g t o organiz e tha t informatio n int o a coheren t outlin e seeme d

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 . Research supporte d b y th e U.S . Departmen t o f Energ y Lo s Alamo s Nationa l Lab -

oratory Megamat h Projec t an d Departmen t o f Compute r Science , Universit y o f Idaho , Moscow, ID .

lrThe Idah o Scienc e Cam p i s sponsore d b y th e Universit y o f Idah o Colleg e o f Engi - neering an d th e U.S . Departmen t o f Energy .

https://doi.org/10.1090/dimacs/036/02

10 NANCY CASE Y

FIGURE 1 . This is the graph that was drawn on the classroom floor with masking tape. No two lines cross. Every vertex of this planar graph has degree 3. The maximum distance be- tween two vertices of the graph is 3. It is not known whether it is possible to draw a graph with more vertices which pre- serves these properties.

impossible. Th e structur e o f th e informatio n i n m y ow n min d mor e closel y resembled th e drawin g o n th e floor tha n a n ordere d lis t wit h indentations . I decide d t o tak e a ris k an d com e t o th e classroo m equippe d primaril y wit h

THREE FOR THE MONEY: AN HOUR IN THE CLASSROOM 11

my understandin g o f graphs . I ha d i n min d a brie f introductio n wit h whic h I woul d begin . Otherwise , m y onl y plan s wer e t o dra w th e grap h i n Figur e 1 o n th e floor ahea d o f time , t o tak e m y cue s fro m th e students , t o b e alert , and t o tr y t o thin k o n m y feet . I wa s mor e tha n a littl e bi t nervou s whe n i t was tim e t o ge t started . I asked :

Do you remember I told you about my friend Mike 2—the Computer Scientist?

"The gu y fro m Sa n Diego wh o invente d th e Orang e Game! 3" someon e called out .

Yes, he's the one. Well, I have learned a lot from him —a lot of mathematics, and also a lot about what mathematicians actually do . One thing that drives him crazy is that people have the mistaken idea that mathematics is primarily about numbers. Really, there is much more to mathematics than that. One day when he was particularly excited, he plucked a drawing from a jumble of papers on the table and exclaimed, cAn object like this is every bit as important as a number — and just as useful, too. Why, this i s a number! This is a psychedelic number!'

Well, this drawing is the one he was referring to. What he called a psychedelic number that day is what most math- ematicians call a graph. It is made of dots (or circles) and lines. You '11 notice right away, I hope, that even though it's called a graph, it doesn't have anything to do with the kind of graph you've probably studied in school —you know, bar graphs, line graphs and such. This is a a whole different kind of a graph, simply made of dots and lines. This might look like an arbitrary configuration of dots and lines to you, but Mike has found it to be very interesting, so much so that this past year he offered students a cash prize if they could solve a certain problem that involves this graph. The problem sounds simple enough: Is it possible to draw a graph that is larger than this one, yet preserves three critical properties?

There wa s a flurry o f excitement—ho w muc h money , ho w soo n d o w e get it , an d quickl y someon e calle d out , " I ca n dra w on e bigger , jus t gimm e the maskin g tape! " I clarify :

2Mike Fellows , Departmen t o f Compute r Science , Universit y o f Victoria, Victori a BC , Canada, mfellows@csr.uvic.c a

3 T h e Orang e Gam e illustrate s routin g an d deadloc k i n networks . An y numbe r o f people ca n play . Al l player s labe l tw o orange s wit h thei r names , the n th e orange s ar e mixed up , a singl e orang e i s removed , an d th e player s eac h pic k u p tw o orange s whic h ar e not thei r own . (On e perso n wil l hav e onl y one. ) Player s the n stan d i n a circle . A playe r may plac e hi s o r he r orang e int o th e empty han d o f a n adjacent playe r only . Th e gam e ends whe n al l player s hav e thei r ow n orange s back .

12 NANCY CASE Y

Bigger means having more nodes or circles; in graph theory they are called vertices. Also, you still need to know what the three critical properties are that you have to preserve. Size is a property. The size of this graph is 12 because it has 12 vertices. Obviously that's not the property that you have to preserve.

"Well, what ar e the properties then? " A t tha t momen t th e ide a occurre d to m e t o tur n th e proble m inside-ou t an d mak e the m gues s wha t th e prop - erties were . I als o decide d t o us e precis e mathematica l term s wheneve r th e opportunity arose , an d no t t o defin e the m unles s someon e aske d m e wha t they meant . I n retrospect , I woul d se e tha t thes e wer e tw o wonderfu l ideas . Scrutinizing an d manipulatin g a mathematica l objec t s o a s t o understan d its propertie s i s so much mor e lik e wha t mathematician s d o tha n bein g pre - sented wit h a lis t o f th e propertie s tha t som e objec t ha s (an d bein g aske d to memoriz e them) . Becaus e I use d th e specia l terminolog y o f grap h theor y right off , th e student s hear d th e word s an d bega n usin g the m themselves , long befor e the y wer e absolutel y sur e wha t the y meant . Thes e word s wer e with u s fro m th e beginning , an d a s th e clas s perio d progressed , student s attached meaning s t o the m tha t the y clarifie d an d shared . I answered :

You have to guess what the properties are. Name all the properties that you can think of and each time you name one of the three critical ones, I will tell you that you have guessed one.

Students bega n talkin g al l a t once , on e o f th e man y chaoti c eruption s that thi s class period woul d have. I doubted tha t th e students ha d mor e tha n a fogg y notio n o f wha t I mean t b y th e wor d property, bu t sinc e thi s didn' t dampen thei r enthusiasm , I wasn' t goin g t o interrup t an d tel l the m that . I als o didn' t thin k tha t the y woul d b e abl e t o gues s th e thre e propertie s without help . I waite d unti l th e excite d conversatio n die d dow n befor e I continued.

The prize-winning graph will be of size greater than 12 and also be planar, 3-regular , and have a diamete r of 3. That's a hint. All you have to do is figure out what that means and you will know what the three properties are.

A fe w peopl e wrot e thos e word s own . Quit e a fe w student s wer e copyin g the graph . Other s wer e excite d abou t winnin g th e money .

Next tim e I ha d everyone' s attention , I tol d the m abou t Pau l Erdos , a n elderly Hungaria n mathematicia n wh o spend s mos t o f hi s tim e o n th e road , finding i t fa r mor e interestin g t o trave l an d visi t mathematician s al l ove r the world , t o wor k wit h the m an d ge t the m excite d abou t problem s tha t interest him , tha n t o sta y hom e an d b e famous . I hel d u p a photocopie d article abou t hi m [5] .

He has a great mathematical mind, and would be famous and world-renowned by simply working alone. But he prefers

THREE FO R TH E MONEY : A N HOU R I N TH E CLASSROO M 1 3

working with other people, planting seeds, sharing ideas and questions, and offering prizes for solutions of unsolved prob- lems. As a result, university mathematicians love to have him visit and give talks. These visits often result in a collab- oration between Paul Erdos and a professor or student, and together they share the credit for some breakthrough.

So Mike's offer of a cash prize for solving this problem is an imitation of Paul Erdos. The prize adds a little excitement to the problem, and Mike is always trying to find ways to get other people excited about mathematics. Here is another interesting thing that Mike told me: he thinks that the reason why no one has found the solution for this problem is that no one has fiddled with it enough. It seems like you should be able to figure it out if you spent enough time understanding how graphs that have the same critical properties as these are constructed. All you have to do here is either draw a larger graph, or explain why it is impossible to do it.

Think about it. Imagine someone offered you a prize if you could find two odd numbers that made an odd number when you added them together . ..

Immediately a voic e calle d out , "Yo u can' t d o that! " Othe r voice s agreed.

Right. You would never say that someone who couldn't add two odd numbers and get an odd number needed to try harder. You would be sure they couldn't do it. With some careful thinking, you can come up with an explanation why no one will be able to add two odds and get an odd.

Some student s bega n explainin g t o on e anothe r wh y thi s i s so , bu t I continued speaking .

But this problem is different. No one has been able to come up with a good logical argument for why it would be impossible to draw a graph larger than this one that preserves the three properties. So if there's no reason that anyone knows of why there can't be a bigger one, it's likely enough that there is one, only no one has discovered it. If that's the case, it doesn't take any special advanced mathematics to just draw it. It just takes a pencil, and a lot of time, patience, and inclination to fiddle with it. Mike shows this problem to grade school kids whenever he can because he thinks one of them might just stumble on the solution.

"Tell u s wha t th e propertie s are , then! " Th e roo m becam e quiet .

OK, I'll give you some hints: Planar, 3-regular, and diameter 3. That's a lot of 3's. That's a hint. The number three. What's with the number 3?

14 NANCY CASE Y

Someone noticed tha t ther e ar e three edge s (lines ) touchin g ever y vertex . This wa s explaine d t o others . The y checke d al l th e vertices . Indee d i t wa s true. I tol d the m tha t th e numbe r o f edge s tha t touc h a verte x i s calle d th e degree of tha t vertex . I n thi s graph , ever y verte x i s o f degre e three . Whe n every verte x o f a grap h ha s th e sam e degree , the n th e grap h i s regular. S o this grap h i s 3-regular. Whe n yo u dra w th e grap h wit h mor e vertices , i t ha s to b e 3-regular .

Immediately som e student s bega n tryin g t o dra w 3-regula r graph s o f size large r tha n 12 . Soo n th e chao s wa s back . A shor t whil e passe d befor e enough student s realize d tha t the y didn' t hav e al l th e informatio n t o solv e the problem . A few more moments passe d befor e the y coul d shus h th e other s so I coul d tel l the m more .

I'll give you hints by showing you some games that you can play on this graph. The games will help you understand the structure of the graph better. No doubt that the person who actually solves this problem will know many more things about the structure of this graph than just the three properties that are preserved. We '11 play some games that will help you see lots of properties. When someone guesses a property that is one of the three critical ones, I will tell you.

I aske d a coupl e o f student s t o labe l th e vertice s wit h littl e strip s o f masking tap e wit h th e number s fro m 1-12 , remindin g everyon e tha t th e numbers weren' t par t o f th e graph . Th e number s woul d serv e a s name s fo r the vertice s an d mak e i t easie r t o tal k abou t them .

Two boy s ha d begu n t o pla y cards . I aske d the m t o loa n m e th e clubs . Surprised, the y shrugge d an d obliged . I remove d th e King , hande d th e remaining club s t o a student , an d aske d hi m t o g o stand i n an y circle . The n I explaine d a game .

Here's the rule: You can travel along the lines and walk from vertex to vertex. Every time you come to a vertex, you must lay a card down in it before you leave. You cannot return to a vertex that already has a card in it. Can you deal out all the cards?

He se t out . I t i s no t ver y har d t o do . Soo n al l th e card s wer e lai d i n circles o n th e floor. "It' s traversible! " crie d a voic e fro m th e crowd . "W e had tha t las t year. "

Right. Traversibl e is a word you can use to describe this graph. You can find a path through the graph that allows you to travel to all of the vertices without touching any of them twice. When you can do that, you can say that the graph is traversible, or that it has a Hamiltonian Path, named for the Irish mathematician William Rowan Hamilton who was very interested in graphs that have this property. This graph is indeed Hamiltonian. It's not one of the critical properties

THREE FOR THE MONEY: AN HOUR IN THE CLASSROOM 15

that has to be preserved, but it is a property. Do you think it will be preserved in the larger one?

Some yes's , som e no's . The y wante d mor e properties . I announce d another game , walke d t o th e cente r o f th e room , picke d u p a car d fro m th e graph, an d calle d fo r volunteers .

We need 11 volunteers. One person for each vertex where there is a card.

Students cam e eagerl y forwar d an d scramble d fo r places . Whe n the y were set , I gav e the m directions .

OK, pick up the cards. Look at the number that you have. Jack is 11, Queen is 12, Ace is 1.

"Oh no! " groane d a voice . "Th e Orang e Game! " W e ha d playe d th e Orange Gam e a fe w day s earlier . Thi s wa s indee d a variation .

You're absolutely right! Routing and deadlock in networks. Let's give it a try. Everyone has to get to the vertex with their number on it, but the only way you can move is from the place where you are standing and along a line into an empty vertex.

The fu n begins . Ther e i s onl y on e empt y vertex . Thi s require s a lo t o f maneuvering, plannin g an d workin g together . I t i s muc h harde r tha n th e Orange Game . Onc e I realiz e this , I a m no t eve n sur e i t i s possible , an d wonder i f I haven' t mad e a n awfu l mistak e b y askin g the m t o d o it .

Several leader s emerge . The y argue . Som e student s ar e confused , bu t follow order s willingly . Som e refus e t o d o wha t other s tel l them . N o on e gives up . A fe w student s i n th e audienc e ad d thei r suggestion s too . Other s continue t o por e ove r graph s a s the y hav e fo r th e las t 2 0 minutes . A gir l has gon e t o th e boar d t o demonstrat e t o tw o friend s wh y yo u can' t dra w a 3-regula r grap h wit h 1 3 vertices. ("Wit h 3 9 ends-of-lines , on e lin e woul d just hav e t o han g ther e loose , wit h n o verte x o n it s othe r end, " sh e says. ) When a studen t strugglin g wit h th e puzzl e o n th e floor i s too frustrate d an d wants t o quit , someon e jump s u p an d offer s t o tak e hi s place . Th e studen t who quit s stand s aside , continue s t o watch , an d i s soon makin g suggestions .

With th e student s movin g abou t th e grap h i n front o f me, once agai n th e series o f question s relate d t o thos e tha t leape d t o m y min d whe n I watche d them pla y th e Orang e Gam e come s back . Wha t i s th e fewes t numbe r o f moves i t wil l tak e t o ge t everyon e bac k t o thei r place s whe n ther e i s jus t one empt y slot ? Ho w i s it differen t whe n ther e ar e 2 , 3 , or 4 empty slots ? I f you char t thes e numbers , wil l you se e a pattern ? Won' t tha t depen d o n ho w "mixed up " everyon e wa s i n th e firs t place ? Wha t doe s "mixe d up " mean ? How di d w e know th e number s wer e "goo d an d mixe d up " whe n w e started ? Is i t possibl e t o star t ou t s o mixe d u p tha t everyon e can' t ge t bac k t o thei r places? Ho w woul d th e gam e b e differen t i f yo u didn' t kno w wha t number s other peopl e had ? Wha t i f yo u couldn' t kno w th e numbe r o f a verte x unti l you stoo d i n it ?

16 NANCY CASE Y

Another observe r migh t find thi s classroo m disorderl y an d unfocused . I t is disorderl y an d unfocused . This , however , i s th e momen t a t whic h I rela x and fee l tha t I hav e don e m y jo b a s teache r well . Th e roo m i s filled wit h excitement an d enthusias m abou t mathematics . A s the student s explor e thi s problem, the y ar e participatin g i n a larg e collaboratio n se t u p b y on e o f th e greatest mathematician s o f our time . Eve n thoug h ther e i s a monetar y priz e for a "righ t answer" , thi s i s no t th e sol e focu s o f th e room . I n fact , non e o f the student s understand s th e questio n yet ! The y hav e los t themselve s i n th e preliminary exploration . I pa t mysel f o n th e bac k fo r th e luck y on-the-spo t inspiration tha t ha d m e us e th e technica l term s fo r th e hints . Thos e big , unfamiliar word s have indeed grow n meaning s as they practice d sayin g the m and trie d t o figure ou t wha t the y mean .

I regre t tha t I wil l sa y goodby e t o thes e student s fo r th e las t tim e to - morrow. Whe n wil l be th e nex t tim e tha t I will have 2 0 enthused an d cleve r people t o thin k abou t problem s lik e thi s wit h m e s o tha t w e ca n ac t the m out together ? Nex t week , m y learnin g o f grap h theor y wil l slo w t o it s usua l solitary, ploddin g pac e a s I si t a t a tabl e an d mov e labele d bottl e cap s ove r graphs draw n o n sheet s o f paper , tryin g t o explor e som e o f th e question s raised fo r m e durin g thes e sessions .

At las t th e student s hav e sorte d themselve s o n th e network . The y stan d triumphantly i n th e vertice s whos e number s correspon d wit h thei r cards . A student wh o i s i n hi s sea t look s u p fro m hi s noteboo k wit h a n expressio n that i s bot h confuse d an d intense . H e asks , "Wha t ar e thos e propertie s again?"

You are trying to find a planar graph of size larger than 12 that is 3-regular, and has a diameter of 3.

I coun t t o 3 o n m y fingers a s I sa y th e word s planar, 3-regular, an d diameter of 3. Voice s joi n an d sa y th e word s alon g wit h me . I a m read y with anothe r proposal .

Do you want to play a game that is easier than that last one ? When you figure the game out, you will know what the diameter business is.

Of cours e the y do . I hel d u p a n illustrate d storybook 4 I ha d made .

This is a game that I invented to teach this problem to a group of 3-year-olds . . .

Now, with children that age, it takes a lot of effort just to get them to stay on the edges when they walk around the graph. So you can imagine how teaching them about proper- ties takes a lot of doing. I invented this story and the game to go with it after watching them play dress-up one day. The

4 The tex t o f thi s an d othe r storie s wit h Gertrude , Supe r person, an d th e Monste r that illustrat e propertie s an d problem s o n graph s throug h game s ar e availabl e o n th e World Wid e We b h t t p : / / w w w . c 3 . l a n l . g o v / m e g a - m a t h / an d i n Chapte r 3 o f [1] , Game s on Graphs .

THREE FOR THE MONEY: AN HOUR IN THE CLASSROOM 17

characters were ones that they had made up for their game- Gertrude (she's a goose), a Monster and Superperson. We '11 need three volunteers to play.

When the three-year-olds play, they spend the first 10 minutes dressing up in their costumes and capes. But you know that we 're doing this so that you can find out what di - ameter means, so we can get right to the point.

In this game, the circles of the graph are ponds. The lines that connect them are the fly ways that you can use to fly from one pond to the next. Superperson has the most flying power, Gertrude has the least, and the Monster is in the middle. The Monster is frightening and ugly, but harmless. He wants to play with Gertrude, but Gertrude is afraid of him. Since Gertrude is going to try to escape from the Monster, and since the Monster has more traveling power, it's more fair to let the Monster choose where he is going to start first. Then Gertrude can pick a good place to hide from him. Superperson will pick her place last.

The student s wh o hav e volunteere d t o pla y th e role s o f eac h o f the char - acters choos e place s t o stan d o n th e graph , an d th e gam e i s read y t o start .

The game begins because Gertrude is bored, this pond is bor- ing and she wants to be somewhere else. So she flies off to another pond. She's not in very good shape, though, so by the time she gets to the next pond, she crash lands in it and has to rest.

The perso n playin g Gertrud e act s ou t he r part .

Now the monster wants to play with Gertrude. He is so ridiculously large that it takes a huge amount of effort just for him to get off the ground. He is still going u p when he passes the first pond. But soon he is tired and he crashes into the second one with an enormous splash.

The good-nature d Monste r act s ou t hi s ungainl y flight. Thi s time , any - way, h e doesn' t reac h th e pon d wher e Gertrud e i s hidin g fro m him .

Now Superperson flies. She has the most power of all. She can fly three ponds before she gets tired, zip-zip-zoom! She flies over two ponds and lands in the third.

Superperson doe s this .

Now it's Gertrude's turn. All three of them will keep flying around—first Gertrude one, then the Monster two, and Su- perperson three —until the Monster catches up with Gertrude.

It doesn' t tak e lon g befor e th e Monste r land s i n th e sam e pon d a s Gertrude, an d th e tw o student s perfor m th e ensuin g action .

18 NANCY CASE Y

Now the Monster isn't going to hurt Gertrude, but Gertrude doesn't know that, so she attacks the Monster who has to cry to Superperson for help. C'mon Gertrude! Attack! Monster, yell for help!

Can Superperson come and save the Monster before Gertrude drowns him in mis-directed self-defense?

Of cours e sh e can . Th e diamete r o f th e grap h i s three . Bu t i t take s a few time s befor e th e student s figure thi s out . Soo n the y complai n tha t th e the gam e i s borin g an d kin d o f silly . Th e Monste r alway s gain s a ste p o n Gertrude an d wil l alway s catc h u p wit h her , an d Superperso n alway s get s there i n tim e t o sav e him .

Yes! The game is horribly dull because of the properties of the graph that is being used for a game board. Superperson always saves the day, and Superperson always saves the day because the diamete r o f th e grap h i s 3 . So what does that mean? Think about that statement: the graph has a diameter of 3?

It take s a littl e bi t o f arguin g an d som e experimentation , bu t soo n ev - eryone realize s tha t yo u ca n ge t fro m on e verte x (n o matte r which ) t o an y other i n thre e o r fewe r steps .

Now th e student s ar e divide d roughl y int o tw o groups : thos e wh o wan t to kee p playin g th e Gertrud e gam e an d inven t mor e rule s s o i t i s les s pre - dictable, harder , an d mor e interesting , an d thos e wh o wan t t o wor k o n th e prizewinning problem . I insis t tha t everyon e si t dow n an d liste n on e las t time.

Planar. That's the other property. What could that mean?

The student s ar e silent . Ther e i s an uncomfortabl e shiftin g o f feet. The y avoid m y gaze .

It has something to do with a plane. Like points, and lines and planes that you may have learned about in geometry.

There ar e stil l n o guesses. The y nee d a hint . Wha t hin t ca n I give that' s not tantamoun t t o telling them? O n th e spu r o f the momen t I can onl y com e up wit h on e idea .

OK, here's a hint. Imagine that this is the map of a city. The edges are roads, the vertices are intersections. What is something that is expensive to build that this city doesn't have any of?

What a stupid, misleadin g hint, I think, an d a m surprise d whe n a choru s of voices shouts , "Bridges! " Soo n everyon e know s tha t planar mean s yo u can dra w th e grap h o n a plan e o r flat surfac e i n suc h a wa y tha t th e edge s only touc h a t vertices . I tel l the m tha t planarit y i s especiall y importan t when peopl e ar e tryin g t o desig n electroni c circuit s tha t ca n fit o n on e sid e

THREE FOR THE MONEY: AN HOUR IN THE CLASSROOM 19

of a silico n chip . Distanc e an d degre e ar e als o importan t propertie s i n th e graphs tha t compute r scientist s us e t o mode l th e chip s tha t the y design .

About te n minute s remai n i n th e clas s period . O n th e on e hand , i t ha s taken ove r 4 5 minute s simpl y t o stat e a problem . O n th e othe r hand , i t i s mind-boggling t o conside r th e amoun t o f thing s w e hav e don e i n tha t shor t time. Thi s grou p i s heterogeneous , no t "gifted" . I n tw o day s th e Scienc e Camp wil l b e over . The y ar e windin g down , hav e bee n stayin g u p to o late , far mor e intereste d i n gettin g i n th e las t day s o f socializin g tha n th e las t moments o f mathematics . I di d no t anticipat e thi s leve l o f enthusias m an d tenacity. I neve r coul d hav e planne d this . Ha d I no t bee n there , I woul d argue tha t i t i s no t possibl e t o "cover " s o muc h i n on e clas s period . An d yet, i t ha s happened .

Alone, i n pair s o r i n group s som e student s ar e thoughtfull y drawin g graphs. Other s wan t t o kee p playin g game s an d I tur n m y attentio n t o them. I ri p u p five sheet s o f pape r tha t ar e differen t colors , an d randoml y place strip s o f al l five color s o n th e vertices .

You want to make it so that when you walk along the edges and leave from a vertex of one color, the next vertex you come to will not be the same color. You might need these.

I giv e a handfu l o f th e colore d strip s t o eac h student . It' s no t har d t o do a vertex-colorin g o f thi s grap h wit h five colors . Whe n the y finish, I as k them t o tr y t o colo r th e vertice s i n th e sam e wa y an d us e fewe r tha n five colors. The y begi n experimenting .

I tur n t o th e student s wh o hav e queue d u p t o sho w m e graph s the y hav e drawn, tryin g t o wi n th e priz e fo r a planar , 3-regula r graph , wit h maximu m distance 3 and siz e larger tha n 12 . I systematically chec k ou t th e properties , showing everyon e m y thought-processe s ou t loud . Soo n the y ar e checkin g their ow n an d eac h others . I t i s mor e efficien t t o hel p eac h othe r tha n t o wait fo r m y undivide d attention .

When th e game-playing students hav e found a vertex-coloring that work s for thre e colors , I as k the m wh y the y can' t d o i t wit h tw o colors , an d the y don't hav e a har d tim e showin g m e wh y tw o color s just can' t work . I scoo p up th e colore d strip s fro m th e vertice s an d plac e the m o n th e edge s an d ask the m wha t the y thin k the y ar e suppose d t o d o now . The y don' t see m to hea r me . The y ar e alread y working , tryin g t o find a wa y t o arrang e th e colors s o tha t n o tw o edge s o f th e sam e colo r touch .

I interrup t an d as k them ho w many color s they thin k i t wil l take. Three , someone says . Agreemen t i s muttered ; the y al l see m sure . I as k ho w the y know that , bu t n o on e answers . Whe n the y finish th e edge-coloring , ther e is stil l tim e t o colo r th e regions .

A fe w student s wh o hav e bee n drawin g graph s tha t solv e th e origina l problem ar e convince d tha t i t i s impossibl e t o do .

"Tell tha t Orang e Gam e gu y yo u can' t d o it, " someon e says . I explai n that i f the y ca n demonstrat e wh y i t i s no t possible , the y ca n probabl y wi n the prize . The y ar e excited—unti l I insis t tha t a vali d demonstratio n i s no t

20 NANCY CASE Y

FIGURE 2 . The story of mathematics is much more the story of long and fruitless searches for solutions to problems, than it is the story of a discipline full of quick answers that are eas- ily verified as right and wrong —even though the latter is the more common experience of many students. In 1889 after several years of study and experimentation, mathematician P. G. Tait published a table of knots. He believed it was not possible to twist, pull, or otherwise deform (without cutting) any of the knots in the table so that they looked like any of the others. 75 years later, Kenneth Perko, a New York attor- ney and amateur mathematician demonstrated that these two knots from Tait's tables were, in fact, the same knot. Make these knots out of rope and try for yourself to deform one so that it looks like the other. Chances are it won't take you 75 years!

simply sayin g tha t yo u trie d an d trie d an d couldn' t find it . Ho w d o yo u know i f yo u hav e trie d lon g an d har d enough ? I happe n t o hav e a boo k [3 ] with m e tha t ha s a pictur e o f tw o knot s whic h wer e though t t o b e differen t for 7 5 years. (Se e Figur e 2. ) The n afte r al l tha t time , someon e showe d tha t they wer e reall y th e same . I war n them :

Even if you try for a long time and can't do it, that doesn't mean that someone else won't get lucky and figure it out.

We review th e reaso n wh y w e are sur e tha t yo u can' t dra w a grap h wit h 13 (o r an y od d number ) vertice s tha t ha s thes e properties . Mayb e i t won' t work fo r 14 , bu t ho w ca n yo u b e sur e i t doesn' t wor k fo r 50 , o r 246 ? An d how do you know i t doesn' t wor k fo r 14 ? Ho w can you be sure you hav e trie d every possibl e combination ? I s ther e a systemati c wa y t o lis t th e differen t things yo u tr y t o tha t yo u ca n tel l i f you hav e lef t an y possibilit y out ? The y want tha t prize . The y ar e talkin g abou t ho w the y wil l spen d th e money . They kee p working .

At th e en d o f th e clas s period , I sho o th e student s awa y s o the y wil l b e on tim e fo r thei r nex t class . Late r i n th e afternoo n an d th e nex t day , th e final da y o f classes, student s wil l bring m e graph s tha t the y hav e draw n an d

THREE FOR THE MONEY: AN HOUR IN THE CLASSROOM 21

F I G U R E 3 . The Peterson Graph is one of the most famous graphs in all of graph theory. Its renown stems from the unusual list of properties it both possesses and fails to possess. It is an interesting graph for students to examine to see what properties it has and does not have.

we wil l g o ove r the m carefully . N o on e solve s th e problem , bu t the y hav e tried hard . I hav e t o examin e the m carefull y t o find th e flaws.

"What wil l yo u d o i f I find it? " someon e asks . "We wil l ge t o n th e phon e righ t awa y an d cal l Mike. " I as k th e student s t o writ e dow n wha t the y hav e learne d abou t math -

ematics thi s week . On e gir l writes , "M y teacher s tr y t o mak e m e thin k abstractly, an d I refuse . Yo u tricke d m e int o doin g it. "

How I wish I had bee n abl e t o teac h fo r both weeks of the Scienc e Camp . This sessio n woul d hav e bee n i n th e first week , an d o n th e weeken d I coul d have accompanie d the m o n thei r Saturda y tri p t o picni c an d swi m a t th e Snake River . The y woul d sho w m e eve n mor e graphs , w e coul d loo k mor e closely a t th e game s an d tal k abou t othe r thing s tha t cam e u p durin g th e week, suc h a s logi c puzzles , an d wha t i t takes , i n mathematics , t o sa y tha t something is true. Perhap s there would be a student o r two tenacious enoug h to pore over Mike's paper o n dense planar network s [2 ] with m e and deciphe r it. Certai n element s o f his method s fo r drawin g graph s wit h minima l degre e and diamete r ar e accessibl e an d migh t hel p the m i n thei r effort s t o find a prize winning graph .

If w e wer e togethe r a secon d week , w e woul d surel y gathe r wit h a dif - ferent grap h draw n o n th e floor i n fron t o f us . Th e Tutt e Grap h perhaps , or th e Peterso n Graph—the y ar e prett y an d symmetri c an d hav e differen t properties t o discove r an d tal k about . (Se e Figure s 3 & ; 4.) W e coul d eve n draw i t i n colore d chal k o n th e sidewal k i n fron t o f the librar y o r somewher e downtown; o r w e coul d us e law n pain t t o spra y pain t i t o n th e grass .

If w e ha d mor e time , w e coul d retur n t o Mike' s ide a o f a grap h a s a psychedelic number , an d tr y t o figure ou t wha t h e meant . I woul d b e abl e

22 NANCY CASE Y

FIGURE 4 . When Canadian mathematician William T. Tutte drew this graph in 1946, a question which had been open for over 60 years was solved. The Tutte graph is 3-regular and has the property of being 3-connected , which means that in order to break the graph into two disconnected components, you must delete at least 3 vertices, along with the edges that are connected to them.. In 1880, the English mathemati- cian P.G. Tait (see also Figure 2) conjectured that every 3- connected, planar, 3-regular graph is Hamiltonian. (A graph is Hamiltonian if it is possible to find a path that allows you to walk from vertex to vertex, touching each vertex exactly once and return to the vertex where you began.) The Tutte graph is not Hamiltonian, so it provides the counterexample which disproved TaiVs conjecture —even though TaiVs con- jecture seemed to be true for all those years.

to talk abou t mathematica l objects , thei r propertie s an d operations o n them . I coul d watc h thes e term s acquir e meanin g fo r th e student s a s w e looked fo r properties o f objects suc h a s graphs an d numbers , knot s an d map s an d the n

THREE FO R TH E MONEY : A N HOU R I N TH E CLASSROO M 2 3

discussed wha t i t mean t t o d o operations—suc h a s add , subtract , multipl y and divide—o n them .

No matte r ho w engagin g m y presentations , I woul d expec t the m t o sto p listening t o m y teacherl y lecture s an d begi n t o experimen t an d tal k abou t properties tha t the y suspec t o r discover . The y ma y adap t version s o f th e games w e have playe d t o th e mathematica l objec t tha t i s at hand . The y wil l get i n eac h others ' way , interrup t on e anothe r an d mak e eac h othe r angry . Some student s ma y tr y t o tak e charge . Other s wil l ignor e the m an d wor k furiously i n their seat s o n problem s the y ma y o r ma y no t understand . Som e students wil l stan d asid e an d appea r t o b e doin g nothin g a t all . This , I a m learning, i s ofte n a sympto m o f carefu l reflection .

I wil l remin d mysel f ho w hoars e I ge t wheneve r I rais e m y voic e i n th e classroom. A s I (an d m y agenda ) sli p fro m th e cente r o f thei r attention , I can b e confiden t tha t thei r attentio n wil l com e back . Perhap s someon e o n the othe r sid e o f th e roo m wil l cal l ou t a questio n t o me . I wil l pantomim e my inabilit y t o tal k lou d enoug h t o b e heard . Perhap s I wil l adop t a moc k professorial attitud e an d say , "Ahem , I thought I was aske d t o impar t som e valuable informatio n here. "

Eventually someon e wh o like s t o giv e order s wil l wan t t o hea r wha t I have t o sa y an d shout , "Hey ! Psssh ! He y yo u guys , shut up! Can' t yo u se e she's tryin g t o tell yo u something?! "

I will begin talkin g i n a quiet voice , but befor e I have finished a sentence , all th e othe r conversation s i n th e roo m wil l resume . I wil l sto p talking , an d the student s wh o hav e accepte d th e responsibilit y o f makin g th e other s b e quiet an d liste n wil l agai n tak e th e floor an d mak e i t quie t s o tha t I ca n talk. Eventually , the y wil l b e read y t o listen , an d I wil l dra w o n wha t ha s happened t o pic k u p th e threa d o f th e stor y tha t ha s wove n throughou t al l of ou r clas s sessions—th e stor y o f wha t i t mean s t o d o mathematics .

I wil l poin t ou t tha t wheneve r I tr y t o tel l the m abou t a goo d mat h problem, i f i t is a goo d problem , i t i s s o ric h an d many-tentacle d tha t im - mediately, whe n I begin talkin g abou t it , thei r mind s dar t t o interestin g an d compelling places . The y notic e things , the y wan t t o experiment , the y mak e guesses an d argue ; the y wan t t o tel l someon e els e wha t the y ar e thinking . This happen s becaus e they ar e doin g mathematics: churnin g u p ideas , filling the roo m wit h creativ e thought s an d guesse s tha t eventuall y the y ca n pi n down, refine , an d eithe r kee p o r reject , b y marchin g carefull y throug h thei r thoughts wit h rigor , logic , an d proof . The y ar e doing mathematics , an d that i s fa r mor e stimulatin g tha n sittin g stil l an d listenin g t o a teache r tal k about it . I hope I will have to sho o the m ou t th e doo r t o b e o n time fo r thei r next class . I wil l leav e them , a s I alway s do , excite d abou t learning , abou t mathematics an d abou t question s o f m y own , becaus e thei r enthusias m an d excitement ar e s o very , ver y contagious .

24 NANCY CASE Y

References

[1] Casey , Nanc y an d Michae l Fellows , This is MegaMathematics: Stories and Activities for Mathematical Thinking and Problem Solving, Lo s Alamo s Nationa l Laboratory,Lo s Alamos NM , 1993 . Availabl e vi a anonymou s ft p fro m f t p . c s . u i d a h o . e d u i n th e di - rectory pub/mega-math/workbk .

[2] Fellows , M. , P . Hell , an d K . Seyffarth , "Construction s o f Dens e Plana r Networks" , Unpublished paper , Universit y o f Victoria , 1993 . Availabl e fro m Mik e Fellows , De - partment o f Compute r Science , Universit y o f Victoria , Victoria , BC , Canad a V8 W 2Y2 o r m f e l l o w s Q c s r . u v i c . c a

[3] Peterson , Ivars , Islands of Truth: A Mathematical Mystery Cruise, W . H . Freeman , New York , 1990 , p . 55 .

[4] Tait , P.G. , "O n Knot s I , II , an d III. " Scientific Papers, Cambridg e Universit y Press , 1990.

[5] Tierney , John , "Pau l Erdo s i s in Town . Hi s Brai n i s Open." Science '84, October 1984 , pp. 40-47 .

[6] Trudeau , Richar d J. , Dots and Lines, Ken t Stat e Universit y Press , Ken t OH , 1976 . (For informatio n abou t othe r accessibl e grap h theor y problem s an d thei r applications )

D E P A R T M E N T O F C O M P U T E R S C I E N C E , U N I V E R S I T Y O F I D A H O , M O S C O W , I D 8384 3

E-mail address: casey931Qcs.uidaho.edu , h t t p : / / w w w . c s . u i d a h o . e d u / ~ c a s e y 9 3 1

DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Fibonacci Reflections—It' s Elementary !

Janice C . Kowalczy k

Background

While I am presently i n the position of the assessment coordinato r fo r th e Leadership Progra m i n Discret e Mathematics , I have se t asid e som e tim e t o practice th e mathematica l idea s an d teachin g methodologie s tha t I gleane d from bein g a participan t i n th e firs t progra m fo r middl e an d elementar y school teacher s i n 1992 . Durin g recen t years , I hav e conducte d a numbe r o f teacher workshop s an d session s wit h students . Th e followin g i s a n accoun t of a recen t experienc e i n a fourth-grad e class .

Connections

In Decembe r o f 199 3 I conducte d a serie s o f one-hour session s o n th e Fi - bonacci number s wit h my daughter's fourth-grad e clas s at th e Fores t Avenu e School i n Middletown , Rhod e Island . M y daughte r create d th e opportunit y when sh e recognize d th e Fibonacc i number s i n a clais s scienc e uni t whe n they wer e studyin g monocot s an d dicots . He r Fibonacc i comment s caugh t the attentio n o f he r curiou s teache r an d befor e lon g a connectio n wa s made , and a dat e se t fo r thi s series .

The Fibonacci sequence begins with the numbers 1 and 1. Each number that follows is the sum of the previous two numbers, hence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, • • • are the first 12 terms of this infinite sequence.

T h e B e g i n n i n g — S o m e T h o u g h t s o n M a t h e m a t i c i a n s

On ou r firs t da y togethe r w e investigate d th e classi c rabbi t question ; however, befor e w e bega n tha t activity , w e too k som e tim e t o discus s wha t mathematicians loo k like and what mathematician s do . Non e of the student s had an y clea r image s o f a mathematician; however , man y o f the students fel t

1991 Mathematics Subject Classification. 00A35 , 00A05 . © 199 7 America n Mathematica l Societ y

https://doi.org/10.1090/dimacs/036/03

26 JANICE C . KOWALCZY K

that mathematician s add , subtract , multiply , an d divid e fo r a living . On e student di d repl y tha t mathematician s solv e problem s usin g numbers .

Some Thought s o n D o i n g M a t h

Next, w e pursue d a numbe r o f rea l mathematica l question s an d looke d at th e proces s associate d wit h them . Fo r example : I f I gav e yo u 1 0 cent s every da y t o pu t i n you r bank , ho w muc h mone y woul d yo u hav e a t th e en d of a week ? Wha t informatio n di d yo u hav e t o kno w first ? Wha t woul d yo u need t o kno w i f I aske d yo u ho w muc h mone y yo u woul d hav e afte r a yea r (with n o interest) ? I wa s surprise d a t thi s point , t o find ou t tha t mos t o f the student s di d no t kno w ho w man y day s wer e i n a year ; however , the y were cleve r enoug h t o questio n ho w muc h mone y wa s i n th e ban k whe n th e problem began . I gave them th e bank questio n fo r a homework challeng e an d added tha t the y ha d t o tel l m e ho w the y go t th e informatio n the y neede d as wel l a s ho w the y go t thei r answer .

W h e n A m I Eve r Goin g t o U s e This ?

Next, w e took a loo k a t th e classi c rabbi t questio n below . I teased the m with the ide a that th e rabbit questio n di d not appea r t o be a significant ques - tion wit h an y "rea l world " applicatio n whe n i t wa s first pose d b y Leonard o of Pisa (bette r know n b y his nickname Fibonacci) . The n I assured the m tha t we woul d se e tha t th e exploratio n o f thi s questio n ha s le d t o a n incredibl e number o f importan t "rea l world " connection s i n a numbe r o f othe r fields.

T h e Rabbi t Questio n

/ / a pair of rabbits were put into a walled enclosure to breed, how many pairs of rabbits would there be after one year if it is assumed that every month each pair produces a new pair, which, in turn, begins to bear young two months after its own birth? [1 ]

With th e overhea d projector , th e clas s helped m e develop a rabbit popu - lation diagram, a s in Figure 1 , month b y month, fro m Januar y throug h April . To mak e th e experienc e mor e concrete , I create d 1 2 envelopes containin g 1 8 cardboard pair s eac h o f bab y rabbits , matur e rabbits , an d "married " pro - ducing rabbits ; an d I gav e on e envelope t o eac h pai r o f students i n the class . We the n brok e th e clas s u p int o pair s t o recreat e th e growt h o f th e rabbi t population throug h th e month s o f Ma y an d June . Eac h pai r o f student s used th e envelope s o f rabbit s t o buil d a mode l o f th e problem . Rabbit s wer e taped o r glued ont o large sheets o f paper whil e labels an d arrow s were draw n to clarif y th e growt h patter n o f th e rabbits .

FIBONACCI REFLECTIONS—IT' S ELEMENTARY ! 2 7

FIGURE 1 . Th e Fibonacc i rabbit-breedin g model , Januar y through April .

Building a M o d e l

About hal f o f th e group s di d extremel y wel l an d wer e read y t o hel p others trac e th e growt h o f th e rabbit s throug h th e nex t month , whil e on e group seeme d t o mis s th e ide a o f tracin g th e growt h o f th e populatio n fro m one mont h t o th e next . Th e student s tha t wen t of f trac k ha d a tendenc y to trac e th e growt h o f onl y on e o r tw o rabbit s throug h a numbe r o f month s and los t trac k o f th e othe r rabbits . Thes e student s wer e abl e t o understan d readily ho w eac h kin d o f rabbi t progresse d i n thei r developmen t bu t wer e not abl e t o organiz e themselve s aroun d th e ide a tha t i n eac h mont h th e rabbit populatio n changed . However , afte r abou t 2 0 minutes an d hel p fro m either me , thei r teacher , o r thei r peers , al l o f the clas s member s wer e abl e t o complete their models . O n our chart o n the overhead projector, w e continued to trac e an d dra w th e growt h o f th e rabbi t populatio n together . Thi s tim e we continue d solvin g th e proble m throug h th e month s o f Ma y an d June . A few student s ha d alread y successfull y complete d th e proble m throug h Jun e with thei r cardboar d rabbit s an d ha d commente d tha t July' s rabbit s woul d probably no t fi t ont o thei r larg e piece s o f paper . Next , a s a group , w e discussed way s t o answe r th e origina l question , "Ho w man y rabbit s woul d there b e afte r a year? "

28 JANIC E C . KOWALCZY K

Thinking A h e a d

Month Pair s o f Babie s Pair s o f Adult s Tota l Pair s January 1 February 1 March 1 April 1 May 1 June 1

0 1 1 1 1 2 1 2 3 2 3 5

T A B L E 1 . Char t wit h number s o f pair s o f rabbit s b y month ,

We organized th e number s o f eac h kin d o f rabbit s eac h mont h o n chart s that wer e hande d out , simila r t o Tabl e 1 . Th e student s wer e quic k t o recog - nize patterns . On e studen t realize d tha t sh e coul d predic t th e nex t numbe r of pairs o f bab y rabbit s b y lookin g a t th e sequenc e o f number s produce d b y the pair s o f al l th e rabbits . Anothe r studen t commente d tha t h e though t that th e differenc e betwee n successiv e number s wa s 1 , 2 , 3 , 4 , 5 , an d s o on . We looke d a t th e serie s t o se e i f w e coul d verif y thi s an d foun d ou t tha t w e could no t ge t a differenc e o f 4 i n thi s expecte d series ; bu t thi s ide a di d lea d the clas s t o discove r tha t th e differenc e betwee n successiv e term s wa s als o a Fibonacc i sequence . Interes t wa s climbing ! Eventually , on e o f th e girl s i n the clas s realize d tha t eac h ter m wa s determine d b y th e su m o f th e 2 previ- ous terms . Sh e expresse d thi s as , "Yo u ca n ge t th e ne w numbe r b y addin g the las t tw o number s together. " Th e group s worke d togethe r t o verif y he r idea an d reporte d bac k tha t i t seeme d t o b e true . Th e homewor k challeng e was t o us e thi s metho d o r an y othe r metho d tha t the y coul d justif y t o tr y to answe r th e rabbi t questio n throug h th e firs t year .

"Thirty D a y s H a t h S e p t e m b e r . . . "

The nex t da y w e warme d u p b y revisitin g th e firs t homewor k challeng e to explai n th e answe r an d th e proces s t o th e "1 0 cent s ever y da y fo r a year" question . Mos t student s ha d aske d other s ho w man y day s wer e i n a year. A fe w ha d looke d a t calendars . I commente d tha t whil e ther e wa s nothing wron g wit h gettin g th e answe r fro m others , man y time s w e hav e the knowledg e t o fin d a n answe r withi n ourselve s an d d o no t eve n realiz e it . I tol d the m I expected tha t thi s wa s the cas e wit h th e question , "Ho w man y days ar e i n a year? " an d reminde d the m o f th e verse , "Thirt y day s hat h September". W e spen t a fe w minutes , recitin g an d recordin g th e number s from th e vers e an d the n pu t togethe r al l th e number s t o ge t 365 . Whil e all o f th e student s kne w th e verse , n o on e ha d though t t o cal l upo n i t a s a resource.

FIBONACCI REFLECTIONS—IT' S ELEMENTARY ! 29

Revisiting t h e Rabbit s

Next, w e revisite d an d retrace d th e rabbi t proble m t o th e en d o f th e year usin g the chart s I had hande d ou t t o kee p track o f the numbe r patterns . Many student s cam e t o clas s wit h th e correc t answe r o f 377 , bu t a fe w ha d numbers lik e 38 1 or 367 . I aske d th e clas s t o gues s wh y som e answer s migh t be of f b y a little . I aske d them , "Wha t migh t hav e happene d i n th e proces s to caus e answer s suc h a s 381 , etc?" Som e student s conclude d tha t a smal l addition erro r coul d have caused the discrepancy . Th e students wit h answer s around 37 7 were aske d t o tr y t o retrac e thei r proces s an d repor t bac k i f th e "addition error " theor y wa s correct . I n al l case s th e student s wer e please d to respon d tha t thi s theor y ha d bee n correct .

Spiraling I n

The mai n activit y fo r th e secon d da y wa s t o investigat e th e item s tha t had bee n se t u p o n th e first da y i n a displa y area . Th e purpos e fo r thi s activity wa s t o hel p student s mak e connection s t o Fibonacc i number s i n na - ture an d t o hel p the m recogniz e a connectio n wit h spiral s [5] . Thes e item s included: Th e poster , "Fibonacc i Number s i n Nature " [2] ; five different va - rieties o f pinecones ; a n artichoke ; a cactus ; a pineapple ; a larg e sunflower ; the book , Fascinating Fibonaccis [1] ; an d som e se a shells . Man y o f thes e items wer e marke d wit h eithe r white-ou t o r colore d Elmer' s glu e wher e spi - rals coul d b e seen . Whil e passin g th e item s around , w e trie d t o determin e what the y al l ha d i n common . Afte r som e discussion , spiral s wer e agree d upon. Th e wor d helix wa s introduce d an d student s wer e aske d t o thin k o f other thing s i n natur e tha t displa y thi s shape . Tornadoes , whirlpools , an d seahorse tail s wer e thre e o f th e responses .

Seeing i s believin g

Students wer e pu t togethe r fo r a "pai r share " activity , a n arrangemen t in whic h eac h pai r o f student s explor e a proble m an d the n tw o pair s ar e pu t together t o compare , discuss , o r verif y eac h other' s answers . Eac h pai r o f students wa s aske d t o coun t th e numbe r o f clockwis e an d counterclockwis e helices o n a t leas t tw o varietie s o f pinecon e (se e Figur e 2) . Som e o f th e students coul d se e th e helices , bu t ha d difficult y countin g them , other s ha d difficulty seein g th e helice s whe n the y wer e give n th e unmarke d pin e cones . The teacher , myself , an d m y daughte r wer e kep t ver y bus y rotatin g fro m group t o grou p helpin g student s visualize , mark , an d coun t th e helices . Suc - cessful group s wer e employe d t o hel p others . Eventually , everyon e seeme d to b e abl e t o accomplis h th e tas k and , i n th e process , foun d ove r an d ove r again th e presenc e o f th e Fibonacc i numbers . I lef t th e student s wit h th e challenge o f countin g th e helice s o n th e larg e sunflower .

30 J A N I C E C . KOWALCZY K

F I G U R E 2 . Eigh t clockwis e helice s on a pinecone , colore d al - ternately fo r clarity . Ther e i s a set o f thirteen helice s windin g counter-clockwise, whic h i s harde r t o see . F r o m : F O R AL L P R A C T I C A L P U R P O S E S 3r d E d i t i o n b y C O M A P (p . 662) .

C o p y r i g h t 199 4 b y C O M A P , I n c . Use d w i t h p e r m i s s i o n o f W . H . F r e e m a n a n d

C o m p a n y .

Is Al l Fai r I n Lov e an d War ?

For a closing , cardboar d bookmark s wer e passe d out . Thes e marker s were decorate d wit h flowers tha t contai n peta l count s wit h Fibonacc i num - bers fro m 3 t o 8 9 [5] . Example s o n th e bookmar k includ e th e Oxey e dais y with 3 4 petal s an d th e Africa n dais y wit h eithe r 5 5 o r 8 9 petals . W e ende d the sessio n b y thinkin g abou t eve n an d od d i n th e "Love s me , Love s m e not" activit y tha t i s classicall y playe d wit h daisies . W e discusse d whethe r the activit y wa s fair . (I f yo u hav e neve r though t abou t thi s before , no w i s the tim e t o d o so. )

Fibonaccis ar e Ver y Prolifi c

On th e thir d day , w e reviewe d th e Fibonacc i sequence . On e studen t asked how long it would take for the rabbit populatio n to grow to one million. Other student s showe d thei r interes t i n thi s questio n immediately , an d a s I quickly discovered , thes e fourth-grad e student s wer e in love with th e concep t of " a million" . I aske d student s t o writ e dow n thei r tim e estimates . O n th e overhead I uncovere d th e populatio n number s slowly , mont h b y month . A t the en d o f th e secon d year , student s wer e give n th e opportunit y t o revis e their estimates . Shortl y after , the y wer e amaze d t o fin d ou t tha t th e walle d enclosure containe d ove r 1 millio n rabbit s afte r onl y 3 1 month s an d tha t their estimate s ha d bee n to o high . T o ge t a n additiona l sens e o f 3 1 months , students wer e asked t o conver t thei r answer s t o the numbe r o f days an d the n to year s an d t o shar e thei r answer s an d thei r thinkin g wit h th e class .

FIBONACCI REFLECTIONS—IT' S ELEMENTARY ! 3 1

Fleas, B e e s , Trees , an d P i a n o K e y s

I followe d thi s activit y wit h a presentatio n intende d t o quickl y mak e many othe r Fibonacc i connections . I n th e presentation , I proceede d to :

• trac e th e mal e honeybe e famil y tre e t o th e blac k key s o f th e piano ; • loo k a t thre e commo n musica l scale s tha t us e 5 , 8 , an d 1 3 keys; • examin e a scal e o f th e keyboar d wit h 1 3 keys includin g 8 whit e key s

and 5 blac k key s wit h th e blac k key s arrange d i n 2 s an d 3s ; • cla p ou t th e rhyth m o f a limeric k abou t fleas wit h beat s o f 2 s an d 3 s

and a tota l o f 1 3 beats complete d i n 5 lines ; • demonstrat e th e Fibonacc i (fractal ) growt h patter n o f some trees an d

root systems ; • tr y t o explai n th e connectio n wit h th e "Ellio t Wav e principle " an d

the Do w Jone s averages .

In th e 1930's , Ralp h Ellio t studie d pattern s i n th e stoc k market , real - izing tha t busines s swing s ar e a resul t o f huma n pattern s o f optimis m an d pessimism. Hi s observation s lea d t o th e predictio n tha t th e stoc k marke t follows Fibonacc i patterns . I carefull y connecte d thi s ide a t o som e issue s that student s coul d connec t t o thei r lives . W e talke d abou t optimis m an d pessimism i n spendin g an d relate d thi s t o th e recessio n i n Rhod e Island . I explained tha t whe n th e econom y i s i n a downswin g peopl e ten d t o hol d o n to thei r money , an d whe n i t i s i n a n upswin g peopl e ar e mor e ap t t o spend . I relate d thi s t o th e schoo l bon d issu e whic h ha d bee n defeate d twic e i n th e last fe w years , bu t i n th e recen t election s i t ha d bee n passe d b y th e voters . I explaine d tha t th e econom y wa s beginnin g t o improv e an d therefor e th e voters wer e finally agreein g t o spend . Sinc e thi s particula r schoo l i s on e o f the tw o school s tha t wil l b e mos t affecte d b y thi s bon d issue , student s wer e keenly awar e o f th e electio n an d therefor e abl e t o mak e th e connection . B y this poin t i n th e presentation , student s wer e fascinate d an d literall y o n th e edge o f thei r chairs . Afte r som e discussion , I aske d student s i f the y though t that Fibonacc i kne w th e significanc e o f thi s seemingl y sill y questio n abou t rabbits, whe n h e se t ou t t o explor e an d writ e abou t it . Whil e som e di d no t respond, man y doubte d tha t Fibonacc i ha d thes e application s i n mind .

T h e Final e

For th e final activity , w e organize d ourselve s int o group s t o explor e on e last Fibonacc i connection . Th e student s wer e tol d tha t the y woul d b e di - rected t o us e buildin g block s whos e dimension s wer e Fibonacc i number s t o create puzzl e piece s tha t coul d b e reassemble d int o a pictur e o f somethin g that migh t b e foun d i n Jurassi c Park . I als o sai d tha t I ha d a fossi l i n my pocke t an d tha t i t wa s 35 0 millio n year s old . The y wer e tol d tha t th e check o f thei r wor k woul d b e comparin g thei r complete d puzzl e t o th e fossi l in m y pocket . Afte r discussin g th e ide a o f 35 0 millio n fo r a fe w minutes , each pai r o f student s measure d an d cu t out : tw o 1-uni t squares ; on e 2-uni t square; on e 3-uni t square ; on e 5-uni t square ; an d on e 8-uni t square . The y

32 JANICE C . KOWALCZY K

used compasse s t o swin g a n ar c joinin g diagonall y opposit e corner s o f eac h square an d the n cu t ou t al l si x squares . The y wer e the n aske d t o assembl e these si x puzzl e piece s int o a n picture . I als o tol d the m tha t th e complete d puzzle woul d b e rectangular . Grou p b y group , the y verifie d thei r complete d puzzles wit h th e Ammonit e fossi l tha t I had i n m y pocket . Thi s i s similar t o a Nautilu s shell , show n i n Figur e 3 . Onc e verified , eac h grou p wa s give n a handout wit h a pictur e an d stor y abou t Ammonite s an d th e creatio n o f fos - sils. (Thi s activit y coul d b e a springboar d fo r exploration s wit h th e Golde n Ratio.) Thi s puzzl e activity becam e the connectin g an d culminatin g activit y of th e thre e days , i n tha t i t dre w togethe r th e theme s o f Fibonacc i number s and helice s discusse d earlier .

F I G U R E 3 . Cross-sectio n o f a Nautilu s shell , simila r t o tha t of a n Ammonite . F r o m : T a n n e n b a u m / A r n o l d , E X C U R S I O N S I N M O D E R N M A T H E M A T I C S ( p .

285), C o p y r i g h t 1992 . R e p r i n t e d b y p e r m i s s i o n o f P r e n t i c e - H a l l , I n c . , N J .

T h e B e a t G o e s O n

We close d wit h som e discussio n abou t helice s i n natur e an d th e worl d around us , an d abou t th e occurrenc e o f th e Fibonacc i number s tha t w e had see n ove r th e las t thre e days . I showe d som e spira l image s tha t I ha d gathered fro m book s an d a n invitatio n wa s extended t o student s t o continu e to mak e Fibonacc i discoverie s an d t o shar e an y ne w findings. Th e Fibonacc i display wa s lef t u p wit h a dinosau r boo k an d th e Ammonit e fossi l fo r abou t a wee k (th e student s wer e abl e t o cu t u p an d enjo y th e pineapple) .

Afterwords

I am no t sur e wha t m y expectation s o f fourth-grade student s wer e whe n I began , bu t I kno w no w that the y ca n understan d an d connec t wit h th e Fi - bonacci numbers . Th e fourth-grad e teacher , wh o i s a ver y hones t an d ope n person, mad e tw o particula r comment s tha t caugh t m y attention . Firs t sh e noticed tha t student s wh o usuall y d o no t d o wel l i n math , wer e doin g wel l

FIBONACCI REFLECTIONS—IT' S ELEMENTARY ! 3 3

with thi s serie s o f activitie s an d tha t som e o f thes e sam e "weak " mat h stu - dents were particularly stron g i n their visualizatio n o f the helice s (somethin g which sh e hersel f foun d challenging) . He r secon d commen t ha d t o d o wit h her teachin g o f math . Sh e sai d tha t whil e sh e feel s ver y comfortabl e teach - ing scienc e an d writing , sh e doe s no t fee l comfortabl e helpin g student s wh o have difficultie s i n math , becaus e sh e can no t see m t o comprehen d wh y the y don't ge t it . Sh e sai d he r teachin g o f mathematic s ha s bee n rathe r cut-and - dried, sinc e sh e ha s alway s see n mathematic s a s eithe r righ t o r wrong . Sh e felt tha t I ha d give n he r a ne w vie w o f teachin g math . Sh e sai d sh e notice d how muc h emphasi s I pu t o n th e proces s o f gettin g th e answe r rathe r tha n the answe r itsel f an d tha t sh e had neve r though t t o g o about teachin g math - ematics tha t way . Sh e sai d tha t th e session s wer e inspirin g t o he r an d tha t she learne d a lo t fro m havin g m e visi t he r classroom .

T h e Cal l o f t h e Classroo m

Personally I foun d thes e thre e day s valuabl e an d enjoyabl e an d onc e again hea r th e "cal l o f th e classroom" . Teachin g mathematic s aroun d a theme i s a grea t dea l o f fun . I expec t tha t m y approac h t o teachin g math - ematics i n thi s fourth-grad e classroo m woul d hav e bee n somewha t differen t if I ha d bee n th e permanen t teacher . Becaus e I wa s invite d t o thi s clas s t o introduce th e Fibonacc i numbers , an d becaus e th e tim e limi t wa s predeter - mined, m y approac h wa s muc h mor e guide d tha n I woul d expec t i t t o b e i n an ongoin g self-containe d classroom . I n a year-lon g classroo m experience , I imagin e tha t Fibonacc i connection s coul d b e wove n int o th e classroo m experience man y time s an d i n man y places . I hav e com e t o believ e tha t mathematics i s part o f ou r live s an d I hop e tha t throug h m y enthusias m fo r richly connecte d topic s such a s the Fibonacc i number s thi s belie f wil l also b e contagious t o my students. A s I guided ou r three-hou r exploration , I allowed some mathematica l sid e trip s a s studen t interest s an d question s sometime s exposed thei r "nee d t o know" . Creatin g an d guidin g thi s "nee d t o know " will b e fo r m e th e mos t excitin g tas k whe n I retur n t o th e classroom .

T h e Impac t o f t h e Leadershi p P r o g r a m i n D i s c r e t e M a t h e m a t i c s

My ow n curiosit y an d awarenes s o f th e Fibonacc i number s wa s devel - oped a fe w year s ag o when I first hear d abou t the m i n a n articl e b y Michae l Tempel i n th e Logo Exchange. I can' t sa y whethe r I full y understoo d th e article a t th e time , bu t i t di d ge t m e t o ope n m y ear s an d eye s t o thi s se - quence. A t th e Leadershi p Progra m i n Discret e Mathematic s a t Rutger s [4], I becam e awar e o f th e connection s o f thi s numbe r sequenc e t o scienc e and nature . Thes e connection s sparke d m y interes t i n th e topic . I spen t th e school yea r 1992-9 3 learnin g mor e abou t th e Fibonacc i number s an d i n th e summer o f 1993 , I collaborate d wit h othe r Leadershi p Progra m participant s to pu t togethe r som e classroo m material s an d a staf f developmen t work - shop o n thi s topic . Th e classroo m activitie s i n thi s articl e ar e draw n fro m

34 JANICE C . KOWALCZY K

these materials . The y ar e designed t o make connections , encourag e con - crete exploratio n an d foste r communication . Th e inspiratio n fo r this kin d of mathematic s teachin g is drawn fro m th e Leadershi p Program .

References

[1] Garland , Trud i Hammel , Fascinating Fibonaccis: Mystery and Magic in Numbers, Dale Seymou r Publications , Pal o Alt o CA , 198 7

[2] Garland , Trud i Hammel , an d Edit h Algood , Poster : "Fibonacc i Number s i n Nature", Dale Seymou r Publications , Pal o Alt o CA , 1988 .

[3] Kappraff , Jay , Connections, McGra w Hill , Ne w York , 1990 . [4] Rosenstein , Josep h G. , an d Valeri e A . DeBellis , "Th e Leadershi p Progra m i n Discrete

Mathematics", thi s volume . [5] Tannenbaum , Peter , an d Rober t Arnold , Excursions in Modern Mathematics, Prentic e

Hall, 1992 . [6] Wahl , Mark , A Mathematical Mystery Tour: Higher-Thinking Math Tasks, Zephy r

Press Learnin g Materials , Tucso n AZ , 1988 .

R H O D E ISLAN D SCHOO L O F THE F U T U R E , P . O . B o x 4692 , M I D D L E T O W N , R I 0284 2

E-mail address: k o w a l c j n Q r i d e . r i . n e t

DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Using Discret e M a t h e m a t i c s t o Giv e Remedia l S t u d e n t s a Secon d Chanc e

Susan H . Picke r

In th e summe r o f 1990 , a s a participan t i n th e Leadershi p Progra m i n Discrete Mathematics a t Rutger s University [2] , I first cam e to study discret e mathematics. Whe n I returne d t o m y classroo m a t Murr y Bergtrau m Hig h School i n lowe r Manhatta n tha t fall , I wa s excite d an d enthusiastic , eve n though I woul d onl y b e teachin g discret e topic s t o th e 10t h grad e remedia l classes. The y wer e ofte n referre d t o a s th e "classe s fro m hell" . I n fact , th e "classes fro m hell " gav e m e th e mos t rewardin g ter m I' d eve r experienced .

As th e ter m began , I define d som e goal s fo r myself . The y ar e basi c questions whic h helpe d m e t o clarif y wha t I hope d t o accomplish , an d the y were puttin g discret e mathematic s t o a test . Th e first questio n was : ca n students b e encourage d t o com e to clas s regularly? I n Ne w York City , wher e I teach , w e hea r storie s o f student s wh o wil l literall y clim b ou t th e window s to ge t ou t o f class . An d thi s attitud e wa s widesprea d amon g th e remedia l students. I fel t an d hope d tha t wit h th e materia l I wa s bringin g bac k t o them, student s woul d wan t t o com e t o clas s o n a regula r basis .

Next, I wondered , ca n student s b e encourage d t o believ e tha t there' s more t o mathematic s tha n arithmeti c computation ? A t tha t poin t I' d ha d nine year s o f hearin g student s proclaim : " I hat e math! " Bu t I kne w tha t this wa s goin g t o chang e becaus e a t Rutger s tha t summe r I ha d com e t o understand finally tha t whe n student s sai d the y hate d mathematic s the y were reall y sayin g the y hate d arithmetic . The y didn' t kno w anythin g abou t mathematics; durin g thei r entir e studen t career s mos t o f them ha d onl y see n the fou r operation s o f arithmetic , ove r an d ove r an d over . An d I kne w tha t discrete mathematic s wa s rea l mathematic s an d no t som e watered-dow n version o f mathematics .

My thir d questio n wa s mor e specific : Ca n th e student s b e encourage d to lik e a t leas t som e topic s whic h the y clearl y kne w t o b e mathematics ? I saw thi s a s a ke y t o changin g thei r mind s —havin g the m reconside r thei r attitude t o mathematic s a s a whole .

1991 Mathematics Subject Classification. Primar y 00A35 , 00A05 , 05C15 .

https://doi.org/10.1090/dimacs/036/04

36 SUSAN H . PICKE R

The las t questio n I pose d wa s base d o n thei r nee d t o b e abl e t o tak e some additional mathematics a s preparation fo r colleg e —a goal in my schoo l and i n Ne w Yor k City : Ca n student s b e encourage d t o stud y mathematic s further —i n thi s cas e t o stud y algebra ?

The result s wer e beyon d anythin g I coul d hav e imagined . I foun d i n time tha t I coul d answe r ye s t o al l thes e questions . I foun d mysel f enjoyin g being i n th e classroo m eac h day , an d s o di d th e students .

I starte d slowly . I didn' t hav e a lo t o f material s s o I bega n buyin g o r getting publisher s t o sen d m e everythin g tha t ha d th e wor d "discrete " i n the title . Eventuall y I bega n developin g m y ow n material s —suc h a s takin g the ma p o f Ne w Yor k Cit y o r th e ma p o f Pari s an d creatin g a proble m which wa s a variatio n o f th e famou s Konigsber g Bridg e problem . Bu t a t a certain poin t I needed eve n more materials, s o I found mysel f goin g to colleg e textbooks an d eve n graduate-leve l texts . Becaus e o f th e natur e o f discret e mathematics an d th e students ' increasin g interest , I foun d tha t i t didn' t matter tha t thes e wer e intende d a s college-leve l problems . Th e student s could understan d the m an d solv e them .

An exampl e o f th e typ e o f proble m student s cam e t o b e abl e t o solv e b y December o f tha t term , come s fro m [1] . Th e proble m (Figur e 1) , whic h i s sophisticated fo r hig h schoo l students , give s a matri x indexe d b y chemicals , many o f whic h ar e unsaf e t o transpor t together , an d set s u p a situatio n where a trai n ha s t o b e assemble d wit h on e chemica l pe r car . Becaus e o f the incompatibilit y o f som e chemical s wit h eac h other , thei r car s ca n onl y travel nex t t o eac h othe r i f tw o ope n gondola s o f san d separat e them . Th e challenge i s t o find th e minimu m numbe r o f gondola s o f san d tha t ca n b e used i n settin g u p thi s trai n wit h twelv e differen t chemicals .

As preparatio n fo r understandin g th e problem , w e talke d first i n clas s about grocer y product s travellin g togethe r t o th e supermarke t an d I aske d students what the y would not wan t travellin g together say , in the same truck . Students sai d suc h thing s a s mea t an d bleach , strawberrie s an d onions-the y saw tha t thes e woul d no t affec t eac h othe r well . Nex t w e talke d abou t th e chemicals. Student s commente d o n wha t the y kne w o f th e chemical s wit h which they wer e familiar, lik e acetone, which some of the students recognize d from nai l polis h remove r bottles . Student s notice d tha t i n th e proble m nitrogen travel s safel y togethe r wit h everythin g an d the y wer e curious abou t why tha t was . W e als o discusse d variou s chemica l spill s whic h ha d bee n i n the news ; train s whic h ha d deraile d carryin g chemical s an d ho w tha t ha d affected th e environment . I t becam e a n interdisciplinar y discussio n abou t a real-world situation .

In studyin g th e topi c o f grap h coloring , student s ha d alread y solve d many smalle r an d simple r problem s involvin g schedulin g conflicts bu t non e as larg e o r EL S complicated a s this . The y no w kne w tha t whe n ther e i s a possible schedulin g conflic t as , say , subcommittee s o f a legislatur e havin g members i n commo n an d therefor e no t abl e t o mee t a t th e sam e time , tha t the wa y t o begi n a n efficien t schedulin g i s t o creat e a conflic t grap h o f

GIVING REMEDIA L STUDENT S A SECOND CHANC E 37

vertices an d edges . Th e vertice s represen t thos e thing s whic h coul d b e i n conflict, lik e the subcommittees , o r i n this case , chemicals. A n edge is draw n between tw o vertice s i f an d onl y i f ther e i s a conflic t betwee n them . I n thi s case each "U " i n the matrix indicated t o students that thos e chemicals woul d

A manufacturer of chemicals ships its products by railroad tank cars. To reduce the danger that might occur through accidental spills of chemicals, the company specifies that the train must be made up in segments in such a way that

a: no two chemicals in the cars of each segment react dan- gerously with each other,

b : two open gondola carloads of sand must precede each seg- ment to separate dangerously reactive chemicals in case of a derailment or other emergency,

c: two open gondolas of sand must separate the last segment of chemical cars from the caboose.

Determine the smallest possible number of gondolas of sand needed to make up a train that carries one tank car of each of the following 12 chemicals. Show your work and reasoning.

Chemicals 1. Toluene 2. Acetone 3 . Phosph 4. Sulfuric

oric acid : acid

5. Potassium cyanide 6. Sodium hydroxide

1 2 3 4 5 6 7 8 9

10 11 12

1 - S S

u s s s u u s u u

s -

u u u u u u u s u u

3 S

u -

s u u u s s s s s

4 U

u s -

u u u s s s u s

Reaction Table:

s u u u -

s s u u s u u

7. 8. 9.

10. 1 1 . 12.

6 S

u u u s -

s u u s u s

7 S U

u u s s -

u u s u u

Dimethyl hydrazine Dinitrogen tetroxide Chromic anhydride Nitrogen Chlorine Potassium dichromate

8 U U

s s u u u -

u s u u

S (relatively saf U (unsaj e when

9 1 0 1 1 1 2

u s u u u s u u s s s s s s u s u s u u u s u s u s u u u s u u - s u s s s s s u s - u s s u -

e when mixed); mixed)

F I G U R E 1 . Chemica l transportatio n proble m [1 ]

38 SUSA N H . PICKE R

1 (red ) 12 ( w h j t e j ^ ^^ 2 (red )

11 (green) / / j t \ / \ [ ^^ /K \ \ \ \ ^ ( ^ u e )

10 (red) | ^ 4 / A / / M XJ\

9 (white) m*±J\ / i x ̂ ^ C x f /(r^ 4(blue)

8 (blue) ^ \ V V K ^ X \̂ / / 5 (P u rPl e)

7 (purple ) 6 (purple )

F I G U R E 2 . Th e conflic t graph , wit h a 5-colorin g

react unsafel y an d woul d therefore b e i n conflict wit h eac h other, s o student s rightly conclude d tha t chemical s with a "U " betwee n the m wer e to b e joined by a n edg e i n th e graph .

After creatin g th e grap h (Figur e 2) , student s colore d th e vertice s usin g the fewes t color s possible to determine the chromati c number . Student s wer e aware tha t sinc e th e edge s joined conflictin g things , thos e vertice s joined b y an edg e woul d hav e t o hav e differen t colors . Therefor e thos e vertice s whic h had th e sam e colo r wer e compatibl e an d coul d b e schedule d o r groupe d to - gether. Throug h discussion , student s sa w tha t i n thi s problem , th e colorin g would separat e th e chemical s int o group s whic h coul d trave l safel y together . The chromati c numbe r tol d the m ho w man y group s ther e woul d be . I t wa s these grouping s —th e student s determine d ther e wer e 5 o f the m —whic h then ha d t o b e separate d b y th e tw o ope n gondola s o f sand . I n th e grap h in Figur e 2 , whic h accompanie d model s student s dre w o f th e trai n (se e Fig - ure 3 , fo r example) , student s indicate d th e chromati c numbe r o f 5 with th e Greek characte r x- The y becam e enthusiasti c abou t usin g mathematica l symbols whic h the y hadn' t see n before .

Students a t thi s point i n the term ha d becom e used to working in groups, for problem s i n discret e mathematic s ofte n ar e solve d mor e successfull y b y a grou p tha n b y a studen t workin g alone . A numbe r o f student s go t s o involved tha t the y starte d comin g t o schoo l regularly , especiall y whe n w e began th e uni t o n grap h coloring . M y chairma n wa s pleased .

This chemicals-carrying-trai n proble m wa s th e culminatin g projec t in - volving grap h colorin g an d i t too k a wee k fo r th e student s t o d o it . Thoug h the proble m migh t hav e take n othe r student s a muc h shorte r tim e t o solve , the importan t thin g i s that thes e student s understoo d wha t th e proble m de - manded an d wha t th e constraint s wer e an d wer e abl e t o wor k methodicall y

GIVING R E M E D I A L S T U D E N T S A SECON D CHANC E 3 9

to arriv e a t a solution . Fo r nearl y al l o f the m i t wa s i n thi s clas s tha t the y did rea l problem-solvin g i n a mathematic s clas s fo r th e first time .

What the y wer e abl e t o accomplis h surpasse d wha t I ha d believe d stu - dents wh o wer e apparentl y s o low-leve l coul d produce . An d ye t thes e stu - dents weren' t reall y s o low-leve l anymore . Ther e wer e ne w concept s i n mathematics an d approache s t o problem-solvin g tha t the y wer e comin g t o understand an d utilize .

FIGURE 3 . "Th e Picke r Express " (wit h th e autho r a s en - gineer) wa s draw n b y a studen t i n th e clas s t o illustrat e th e solution t o th e chemica l trai n problem . Freigh t car s pile d high wit h san d ar e situate d betwee n saf e combination s o f chemicals. Eac h ca r i s labele d wit h it s cargo , an d onl y th e first te n car s ar e picture d here .

The mont h afte r thi s project , i n January , a s th e ter m wa s ending , I wa s invited t o guest teac h fo r a week in a couple of the calculu s classes. I planne d to d o a s muc h a s I coul d o f grap h theor y an d coloring . Thes e wer e classe s with ver y hig h registratio n —mayb e 3 4 student s each , an d I realize d tha t I would no t b e abl e t o b e everywher e a t once . I t occurre d t o m e tha t perhap s it woul d serv e a dua l purpos e t o brin g fou r differen t discret e mathematic s students wit h me each day to assis t an d answe r questions . Whe n I suggeste d this t o them , m y student s wer e terrified . "Ho w ca n w e g o int o classe s wit h kids wh o ar e takin g calculus ? We'r e jus t tent h graders. "

It wa s no t eas y fo r thes e student s t o understan d tha t the y kne w thing s of whic h th e twelft h grader s ha d n o ide a —tha t the y ha d neve r see n befor e

40 SUSAN H . PICKE R

and coul d no t do . T o see themselves a s the "experts " an d other s a s learnin g from the m wa s a completel y ne w wa y o f thinkin g fo r them , bu t i n th e en d they cam e wit h m e t o th e calculu s classes . I t turne d int o a grea t experienc e for them . A s the hand s bega n goin g up, th e tent h grader s starte d goin g over to th e calculu s student s an d answerin g thei r questions . M y student s wer e amazed tha t the y coul d d o such a thing. An d i t gav e them a confidenc e tha t they ha d neve r ha d before .

About hal f o f th e class , eleve n students , wen t fro m m y discret e mathe - matics clas s int o algebr a th e followin g term . Thes e wer e student s wh o wer e not i n th e "academi c mat h track " an d i t i s unlikel y tha t the y woul d hav e left hig h schoo l havin g studie d an y algebra , ha d the y no t bee n encourage d in thi s ter m t o se e tha t the y coul d succee d i n a mathematic s class . I fol - lowed u p wit h thes e student s th e bes t I could durin g th e ter m tha t followed . Meeting the m i n th e halls , I aske d the m ho w the y wer e doin g i n thei r first term o f algebra , an d th e response s wer e generall y favorable . I als o hear d about thei r progres s fro m othe r teacher s wh o no w ha d the m i n thei r classes . But i t wa s har d t o continu e t o kee p trac k o f th e student s an d continu e t o encourage the m becaus e I lef t th e schoo l a t th e en d o f tha t year , i n June .

The followin g fal l I wa s periodicall y i n th e schoo l an d I happene d t o se e a videotap e o f on e o f th e second-ter m algebr a classes . Ther e wer e fou r o f my forme r discret e mathematic s student s activel y participatin g an d clearl y doing well . I ra n int o on e o f th e student s an d tol d he r tha t I ha d see n th e tape. "You r clas s mad e al l th e difference, " sh e said .

At th e beginnin g o f th e ter m I alway s giv e student s a questionnair e about thei r attitudes . I alway s as k th e student s "D o yo u thin k you'l l eve r be a mathematician?" , becaus e I a m concerne d tha t thei r imag e o f mathe - maticians lack s a clea r understandin g o f what a mathematicia n i s an d does . Students alway s write , "NO! " —bi g letters ; exclamatio n points . " I don' t think so! "

At th e en d o f tha t ter m a fe w day s befor e ou r las t meetin g I aske d m y students i f they remembere d thei r answer s t o tha t questio n o n th e question - naire. An d student s sai d —"Yo u know...w e lik e this , bu t w e don' t thin k we're goin g t o b e mathematicians! " An d I sai d well , I hav e t o tel l you , I've learne d tha t yo u ar e mathematician s becaus e you'v e bee n doin g math - ematics. An d a mathematicia n i s a perso n wh o doe s mathematics . Wha t happened nex t too k m e completel y b y surprise : th e student s spontaneousl y burst int o applause . I wa s neve r move d i n a mathematic s clas s before . An d it move s m e stil l whe n I thin k o f i t —tha t thes e student s ha d com e t o hav e such a differen t sens e o f themselve s an d suc h a differen t attitud e i n class .

As I loo k bac k o n tha t ter m I ca n se e tha t I ha d changed , too . Discret e mathematics ha d give n me a wider view of mathematics a s a live and growin g subject wit h ne w area s fo r exploration . Perhap s becaus e o f thi s I wa s als o open t o se e ne w thing s i n m y student s includin g strength s I ha d previousl y overlooked. Man y wer e ver y visual , som e mor e visuall y adep t tha n th e calculus students , an d the y ha d a n easie r tim e wit h grap h theor y an d othe r

GIVING REMEDIA L STUDENT S A SECON D CHANC E 4 1

topics becaus e o f it . The y showe d a grea t creativit y i n thei r approache s to problems , an d i n presentin g solution s t o thos e problems , a s i n th e trai n problem. The y go t ver y involve d i n topic s whic h I hadn' t though t woul d interest them , lik e chromatic polynomials , whic h they wer e able to work wit h without th e algebrai c notation . I came t o respec t m y students mor e throug h this, an d t o believ e eve n mor e i n thei r abilit y t o learn . I t ha d neve r bee n clearer t o m e tha t a hug e obstacl e t o thes e students ' progres s an d succes s was thei r dislik e o f mathematics , an d thi s dislik e i s wha t I sa w lesse n an d change.

References

[1] Finkbeiner , D.T . II , an d Lindstrom , W.D. , A Primer of Discrete Mathematics, W . H . Freeman an d Company , 198 7 (proble m use d wit h permission) .

[2] Rosenstein , Josep h G. , an d Valeri e A . DeBellis , "Th e Leadershi p Progra m i n Discret e Mathematics", thi s volume .

O F F I C E O F T H E S U P E R I N T E N D E N T , MANHATTA N H I G H SCHOOLS , 12 2 A M S T E R D A M

AVENUE, N E W Y O R K , N Y 1002 3

E-mail address: s p i c k e r Q d i m a c s . r u t g e r s . e d u

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

W h a t We'v e Go t Her e i s a Failur e t o C o o p e r a t e

Reuben J . Settergre n

1. T h e S e t u p

At twelv e year s old , yo u fee l terribl y matur e an d independen t livin g i n a rea l colleg e dorm . A t th e moment , i n th e sunn y Lo s Angele s afternoon , at th e John s Hopkin s Cente r fo r Talente d Yout h (CTY) , yo u ar e stretchin g your min d an d body , tryin g to master th e strategy an d technique of Ultimat e Prisbee. Suddenly , fro m ou t o f nowhere , you r Residentia l Adviso r (RA ) grabs yo u an d stuff s a n envelop e int o you r hands , saying , "Tel l n o on e o f this." Ho w intriguing ; th e R A usuall y hand s ou t studen t mai l a t th e nightl y hall meetings . A t th e first opportunity , th e nex t brea k betwee n activities , you tea r ope n th e envelop e (marke d onl y wit h you r nam e an d CTY' s logo) , and rea d th e followin g letter: 1

Dear , / am sending this letter out via Special Delivery to fif-

teen of 'you' (namely, various friends of mine at CTY). I am proposing to all of you a game, the payoffs to be in rea l mone y (provided by me). It's very simple. Here is how it goes.

Each of you is to give me a single letter: 'C or 'D\ stand- ing for 'cooperate' or 'defect \ This will be used as your move in a game against eac h of the other players. Here is the payoff scheme for each of the two-player games. If both players coop- erate, each gets 5 cents. If both defect, each gets 1 cent. If one cooperates and one defects, though, the defector gets 9 cents, while the cooperator gets nothing.

Thus, if everyone sends in 'C\ everyone will get 1 4 x 5 = 70 cents, while if everyone sends in 'D\ everyone will get 1 4 x 1 = 1 4 cents. You can't lose! And of course, anyone who sends

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 . An earlie r versio n o f thi s articl e appeare d a s " A Classroo m Dilemma " i n [5] . 1 T h e tex t o f thi s lette r i s taken fro m [2] , and wa s slightl y modifie d t o fi t m y audienc e

and financia l resources .

https://doi.org/10.1090/dimacs/036/05

44 REUBEN J . SETTERGRE N

in 'D' will get at least as much as everyone else will. If, for example, 8 people send in 'C and 7 send in 'D', then the 8 C- ers will get 5 cents apiece from each of the other C-ers (making 35 cents), and zero from the D-ers. The D-ers, by contrast, will pick up 9 cents from each of the C-ers, making 12 cents, and 1 cent from each of the other D-ers, making 6 cents, for a grand total of 18 cents. No matter what the distribution is, D-ers always do better than C-ers. Of course, the more C-ers there are, the better everyon e will do!

By the way, I should make it clear that in making your choice, you should not aim to be the winner, but simply to get as much mone y for yourself as possible. Thus you should be happier to get 35 cents (say, as a result of saying 'C along with 1 others, even though the 1 D-sayers get more than you) than to get 14 cents (by saying ( D' along with everybody else, so nobody 'beats' you). Furthermore, you are not supposed to think at some subsequent time you will meet with and be able to share the goods with your co-participants. You are not aiming at maximizing the total amount of money I shell out, only at maximizing the amount that comes to you /

Of course, your hope is to be the uniqu e defector, thus really cleaning up: with 14 C-ers, you'll get $1.26, and they'll each get 13 times 5 cents, namely $0.65! But why am I doing the multiplication or any of this figuring for you? You're very bright. So are all of you! All about equally bright, I'd say, in fact. So all you need to do is tell me your choice.

It is to be understood (it almos t goes without saying, but not quite) that you are not to try to get in touch with and con- sult with others who you guess have been asked to participate. In fact, please consult with no one at all. The purpose is to see what people will do on their own, in isolation. Finally, I would very much appreciate a short statement to go along with your choice, telling me wh y you made this particular choice.

Sincerely, Hank D. Bank

"Hank D . Bank?" yo u think, "Gimm e a break! Ho w stupid doe s Reube n think I am? " Wha t a transparen t guis e fo r a classroo m exercise . Obviously , the righ t thin g t o do , th e nic e thin g t o do , i s t o cooperate . Bu t you'r e n o sheep; wh y no t tr y fo r th e bi g bucks ? Bu t wha t i f everybod y els e defect s too? Yo u mentally enumerate you r classmates , an d decid e that enoug h othe r players wil l cooperat e tha t yo u wil l ge t quit e a sizabl e payoff . Later , yo u chuckle t o yoursel f a s yo u embellis h th e big , dar k "D " o n you r paper , know - ing tha t man y o f you r classmate s wil l foolishl y coun t o n yo u t o cooperate , because o f you r generall y mee k demeanor . Ha , ha , h a

WHAT WE'V E GO T HER E I S A FAILURE T O COOPERAT E 4 5

2. Th e S e t t i n g

Every summer , th e John s Hopkin s Universit y Cente r fo r Talente d Yout h (CTY) offer s tw o three-wee k session s o f residentia l instructio n a t colleg e campuses u p an d dow n th e eas t coast , i n Lo s Angeles , an d eve n i n Europe . Their course s includ e no t onl y mathematica l an d physica l sciences , bu t als o a larg e arra y o f innovativ e humanitie s courses , fro m archaeolog y t o politica l science t o etymology . I n 1994 , CT Y offere d a ne w clas s calle d Application s of Contemporar y Mathematic s (ACOM) . ACO M wa s intende d fo r CTY' s youngest, leas t experience d students , thos e enterin g eight h grade—possibl y without eve n pre-algebr a unde r thei r belts .

I eagerl y accepte d CTY' s offe r t o teac h tw o session s o f ACO M i n Lo s Angeles. CT Y ha d chose n a textbook fo r ACOM , For All Practical Purposes [1], bu t als o allowe d latitud e i n developin g a syllabus . So , whil e waitin g for m y cop y o f th e text , I brainstorme d an d searche d fo r activities . I ra n across an articl e [2 ] I had rea d year s before b y Douglas Hofstadter (autho r o f Godel, Escher, Bach: an Eternal Golden Braid) i n his collection of Scientific American article s [4] . Hofstadte r describe d ho w h e devise d th e cooperatio n game above, an d discusse d the result s of playing it wit h twent y of his friends . The result s wer e s o compellin g tha t I decide d t o us e th e gam e t o teac h m y students abou t th e Prisoner' s Dilemma .

In thi s famou s an d fundamenta l gam e theor y situation , tw o partner s i n crime ar e detaine d b y th e polic e an d hel d incommunicado . Th e polic e tel l each suspec t "I f yo u giv e a ful l confession , an d you r confessio n lead s t o th e conviction o f you r partner , the n yo u ca n g o free. " Eac h partne r know s tha t the evidenc e agains t the m i s slim , an d silenc e b y bot h woul d mea n smal l sentences, bu t ca n eac h partne r trus t th e othe r no t t o confes s an d stic k him wit h a lon g sentence ? B y havin g m y student s pla y a gam e simulatin g the Prisoner' s Dilemma , I coul d forc e the m t o wrestl e wit h th e dilemm a themselves, no t jus t liste n t o m e tal k abou t it .

I timed this activity to coincide with classroom material on Game Theor y (Chapter 1 5 i n [1]) ; I distribute d th e letter s o n Tuesda y an d Wednesda y of th e secon d week , an d announce d th e results , gav e ou t money , an d hel d discussion o n Frida y afternoon . I distributed th e letter s individually , outsid e of th e classroom , an d a s mysteriousl y a s possible , i n orde r t o establis h th e atmosphere o f secrec y an d isolatio n necessar y fo r prope r pla y o f th e game . In th e secon d session , whe n ther e wer e tw o section s o f ACO M an d anothe r instructor wh o wante d he r clas s t o pla y a s well , I divide d th e large r numbe r of player s int o thre e games , wit h gam e group s cuttin g acros s clas s divisions .

3. Th e S e t t l e m e n t

On Friday, just befor e th e weekend, w e gathered t o reveal and discus s th e outcome o f th e game . No t surprisingly , i n al l case s ther e wer e dismall y fe w cooperators. Eac h o f th e secon d sessio n game s ha d tw o o r thre e cooperator s

46 REUBEN J . SETTERGRE N

each (ou t o f fifteen o r sixtee n players) , an d i n th e firs t session , ther e wa s one lon e cooperato r amon g fiftee n (wh o o f cours e receive d nothing) !

Student respons e wa s varied . No t onl y cooperators , bu t als o som e de - fectors wer e shocke d tha t thei r classmate s wer e s o greedy . Som e defector s expected tha t mos t o f thei r classmate s woul d defect , bu t lemming-like , de - fected themselve s — hoping t o squeez e th e mos t ou t o f th e fe w sucke r coop - erators, an d fearfu l o f being counte d a s one o f them. On e defector , wh o ha d obviously agonize d ove r hi s decision , responde d t o th e result s b y lamenting , "I fee l crummy!" , an d becam e m y mos t arden t advocat e fo r cooperatio n in th e ensuin g discussion . Anothe r student , mor e Machiavellian , brok e th e game's premis e o f secrec y t o for m a pac t o f cooperatio n wit h tw o hench - men — i n orde r t o sweete n hi s defection ! Durin g th e discussion , however , he repented , an d als o embrace d cooperation . O f course , thi s wa s jus t dur - ing th e discussion . I' m sur e tha t h e woul d b e amon g th e wil y student s tha t would b e quic k t o tak e advantag e o f th e lesson s othe r student s learned , i f the gam e wer e t o b e playe d a secon d time. 2

So far i n this article , I have tried onl y feebl y t o concea l that th e lesso n t o be learned i s that cooperatio n i s the "right " answe r to this game; tha t i n thi s type o f situation , th e player s mus t thin k no t individually , bu t collectively . Hofstadter explaine d i t best : i f al l th e player s ar e equall y an d perfectl y rational (an d thu s thei r move s ar e "right") , the y wil l al l mak e th e sam e move. Sinc e eac h an d al l d o bette r whe n th e collectiv e mov e i s "C " tha n when i t i s "D" , eac h playe r wil l cooperate .

Even i f m y student s didn' t se e th e gam e i n exactl y thes e terms , mos t o f them instinctivel y understoo d thi s logic . Ironically , thi s wa s the see d o f thei r downfall. Sinc e mos t kne w tha t everybod y "should " cooperate , fe w coul d resist th e temptatio n t o tr y t o rea p bi g profits b y relyin g o n th e cooperatio n of thei r classmates .

After initia l reaction s an d handin g ou t o f money , I passe d ou t copie s of Hofstadter' s article , an d w e wen t throug h it , discussin g Hofstadter' s rea - soning, an d comparin g th e response s o f th e origina l player s t o thos e o f th e class. Hofstadte r als o provide s a numbe r o f variation s o n th e game , wit h more extrem e payoffs , an d eve n penalties . Wit h mor e a t stake , student s have t o rethin k thei r moves , becaus e i t i s simultaneousl y mor e important , and ye t mor e dangerou s t o cooperate .

The simples t extrapolatio n i s t o conside r wha t th e student s woul d d o if they wer e offere d Hofstadter' s origina l game , whic h use d dollar s instea d o f cents? O r ho w abou t million s o f dollar s instea d o f dollars ? Thi s lead s t o a n excellent demonstratio n o f th e nonlinearit y o f value . Psychologically , whil e

2 This bring s u p a separat e issu e o f repeate d pla y o f th e Prisoner' s Dilemma , abou t which Hoftsadte r ha s writte n anothe r articl e [3 ] (whic h als o appears i n [4]) . I n a computer - run contest , th e consistentl y bes t strateg y (calle d T I T FO R T A T ) wa s t o cooperat e th e firs t time, an d thereafte r repea t you r opponent' s las t move . Incidentally , th e administrato r of th e compute r tournamen t unhesitatingl y playe d ' D ' himsel f i n Hofstadter' s one-tim e

game.

WHAT WE'V E GO T HER E I S A FAILURE T O COOPERAT E 4 7

three dollar s migh t b e thre e time s a s valuabl e a s one , thre e millio n dollar s are no t thre e time s a s valuabl e a s on e million . Anothe r wa y t o sa y thi s is , a playe r woul d b e mor e willin g t o hav e thei r payof f cu t fro m thre e millio n dollars t o on e millio n tha n t o hav e i t cu t fro m thre e dollar s t o one . Thus , it i s likel y tha t mor e player s woul d cooperat e i n a millio n dolla r gam e tha n in a penn y game , wher e th e mos t significan t par t o f th e payof f i s i n statu s and eg o reinforcement .

There ar e man y nearl y equivalen t socia l situation s tha t student s ca n probably discove r themselves , i f sparke d wit h a n exampl e o r two .

• Durin g th e onse t o f rus h hou r traffic , fast-movin g driver s notic e a n accident a t th e sid e of the road. " I don't hav e to slo w down very muc h to ge t a goo d loo k a t th e carnage, " the y eac h think . Prett y soon , frustrated driver s i n bumper-to-bumpe r traffi c ar e thinkin g instead , "Why doe s everybody hav e to sto p an d stare ? . . . Bu t sinc e I've bee n waiting s o long , whe n I ge t u p t o th e accident , I migh t a s wel l tak e a look, sinc e traffi c i s movin g s o slowl y anyway" .

• A local museum i s exhibiting a painting b y Van Gogh , an d yo u are en- thralled b y th e textur e o f th e paint ; s o chunky , s o three-dimensional , so—"Hey, I wonde r wha t i t feel s like ? I t wil l b e al l righ t i f I' m th e only on e tha t touche s it. "

• It' s a beautifu l day , an d you'r e ou t fo r a driv e wit h you r family , eve n though ga s price s ar e sky-high , an d th e ai r coul d b e a lo t cleaner . You sur e d o loo k goo d drivin g tha t powerful , eight-cylinde r Lincol n Continental, though .

• It' s a Tuesda y i n November , an d you'r e curle d u p fo r th e evenin g i n your La-Z-Bo y waitin g throug h commercial s fo r you r favorit e sitcom , and yo u se e tha t th e loca l new s i s pumpin g thei r electio n coverage , which wil l begi n i n tw o hours , whe n th e poll s close . "M y on e vot e wouldn't matter, " yo u think , a s yo u pu t u p th e footres t an d di p int o a ba g o f chees e snacks .

There ar e many othe r example s which your student s ca n devise . Yo u ca n hope tha t afte r the y hav e played thi s gam e an d discusse d real-worl d applica - tions, the y wil l be abl e to recognize situations wher e this kin d o f cooperatio n can mak e a difference , an d decid e mor e rationall y ho w t o respond .

R e f e r e n c e s

[1] COMAP , For All Practical Purposes: Introduction to Contemporary Mathematics, 3r d ed., W . H . Freeman , Ne w York , 1994 .

[2] Hofstadter , Douglas , "Dilemma s fo r Superrationa l Thinkers" , an d "Th e Tal e o f Hap - piton", Scientific American, Jun e 1983 .

[3] Hofstadter , Douglas , "Th e Prisoner' s Dilemm a an d th e Evolutio n o f Cooperation, " Scientific American, Ma y 1983 .

[4] Hofstadter , Douglas , Metamagical Themas, Basi c Books , Ne w York , 1985 , ch . 29 , 30 , 32.

[5] Settergren , Reuben , " A Classroo m Dilemma, " In Discrete Mathematics: Using Dis- crete Mathematics in the Classroom, # 6 , Spring/Summe r 1995 , p . 2 .

48 REUBE N J . SETTERGRE N

R U T G E R S C E N T E R FO R O P E R A T I O N S R E S E A R C H , P . O . B o x 5062 , N E W B R U N S W I C K ,

N J 0890 3 E-mail address: r e u b e n Q r u t c o r . r u t g e r s . e d u

Section 2

T h e Valu e o f Discret e M a t h e m a t i c s : Achieving Broade r Goal s

Implementing th e Standards : Let' s Focu s o n th e Firs t Fou r N A N C Y C A S E Y AN D M I C H A E L R . F E L L O W S

Page 5 1

Discrete Mathematics : A Vehicl e fo r Proble m Solvin g an d Excitemen t M A R G A R E T B . COZZEN S

Page 6 7

Logic an d Discret e Mathematic s i n th e School s SUSANNA S . E P P

Page 7 5

Writing Discret e (ly) R O C H E L L E L E I B O W I T Z

Page 8 5

Discrete Mathematic s an d Publi c Perception s o f Mathematic s J O S E P H M A L K E V I T C H

Page 8 9

Mathematical Modelin g an d Discret e Mathematic s H E N R Y O . P O L L A K

Page 9 9

The Role o f Application s i n Teachin g Discret e Mathematic s F R E D S . R O B E R T S

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Implementing t h e S t a n d a r d s : Let' s Focu s o n t h e First Fou r

Nancy Case y an d Michae l R . Fellow s

1. Introductio n

The Curriculum and Evaluation Standards for School Mathematics o f the Nationa l Counci l o f Teacher s o f Mathematic s [8 ] ca n b e viewe d a s a n attempt t o shif t attentio n i n th e mathematic s curriculu m t o high-leve l cog - nitive issues , an d awa y fro m th e traditiona l focu s o n th e accumulatio n o f low-level rot e computationa l skill s (task s tha t increasingl y ubiquitou s ma - chines d o quit e well) . A t al l level s o f mathematic s education , an d i n man y different way s throughou t moder n culture , w e see thi s sam e genera l shif t t o higher cognitiv e issue s an d skills . A consensus ha s rightl y emerge d tha t on e of the principa l goal s of mathematics educatio n i s mathematical literac y an d confidence i n mathematica l mode s o f thinking. Th e purpos e o f thi s pape r i s primarily t o discus s th e rol e o f mathematica l conten t i n achievin g thi s goal .

At ever y grad e level , th e followin g fou r standard s appea r a t o r nea r th e head o f th e list :

Standard 1 : Mathematic s a s Proble m Solving . Standard 2 : Mathematic s a s Communication . Standard 3 : Mathematic s a s Reasoning . Standard 4 : Mathematica l Connections .

We wil l cal l thes e the First Four. N o doub t the y appea r a t eac h grad e level becaus e the y addres s directl y wha t i t mean s t o do mathematics. Th e items tha t follo w th e Firs t Fou r i n th e variou s Standard s list s b y grad e leve l describe, fo r th e mos t part , ne w approache s t o ol d conten t wit h a minima l amount o f ne w content . W e argu e tha t mor e need s t o b e don e i n term s o f content — particularl y i n grade s K-4 .

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 , 05C15 . Research supporte d b y th e U.S . Dept . o f Energy , Lo s Alamo s Nationa l Laboratory ,

MegaMath Project . Research supporte d b y th e Nationa l Scienc e an d Engineerin g Researc h Counci l o f

Canada, an d b y th e MegaMat h Projec t o f th e Lo s Alamo s U.S . Nationa l Laboratories .

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52 N. CASE Y AN D M . R . FELLOW S

For example , wha t abou t th e followin g conten t possibilitie s (an d w e will argue , necessities ) i n th e earl y grades : proof , infinity , variable , logic , induction, recursio n an d computationa l complexity ?

And wha t abou t th e followin g mathematica l experiences ?

• Th e experienc e o f a surprisin g mathematica l trut h tha t contradict s intuition.

• Th e experienc e o f understanding a simply-state d mathematica l prob - lem wit h n o know n solution .

• Th e experienc e o f logica l paradox . • Th e experienc e o f wrestlin g wit h th e ide a o f a limit . • Th e experienc e o f mathematica l exploration .

In th e followin g w e wil l describ e way s i n whic h thes e an d othe r math - ematical experience s an d concept s tha t ar e typicall y considere d advanced , can engag e childre n i n grade s K- 4 (age s 5-9) , an d wh y the y shoul d b e in - troduced t o thi s ag e group . Man y o f th e topic s b y whic h thes e idea s an d experiences ar e conveye d ar e relativel y ne w a s mathematic s — man y ar e part o f compute r scienc e an d it s discret e mathematica l roots .

The mai n point s o f ou r argumen t ar e summarize d a s follows :

• Th e Firs t Fou r curriculu m standard s canno t b e meaningfull y imple - mented excep t i n th e contex t o f a significantl y enriche d mathematic s content agenda . The y ar e no t independen t o f conten t issues .

• Ther e i s natura l compatibilit y betwee n th e Firs t Fou r curriculu m standards an d th e goal s an d method s o f effectiv e mathematic s popu - larization.

• Literatur e an d literac y provid e usefu l metaphor s fo r understandin g many o f th e importan t issue s i n mathematic s education .

• Discret e mathematic s an d compute r scienc e hav e a n importan t rol e to pla y a s source s o f conten t enrichmen t fo r th e elementar y grades .

We hypothesiz e tha t al l o f th e problem s wit h mathematic s educatio n a t all level s ar e abundantl y represente d i n th e first five year s o f school , an d for tha t reason , dra w ou r comment s fro m ou r experience s wit h childre n i n classrooms a t thes e grad e levels .

By th e en d o f eve n th e first yea r many , i f no t most , o f th e childre n w e have me t hav e alread y forme d a disma l impressio n o f mathematics , consid - ering it a boring an d intimidatin g disciplin e devoted primaril y t o speed y an d accurate manipulation s o f numbers. B y th e en d o f the fourt h yea r the y hav e typically ha d a n abundanc e o f th e traditiona l experience s o f schoo l math - ematics: th e meaningles s sea t work , th e rot e memorizatio n o f procedures , the stilte d wor d problem s an d pointles s obscur e vocabulary , th e anxiet y o f parents an d teachers , an d th e testin g tha t separate s th e winner s fro m th e losers. The y hav e alread y experience d "mathematic s a s crow d control" 1

where th e rewar d fo r masterin g a dril l shee t i s — anothe r dril l sheet .

See the paranoid theory of mathematics education i n [5] .

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One o f th e ironie s i n thi s ag e grou p i s tha t thei r playgroun d cultur e i s rich wit h combinatoria l games , wit h riddle s an d word-play , wit h informa l discussions o f infinity , space-time , an d th e Lia r Paradox . The y ar e bus y with topologica l an d dynami c amusement s suc h a s tethe r ball , jum p rope , cat's cradl e an d braids . Thes e activitie s an d puzzlement s ar e i n man y way s closer to the spirit o f mathematics a s it know n b y mathematicians tha n wha t is presente d a s "mathematics " i n th e classroom .

2. T h e Firs t Four : W h a t d o t h e y reall y mean ?

Mathematicians understan d tha t makin g connections , communication , problem-solving an d reasonin g ar e a t th e hear t o f thei r discipline . A s mea - surable skills , however , thes e ar e nebulou s — fa r mor e difficul t t o teac h an d track tha n th e abilit y t o count , compare , o r compute . On e o f th e thing s that i s commonl y happenin g i n practic e a s schoo l district s an d curriculu m developers wrestl e wit h th e Standard s i s tha t th e Firs t Fou r ar e i n man y cases bein g spli t of f an d treate d differentl y fro m th e rest . I n particular , the y are i n man y case s bein g interprete d merel y a s proces s standard s havin g n o particular connectio n t o an y kin d o f mathematica l content .

One well-meaning principa l o f an elementar y schoo l in British Columbia , which ha s bee n use d a s a mode l fo r curriculu m reform , pu t i t thi s way . "These fou r standard s ar e reall y importan t — w e handl e the m elsewher e in th e curriculum! " B y thi s wa s mean t tha t communicatio n skill s ar e prac - ticed i n creativ e writing , problem-solvin g skill s ar e practice d i n designin g art projects , etc .

Our centra l argumen t her e i s tha t th e Firs t Fou r canno t b e realize d without a n expande d agend a o f interestin g mathematic s an d mathematica l experiences t o reason , communicat e an d problem-solv e about . W e simpl y cannot realiz e thes e standard s b y mean s o f classroo m discussion s abou t ou r ideas for doin g long division or naming triangles. I f the current impoverishe d K-4 mathematic s agend a i s no t capabl e o f supportin g an y meaningfu l real - ization o f th e Firs t Four , w e mus t loo k t o al l o f mathematic s fo r expandin g the rang e o f idea s tha t ar e brough t t o th e K- 4 classroom .

We believe th e K- 4 conten t curriculu m shoul d includ e anythin g an d ev - erything suitabl e fo r a Mathematica l Science s Museu m an d thu s th e projec t of realizin g th e Firs t Fou r fo r K- 4 i s naturall y allie d wit h th e vita l projec t of mathematic s popularizatio n fo r al l ages . I n thes e first years , a n endurin g sense shoul d b e forme d o f wha t mathematica l scienc e i s abou t an d ho w i t feels t o participat e i n thi s adventur e o f th e huma n spirit , centra l a s i t i s t o all o f moder n scienc e an d technology. 2

In fact , scienc e popularizatio n i s inherentl y concerne d wit h th e K- 4 au - dience becaus e scienc e museu m exhibit s are , mor e o r less , designe d fo r th e

2 Not ice t h a t i f i n thi s sentenc e th e word s "mathematica l science " ar e replace d b y "print literacy, " the n th e resul t i s a common-place . No t onl y d o childre n routinel y maste r the decodin g o f prin t i n K-4 , bu t the y engag e excitin g poetr y an d storie s (includin g thei r own) an d for m a basi c sens e o f wh y on e woul d wan t t o rea d an d write .

54 N. CASE Y AN D M . R . FELLOW S

4th grad e audienc e i n order t o b e just abou t righ t fo r children , grandparent s and everyon e els e i n between .

For scientist s an d mathematicians , th e K- 4 audienc e i s a delight . The y are ful l o f vibrant curiosit y an d enthusiasm . The y ar e endowe d wit h natura l tendencies t o abstrac t representatio n an d th e pla y o f ideas . (Thin k o f th e odd bit s o f woo d the y hav e aske d yo u t o regar d a s a spac e laser. ) I n th e next section , w e describ e som e o f th e "advanced " mathematica l idea s tha t can b e engage d b y thi s ag e group .

3. Som e Conten t Example s fo r t h e Firs t Fou r

The purpos e o f thi s sectio n i s t o describ e som e example s o f ho w ad - vanced mathematica l idea s ca n b e engage d b y childre n i n grade s K-4 . On e of th e mos t fundamenta l appreciation s tha t on e ca n hav e o f mathematic s i s a sens e o f th e power , th e shee r variet y an d th e marvelou s interconnection s of mathematica l model s o f thing s i n th e world . W e ma y begi n wit h th e K-4 audienc e b y exhibitin g an d engagin g a ric h collectio n o f example s o f mathematical models .

3 . 1 . E x a m p l e 1 : M a p an d Grap h Coloring . Th e basi c M a p Col - oring P r o b l e m i s that o f trying t o discove r th e minimu m numbe r o f color s needed t o properl y colo r a map . A ma p i s properl y colore d i f n o tw o coun - tries sharin g a borde r hav e th e sam e color . (Se e Figur e 1. )

FIGURE 1 . A ma p whic h wil l requir e 3 color s t o b e colore d correctly .

This proble m (lik e an y o f hundred s o f suc h combinatoria l optimizatio n problems) ca n b e presente d i n a classroo m settin g b y doin g th e following. 3

1. Beforehand , mak e u p an d photocop y 3 o r 4 map s o f varyin g sizes , such a s wit h 5 , 1 0 and 2 0 regions . (D o no t "solve " them. )

2. I n class , discus s ho w map s ar e ordinaril y colored , ho w region s o n the ma p tha t shar e a boundar y ar e colore d differen t color s s o tha t they ar e no t easil y confused . Discus s also , ho w i t woul d mak e sens e commercially t o colo r map s wit h a s fe w color s a s possible , du e t o th e cost o f ink , th e complexit y o f printin g man y colors , etc .

3Detailed instruction s fo r classroo m us e o f thi s an d othe r problem s alon g wit h sampl e handouts, idea s fo r discussion , an d explanation s o f thei r relationshi p t o th e whol e o f mathematics ca n b e foun d i n [3] .

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3. Pas s ou t th e map s an d invit e students t o find ways to colo r them wit h as fe w color s a s possible , workin g individuall y o r i n groups , a s the y prefer.

4. B e a n attentiv e listene r an d facilitator . Encourag e th e childre n t o describe thei r idea s fo r solvin g this proble m an d t o explai n wha t the y are doin g t o eac h other .

5. Afterwards , hav e th e childre n writ e abou t thei r ideas , dra w map s o f their ow n t o color , and/o r shar e th e activit y wit h a differen t grou p o f children.

The colorin g proble m i s on e o f th e grea t gem s o f discret e mathemat - ical modeling . Som e o f it s application s include : th e assignmen t o f non - interfering frequencie s t o radi o stations , th e timin g o f traffi c lights , th e scheduling o f meeting s an d machines , an d th e schedulin g o f garbag e truc k routes. Colorin g i s also a n activit y t o whic h th e K- 4 audienc e i s alread y na - tively inclined . Ther e i s some satisfactio n i n connectin g thi s ordinar y child - hood artisti c activit y t o th e dee p an d importan t mathematic s tha t concern s it. Surprisin g t o mos t peopl e i s th e fac t tha t colorin g problem s (o f vari - ous kinds ) remai n a subjec t o f vigorou s mathematica l investigation . The y are importan t t o al l kind s o f discret e mathematica l modeling , including , fo r example, th e analysi s o f DN A sequences .

3.2. Coloring : W h a t ' s I n It ? I n K- 4 classrooms , wher e childre n ar e puzzling ove r finding th e minimu m numbe r o f color s fo r variou s maps , al l kinds o f interestin g an d dee p mathematica l issue s naturall y arise . W e ar e concerned tha t i f thes e idea s ar e lef t of f o f th e conten t agenda , teacher s wil l lack a n adequat e referenc e framewor k t o appreciate , stimulat e an d suppor t the problem-solvin g strategie s tha t th e childre n wil l invent . Th e followin g i s an unsystemati c inventor y o f various fragment s o f our classroo m experience s with th e colorin g problem, pointin g to various "advanced " mathematic s con - tent tha t emerged .

3.2.1. "Two is not enough!" Wha t typicall y happen s wit h a hypothet - ical ma p M , lik e th e on e above , wit h chromati c numbe r 3 (tha t is , wher e 3 color s ar e required ) i s tha t someon e first color s i t wit h 7 colors, an d the n someone color s M wit h 5 colors , .. . th e numbe r graduall y improves . Bu t eventually w e ar e lef t wonderin g (publicly , a s w e celebrat e thi s progress ) whether w e ca n d o i t wit h 2 colors . Inevitably , som e chil d wil l figure ou t that (an d explai n energeticall y why ) tw o i s no t enoug h fo r M , typicall y by finding thre e region s eac h o f whic h border s th e othe r two . Th e mo - ment whe n a chil d give s tha t excite d shou t need s t o b e appreciate d a s a "teachable moment " fo r th e fundamenta l topi c o f m a t h e m a t i c a l proof . A teache r no t equippe d wit h th e ide a o f th e importanc e o f mathematica l proof, an d expectin g t o encounte r an d develo p thi s concept , i s not equippe d to full y appreciat e an d empowe r th e problem-solvin g goin g on .

3.2.2. "I did it with two!" Conside r th e sam e scenari o wit h a differen t map M' havin g chromati c numbe r 2 (Figur e 2.) . W e hav e th e sam e gradua l

56 N. CASE Y AN D M . R . FELLOW S

improvements. Firs t ther e i s a solutio n wit h 5 colors , the n on e wit h 4 o r 3 colors. Finall y someon e shout s tha t the y hav e don e i t wit h 2 colors.

F I G U R E 2 . Map s draw n a s close d curve s ca n alway s b e col - ored wit h tw o colors .

If w e intervie w th e childre n wh o hav e foun d a 2-colorin g fo r M\ askin g about thei r metho d an d thei r ideas , w e usually fin d tha t the y hav e hi t upo n the followin g systemati c approach . The y firs t colo r a region , with , say , red . And the n choos e anothe r color , say , blu e fo r thos e neighborin g region s tha t are force d t o b e blu e becaus e the y shar e a borde r wit h th e firs t region . An d then (conservatively ) the y procee d b y coloring red thos e furthe r region s tha t share a borde r wit h th e newly-colore d blu e ones , an d thu s ar e force d t o b e red, an d s o on .

This i s a ver y interestin g strategy ! I n fact , i t i s a n algorith m tha t ca n serve to determin e fo r an y ma p whethe r tw o colors is enough, an d d o so with great algorithmi c efficiency . I t ca n b e compare d t o a differen t (bu t als o interesting an d respectable ) greed y algorith m tha t man y othe r student s will discover : pic k u p a crayo n an d (a t random ) us e i t t o colo r region s until i t can' t b e use d anymore , the n pic k u p a ne w crayo n an d repea t th e process unti l al l region s ar e colored . A teache r (an d a curriculu m agenda ) not equippe d wit h th e ide a o f a n algorith m i s no t equippe d t o appreciat e the problem-solvin g goin g o n here , th e idea s tha t ar e emerging , an d thei r substantial ultimat e significanc e i n mathematic s education .

Here i s als o a n opportunit y t o poin t ou t t o th e childre n on e o f th e mos t important unsolve d problem s i n all of mathematics an d compute r science . This i s tha t whil e ther e i s a simpl e an d efficien t algorith m t o determin e whether tw o color s i s enough (sketche d above) , n o on e know s whethe r ther e is a fas t wa y t o fin d ou t whethe r 3 colors ar e enough . I t seem s a goo d thin g not onl y t o shar e wit h childre n significan t problem-solvin g situation s tha t have n o singl e righ t answer , bu t als o situation s wher e n o one , no t eve n th e adults, presentl y know s th e answer .

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3.2.3. "I want to try to do it with 3!" Whe n th e childre n ar e workin g with crayons , th e natura l thin g t o d o a t first i s to div e in and colo r a s well a s you can . However , onc e students wan t t o experimen t s o as to trul y minimiz e their colorings , a certai n weaknes s i n usin g crayon s become s apparen t — i t is impossibl e t o bac k up ! Onc e a regio n ha s bee n colore d red , it' s mess y (i f not impossible ) t o tr y t o chang e i t t o green . O n thei r own , o r wit h minima l encouragement, som e childre n wil l switc h t o usin g colore d token s t o mar k the color s tha t the y assig n t o th e regions . Thi s provide s a fa r mor e powerfu l means t o tr y t o achiev e a n optima l coloring .

In on e classroom , a teache r observin g th e childre n movin g th e colore d markers aroun d o n th e map s remarked , "That' s a highe r leve l o f abstrac - tion." Th e teache r obviousl y (an d rightly ) fel t th e nee d t o appreciat e thi s more powerfu l problem-solvin g approac h i n som e way . Rathe r tha n rel y on psychologica l concept s fo r this , w e ca n appreciat e what' s goin g o n i n a straightforward mathematica l way : th e region s ar e no w functioning (manip - ulatively) a s variables tha t ca n b e conveniently instantiate d t o a color valu e by a marker . Thi s i s precisel y wh y thi s i s suc h a powerfu l problem-solvin g strategy, an d a goo d demonstratio n o f wh y th e concep t o f a variabl e i s s o fundamental i n mathematics . I f variable i s not o n the conten t agenda , the n teachers ar e lef t t o a d ho c psychologica l appreciations , wit h n o soun d con - nection to the enduring an d importan t mathematica l idea s that ar e emergin g in th e children' s activity .

3.2.4. "These maps can always be done with 2!" I f yo u plac e you r pe n on a piec e o f pape r an d dra w an y sor t o f intersectin g continuou s curve , eventually returnin g you r pe n t o it s startin g poin t withou t liftin g i t fro m the paper , yo u wil l hav e draw n a ma p tha t i s 2-colorable ! (Se e Figur e 2. ) Try i t out . Colored , i t look s like the kin d o f "psychedeli c checkerboard " tha t Salvador Dal i migh t hav e preferred . I t i s generall y regarde d a s somewha t surprising tha t thes e kind s o f map s ar e alway s 2-colorable . Ho w ca n w e b e convinced tha t thi s i s true ?

One wa y to explor e bein g convince d i s to mak e a loo p o f string. Imagin e that i t i s black string , imitatin g th e blac k in k o f a pe n tha t woul d dra w suc h a map . Surel y yo u wil l agre e tha t yo u coul d la y th e loo p o f strin g righ t o n top o f th e curv e tha t yo u drew . I f i t wer e jus t th e string , lyin g lik e tha t on th e whit e paper , w e migh t think , "Wha t a mess! " W e migh t decid e t o gradually, ver y slowly , on e ste p a t a time , mov e th e loop s apart . W e migh t in thi s wa y obtai n a ver y borin g situation : th e strin g i s no w just lyin g i n a loop tha t doe s no t intersec t itself . I f thi s wer e a map , i t woul d jus t b e on e island an d th e se a surroundin g i t — of cours e w e have n o troubl e 2-colorin g this!

Now let' s slowl y g o backwards , graduall y puttin g thing s bac k th e wa y they were . A t eac h ste p o f the wa y w e will notic e tha t w e make on e o f a fe w kinds o f moves , an d i n eac h case , th e propert y o f th e ma p bein g 2-colorabl e is preserved! (Se e Figure 3. ) No w we see why al l these map s ar e 2-colorable : they ar e al l just mixed-u p form s o f the singl e Two-Colorabl e Islan d (an d th e

58 N. CASE Y AN D M. R. F E L L O W S

* /N e 7 • \ © 0 N

F I G U R E 3 . I f th e ma p contaiin g th e lef t diagra m i s 2 - colorable, the n th e map containing th e right diagram , wher e string 1 is pulled ove r strin g 2 , is also 2-colorable .

mixing u p doesn' t hur t anything) . Ther e i s a lo t o f fun an d contemplatio n in thi s fo r youn g children . I t i s really th e essence o f a proo f b y inductio n of thi s surprisin g theorem . (N o matter ho w old you are, you reall y shoul d try thi s ou t with a piece of string an d two kinds o f colored markers , an d see that inductio n is , after all , really quit e suitabl e fo r 7 year-olds.)

F I G U R E 4 . Thi s grap h i s colored correctl y wit h 3 colors.

3.2.5. "This one can be done with 31 See if you can find how!" A prob- lem closel y relate d t o map coloring i s graph coloring . A graph i s a networ k of dot s (calle d vertices ) connecte d b y lines (calle d edges) . Th e vertices o f a graph ar e properly colore d whe n n o two vertices joine d b y a n edg e receive the sam e color . (Se e Figure 4. )

Several childre n i n on e o f ou r second-grad e encounter s spontaneousl y decided tha t th e 3-colorin g puzzle s fo r graph s tha t wer e passe d ou t wer e so muc h fun , the y woul d hav e Grap h 3-Colorin g a s a n activit y a t thei r birthday parties !

Here i s a littl e myster y fo r furthe r exploration . Ho w was it possibl e t o announce t o them , "I t i s eas y t o dra w a grap h tha t ca n be colore d wit h 3 colors so that yo u know exactl y ho w to color it , but other peopl e wil l have a hard tim e figuring ou t how. " Th e puzzle i s solved wit h a ver y entertainin g

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activity: begi n b y makin g a polka-do t patter n o f 3 kind s o f colore d dots . On to p o f thi s la y a fres h piec e of paper , an d tracin g through , mak e a circl e around eac h dot . No w ad d edge s betwee n thes e circles , bu t onl y betwee n circles that surroun d differentl y colore d dots ! I n thi s wa y you hav e created a graph fo r whic h you know a secret 3-coloring , bu t i t migh t b e pretty har d fo r someone else to find one. Thi s is a kind of combinatorial one-wa y function , a topi c o f profoun d importanc e i n moder n mathematica l cryptograph y (fo r further exploration s beginnin g fro m thi s poin t an d involvin g polynomial s as encryption s o f public-ke y message s se e [6]) .

4. Othe r topic s o f interes t fo r t h e earl y grade s

We nex t describ e ( a bi t mor e telegraphically ) a fe w mor e topic s tha t have prove d fruitfu l i n explorin g th e first fou r standard s i n th e earl y grades .

4 . 1 . M i n i m u m Weigh t Spannin g Trees . W e hav e com e t o cal l thi s the "Mudd y Cit y Problem" . (Se e Figur e 5. ) Th e scenari o i s a cit y wit h unpaved road s i n whic h transportatio n become s impossibl e whe n i t rains . The vertice s o f th e grap h represen t house s an d th e edge s ar e roads . Th e labels o n th e edge s o f th e grap h ar e th e cost s o f pavin g eac h segmen t o f road. Th e questio n becomes : Wha t i s the leas t expensiv e wa y t o pav e road s so tha t everyon e ca n ge t t o everyon e else' s hous e whe n i t rain s (eve n i f i t i s by a circuitou s route) ?

F I G U R E 5 . A ma p tha t ca n b e use d fo r th e Mudd y Cit y Problem .

In th e attempt s t o find a n optima l solutio n fo r a give n weighte d graph , a typica l classroo m experienc e invoke s a hurrican e o f arithmeti c a s childre n work t o creat e eve r bette r solutions , an d t o matc h th e bes t tha t hav e bee n found s o far . Th e ar e man y interestin g nontrivia l idea s an d observation s that childre n wil l typicall y mak e an d b e prepare d t o explai n an d argue : such a s th e fac t tha t a n optima l solutio n ha s n o cycles . Her e agai n w e hav e

60 N. CASE Y AN D M . R . FELLOW S

the vita l conten t o f mathematica l proo f arising . Th e fac t tha t ther e i s a (surprisin g an d elegant ) fas t algorith m fo r th e minimu m weigh t spannin g tree proble m raise s the issue s of algorithm an d o f algorithmic efficiency .

Someone will notice that al l the bes t solution s for a given graph (optima l solutions ar e generall y no t unique ) involv e th e sam e numbe r o f edges . W e have her e agai n a n opportunit y fo r a simple , visuall y presente d argumen t by inductio n t o explai n this .

4.2. Kno t Theory . Th e Canadia n Nav y donate d t o u s a numbe r o f large ropes , an d w e hav e ha d wonderfu l experience s presentin g som e o f th e rudiments o f kno t theor y i n elementar y classrooms . Thi s i s a n excellen t topic fo r mathematic s popularizatio n fo r severa l reasons . Firs t o f all , i t i s first-rate mathematic s tha t ha s recentl y move d center-stag e i n th e researc h world i n a ver y excitin g way . Secondly , everyon e use s knots , an d almos t n o one is aware that the y ar e a n objec t o f mathematical investigation . T o shar e the fac t tha t ther e i s a mathematics o f knots i s a powerfu l illustratio n o f th e richness o f mathematica l science . Finally , kno t theor y i s enormousl y ope n to manipulativ e presentation .

F I G U R E 6 . A left-hande d trefoi l knot .

One ca n as k abou t mirror-imag e knots . I s th e left-han d trefoi l th e sam e as th e righ t han d one ? O n som e occasion s w e hav e brough t alon g a larg e portable mirro r t o sho w that th e on e is indeed th e mirro r imag e of the other . Knots (onc e oriented) suppor t a well-defined notio n o f (abelian ) "multiplica - tion" (havin g eve n a prim e factorizatio n theorem ) tha t i s ope n t o engagin g manipulative exploration . Her e w e hav e s y m m e t r y an d m a t h e m a t i c a l o p e r a t i o n s .

4 . 3 . Othe r Examples . Ther e ar e man y mor e suc h mathematica l top - ics supportin g ric h opportunitie s t o realiz e th e Firs t Four . Th e mai n poin t is that i n reall y engagin g i n thes e o r an y othe r opportunitie s fo r mathemat - ical problem-solvin g an d communicatio n worth y o f th e name , "advanced " mathematical idea s wil l naturall y an d inevitabl y aris e an d shoul d b e bot h expected an d deepene d a s muc h a s possible .

IMPLEMENTING TH E STANDARDS : LET'S FOCU S O N TH E FIRS T FOU R 6 1

5. Parallel s B e t w e e n M a t h e m a t i c s an d Literatur e Teachin g

We hav e foun d th e analogie s betwee n prin t literac y an d mathematica l literacy t o b e bot h stron g an d productiv e fo r generatin g idea s an d method s for improvin g educatio n i n mathematic s durin g th e critical , formativ e ele - mentary schoo l years . Lackin g a scapegoa t an d deterren t t o risk-takin g a s formidable a s Th e Debacl e o f th e Ne w Math , elementar y schoo l languag e arts educator s hav e benefite d fro m 2 5 year s o f experimentatio n an d criti - cal evaluatio n leadin g t o teachin g methodologie s an d classroo m structure s aimed les s a t makin g th e studen t a skille d automato n wit h th e structur e o f written language , an d mor e focuse d o n th e developmen t o f th e studen t a s a literate person .

It i s n o les s difficul t t o defin e wha t a literat e perso n i s tha n i t i s t o describe wha t i t mean s t o d o mathematics . Fo r example , i t i s no t sufficien t to sa y tha t someon e i s literat e becaus e the y kno w a lo t o f words , rea d fast , spell an d punctuat e Standar d Englis h accurately , spea k severa l languages , or ca n pas s test s abou t al l o f th e book s o n a certai n prescribe d list . Ye t so-called literat e peopl e ca n d o man y o r al l o f thes e things . Likewise , i n mathematics, developin g a straight-forwar d definitio n o f literac y i s n o les s complicated o r controversial .

Mathematics an d literatur e hav e muc h i n common . Th e construction , examination an d communicatio n o f idea s i s centra l t o bot h disciplines . I n each disciplin e thes e activitie s ar e carrie d ou t withi n forms . Thes e form s are ofte n misconstrue d t o b e th e disciplin e itself . Eac h disciplin e i s s o vast , with suc h a ric h an d lon g tradition , tha t n o individua l ca n clai m t o gras p i t in it s entirety , ye t an y aspec t i s accessibl e t o th e dedicate d participant . I n both mathematic s an d literature , th e participant s i n th e disciplin e for m a community i n which innovation s an d conten t ar e share d an d examined . Th e most renowne d an d influentia l participant s i n th e communit y achiev e thei r position afte r a lon g perio d o f initiatio n an d experience , muc h o f which , i n the earl y years , occur s i n schools .

Elementary schoo l languag e art s teacher s as k th e followin g question s when the y pla n an d evaluat e thei r lesson s [1] :

• Ho w ca n w e prepar e student s t o becom e creativ e participant s i n a community wher e th e formulatio n an d communicatio n o f ideas i s fun - damental?

• Ho w ca n we , wit h material s an d tool s tha t ar e o n han d now , teac h them t o appreciat e th e vas t an d ever-changin g quantit y o f materia l they wil l encounte r i n thei r lifetimes , t o assimilat e ne w things , an d develop tast e a s the y matur e intellectually ?

• Wha t mus t w e d o s o tha t al l student s acquir e th e complex , interre - lated skill s necessar y t o d o al l this ?

Similar question s shoul d b e fundamenta l t o mathematic s teachin g i n th e formative years .

62 N. CASE Y AN D M . R . F E L L O W S

The followin g insight s borrowe d fro m languag e art s teachers ' examina - tion o f their goal s an d methodologie s ove r th e las t 3 decades ar e mos t usefu l for considerin g th e directio n tha t chang e i n mathematic s educatio n shoul d take [4] .

1. Childre n benefi t fro m exposur e t o a ric h variet y o f conten t withou t regard fo r hierarchica l sequencin g o f material . Statement s o f "devel - opmental appropriateness " mus t b e take n i n a larg e context .

2. Student s ar e draw n forwar d b y exposur e t o materia l tha t the y ca n understand bu t whic h i s beyon d thei r capacitie s t o produce .

3. Althoug h skill s matter , experienc e i n th e disciplin e canno t b e sec - ondary t o master y o f them ; teacher s mus t find way s t o monito r an d nurture skil l developmen t withi n th e contex t o f meaningfu l an d stim - ulating (self-selected ) projects .

4. Student s mus t b e steere d toward s matur e an d independen t self-selec - tion o f conten t material s an d individual/smal l grou p project s whic h they undertake .

5. Pee r communicatio n abou t thei r idea s i s no t onl y critical , bu t in - evitable. Teacher s mus t lear n t o exploit , no t suppres s th e classroo m culture.

6. Student s mus t b e give n larg e block s o f tim e t o read , think , tal k t o one another , share , argue , an d writ e dow n thei r ideas . Th e classroo m should b e a microcos m o f th e communit y int o whic h th e student s ar e being initiated .

7. Th e teache r i s neither spectato r no r ambassado r fro m th e communit y into whic h th e student s ar e bein g initiated , bu t a participan t an d a practitioner.

These questions an d insight s are less about languag e teaching than abou t teaching i n general . The y ar e representativ e o f a largel y grass-root s move - ment i n languag e teachin g refor m whic h cam e t o b e terme d W h o l e Lan - guage.

The Whol e Languag e connectio n [2 ] ha s prove d t o u s t o b e a n enor - mously usefu l handl e i n speakin g t o experience d elementar y schoo l teacher s about mathematic s educatio n reform. 4 Man y elementar y schoo l teacher s have spen t man y year s wrestlin g wit h thes e issues . Wha t the y typicall y sorely lac k i s an y sens e tha t mathematic s has a literature , tha t i t support s any kin d o f thinkin g o r activit y remotel y resemblin g literacy . W e ca n hel p teachers an d parent s appreciat e an d understan d refor m i n mathematic s ed - ucation b y appealin g t o thei r understandin g an d experienc e wit h prin t lit - eracy.

In al l kind s o f contexts , th e literac y connectio n ha s prove d useful . Her e are a fe w example s o f wha t w e cal l "standar d conversations " wit h parent s

4 Current controversie s regardin g form s o f Whol e Languag e hav e don e nothin g excep t strengthen thi s connection , a s the y brin g ou t a n importan t an d inevitabl e tensio n betwee n skills development , an d issue s o f motivation , participatio n an d meaningfu l context . Thes e controversies serv e a s a usefu l warnin g abou t trivializin g literac y educatio n o f an y kind .

I M P L E M E N T I N G T H E STANDARDS : L E T ' S FOCU S O N T H E F I R S T F O U R 6 3

and teachers , an d ho w the y ca n b e answere d b y lookin g throug h th e len s of literature.

"Why does my child need to know about coloring or knot theory?"

Does your chil d need to read Charlotte's Web or Huckleberry Finn? Doe s your chil d nee d t o kno w abou t dinosaur s o r oute r space ? (I t i s sa d tha t mathematics i s s o universall y associate d wit h suc h a miserlines s o f spirit. )

"IVs important to teach arithmetic. So now you are saying that it is impor- tant to teach coloring as well?"

It's importan t t o teach spelling , bu t it' s als o important t o read an d enjo y books. Wha t particula r book s these ar e i s not s o important, bu t the y shoul d be ric h an d interestin g stories . I t i s muc h th e sam e wit h mathematics .

Which topic s ar e bes t fo r eac h grad e level ? Goo d mathematica l topics , like goo d stories , ar e appropriat e a t al l grad e levels . A stor y lik e St . Exe - upery's The Little Prince ca n b e enjoye d b y ver y youn g children , bu t i s a source o f profoun d concept s fo r mor e matur e readers . Similarly , a proble m like ma p colorin g ca n b e explore d b y childre n wh o ar e no t ye t abl e t o read , yet i t i s a sourc e o f comple x an d interestin g conjecture s an d question s fo r older students .

6. W h a t I s T o B e D o n e ?

The elementar y schoo l principa l wh o said , "Thes e fou r Standard s ar e really importan t — w e handl e the m elsewher e i n th e curriculum! " wa s no t (as on e migh t firs t suspect ) makin g a n eas y mistake . A t thi s particula r school, meaningfu l context s fo r learnin g an d th e developmen t o f communi - cations skill s ar e highl y valued . A n intelligen t an d demandin g (an d yes , i t includes phonics ) Whol e Languag e approac h t o prin t literac y i s a deeply - rooted practic e a t thi s "charter " schoo l whic h ha s serve d fo r year s a s a n important mode l fo r curriculu m innovatio n i n Britis h Columbia . Th e tradi - tional mathematic s conten t agend a a t th e elementar y grad e levels , however , simply doe s no t provid e adequat e opportunitie s t o realiz e th e obviousl y im - portant Firs t Fou r in mathematics, s o i n orde r t o addres s them , teacher s must tur n t o opportunitie s elsewher e i n th e curriculum .

There ar e severa l thing s tha t w e in th e Discret e Mathematic s an d Com - puter Scienc e communitie s nee d t o do :

• W e nee d t o pa y fa r mor e attentio n t o th e need s an d opportunitie s i n the earl y an d formativ e year s o f schooling .

• W e nee d t o ge t th e messag e ou t t o th e elementar y school s tha t in - tegrating th e intellectua l cor e o f compute r scienc e (an d it s root s i n discrete mathematics ) int o th e curriculu m i s o f far greate r impor - tance tha n worshipin g i n th e expensiv e Carg o Cul t o f computers-in - the-classroom. (Fo r furthe r discussio n o f thi s poin t se e [6]. )

• W e need t o mak e th e connectio n betwee n mathematic s educatio n an d mathematics popularization . I n the areas of discrete mathematics an d

64 N. CASE Y AN D M . R . FELLOW S

computer scienc e we have enormous resource s of important, accessibl e mathematics fo r thi s purpose .

• W e need to establish connection s between mathematic s educatio n an d literacy education , especiall y a t th e K- 4 level . Suc h connection s ar e likely t o significantl y strengthe n both educational agendas . Th e com - munication o f mathematica l thinkin g an d argument , an d th e formu - lation o f mathematical model s an d conjecture s constitut e challengin g and importan t kind s o f writin g tasks .

• W e need to encourage the development o f whimsical, lengthy , content - rich children' s mathematica l literature . W e nee d stor y problem s tha t are rea l stories , no t "Farme r Brow n want s t o buil d a rectangula r fence... . " Fo r example , w e nee d 30-pag e storie s wit h characters , pictures, maps , an d dialogu e tha t incorporat e interactiv e problem - solving. W e nee d t o b e trainin g mathematics/cross-disciplinar y stu - dents (perhap s educationa l compute r game s designers ) a t th e univer - sities t o create this kin d o f literature .

• W e nee d t o suppor t teache r professionalism , an d serv e a s (energy - efficient) catalyst s fo r change , b y organizing an d involvin g mathemat - ical scienc e undergraduat e an d graduat e student s i n outreac h fro m the universitie s (perhap s a s a componen t o f servic e educatio n pro - grams). W e nee d t o similarl y organiz e summe r in-servic e institute s for teachers , an d mathematica l scienc e summe r camp s fo r kids .

• W e nee d t o establis h two-wa y communicatio n wit h undergraduat e departments o f education . W e canno t com e acros s a s th e arrogan t experts o f th e "Ne w Math " era . W e mus t b e prepare d t o enlighte n ourselves abou t th e problem s an d goal s o f elementar y teache r edu - cators an d th e elementar y schoo l classroo m itself . W e mus t see k ou t and wor k t o establis h productiv e relationship s wit h teache r educator s and creat e a commo n groun d wher e w e ca n trul y communicat e th e relevance o f ou r disciplin e an d ou r enthusias m fo r it .

References

[1] Luc y McCormic k Calkins , The Art of Teaching Writing, Heinemann , Portsmout h NH, 1994 .

[2] Nanc y Casey , "Th e Whol e Languag e Connection, " Connections, Car l Swenson , ed. , Washington Stat e Mathematic s Counci l (1991) , 1-13 .

[3] Nanc y Case y an d Michae l Fellows , This is MEG A-Mathematics! Stories and Activ- ities for Mathematical Thinking, Problem-Solving and Communication, Lo s Alamo s National Laboratories , 1993 .

[4] Kennet h Goodman , What's Whole in Whole Language?, Heinemann , Portsmout h NH, 1986 .

[5] Michae l R . Fellows , "Compute r Scienc e i n th e Elementar y Schools, " Mathematicians and Education Reform 1990-1991, N . Fisher , H . Keyne s an d P . Wagreich , eds. , Conference Boar d o f th e Mathematica l Sciences , Issue s i n Mathematic s Educatio n 3 (1993) , 143-163 .

[6] Michae l R . Fellow s an d Nea l Koblitz , "Ki d Krypto, " Proceedings of CRYPTO '92, Springer-Verlag, Lecture Notes in Computer Science, vol . 74 0 (1993) , 371-389 .

IMPLEMENTING TH E STANDARDS : LET' S FOCU S O N TH E FIRS T FOU R 6 5

[7] Alic e Miller , For Your Own Good: Hidden Cruelty in Child-Rearing and the Roots of Violence, Farra r Straus , Ne w York , 1983 .

[8] Nationa l Counci l o f Teachers o f Mathematics, Curriculum and Evaluation Standards for School Mathematics, NCTM , Resto n VA , 1989 .

D E P A R T M E N T O F C O M P U T ER S C I E N C E , U N I V E R S I T Y O F IDAHO, M o s c o w I D 8384 3

E-mail address: casey931Qcs.uidadio.edu , h t t p : / / w w w . c s . u i d a h o . e d u / ~ c a s e y 9 3 1

D E P A R T M E N T O F C O M P U T E R S C I E N C E , UNIVERSIT Y O F V I C T O R I A , V I C T O R I A , B R I T I S H

COLUMBIA, C A N A D A V 8 W 3 P 6 . P H O N E : 604-721-7299 .

E-mail address: m f e l l o w s Q c s r . u v i c . c a

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Discrete M a t h e m a t i c s : A Vehicl e fo r P r o b l e m Solving an d Excitemen t

Margaret B . Cozzen s

Mathematics ha s alway s ha d th e luxur y an d th e responsibilit y o f bein g recognized a s a fundamenta l componen t o f al l schoo l learning , on e o f th e "three R's" . Howeve r th e globa l consideration s o f economic competitivenes s and persona l an d societa l decision-makin g plac e mathematic s educatio n i n a premie r positio n a s w e mov e int o th e 21st century . Th e idea l mathe - matics classroo m o f th e 21st centur y i s on e wher e student s lear n t o valu e mathematics, becom e confiden t i n thei r ow n mathematica l abilities , becom e problem solvers , an d lear n t o communicat e an d reaso n mathematically . I t is on e wher e al l student s hav e th e thril l o f succes s throug h exploratio n an d hands-on experimentation . Opportunitie s fo r learnin g whic h capitaliz e o n curiosity, eagerness , flexibility, variou s level s o f maturity , uniquenes s o f cur - rent knowledge , an d comfor t leve l in mathematics an d technolog y mus t exis t for al l students .

It ha s no w bee n seve n year s sinc e th e Nationa l Counci l o f Teacher s o f Mathematics (NCTM ) release d th e Curriculum and Evaluation Standards for School Mathematics [1] , which describ e wha t student s shoul d kno w an d be abl e t o d o a t variou s age s (o r grad e levels ) i n mathematics . Th e NCT M Standards ar e no t prescription s fo r teacher s an d school s t o follow , bu t the y create a coheren t visio n o f wha t i t mean s t o b e mathematicall y literate . The goal s o f thes e Standards ar e goal s fo r students ; th e implementatio n depends o n teachers , schools , an d material s developers . Thes e Standards are guide s fo r th e revisio n o f schoo l mathematic s curriculu m framework s that hav e student s explore , reflect , an d discuss . Thes e Standards provid e guides fo r th e developmen t o f material s tha t provid e hands-o n experiences , materials tha t ar e open-ende d an d flexible, bu t a t th e sam e time , material s that provid e structur e an d guidanc e fo r studen t learning .

Where doe s discret e mathematic s fit int o th e schoo l mathematic s cur - riculum framewor k an d a t wha t grad e levels ? NCT M Standar d 1 2 call s fo r a mathematic s curriculu m framewor k i n grade s 9-1 2 tha t include s topic s

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 .

https://doi.org/10.1090/dimacs/036/07

68 MARGARET B . COZZEN S

from discret e mathematics . Th e focu s o f thi s standar d i s directe d a t po - tential application s i n compute r technology , bu t th e emphasi s goe s beyon d the need s o f th e informatio n industr y t o al l area s o f investigatio n wher e th e domain o f discours e i s a finite o r a countabl e se t o f objects . Whe n viewe d in thi s light , discret e mathematic s i s no t independen t o f algebr a an d geom - etry, bu t i t become s a powerfu l representatio n too l tha t pervade s al l o f th e K-12 mathematic s standards . Childre n o f al l age s ar e muc h mor e familia r with discret e set s tha n o f thos e whos e cardinalit y i s uncountable . Chil - dren's earlies t experience s wit h countin g ar e littl e mor e tha n matchin g th e elements o f a se t wit h a finite subse t o f th e natura l numbers , eve n thoug h they kno w non e o f thi s terminology . Thei r day-to-da y lif e i s envelope d i n "discrete" number s an d applications . I t i s throug h thes e doorway s o f dis - crete mathematics , an d th e opportunitie s tha t li e beyond , tha t the y lear n to b e proble m solver s an d mathematica l reasoners . I t i s difficul t t o overes - timate th e importanc e o f "engagement " a s a facto r i n learning . Activitie s in th e are a o f discret e mathematic s provid e thi s "engagement " opportunit y at al l educationa l levels .

Long befor e comin g t o th e Elementary , Secondary , an d Informa l Edu - cation Divisio n (ESIE ) a t th e Nationa l Scienc e Foundatio n (NSF ) i n th e area o f K-1 2 mathematics , science , an d technolog y education , I believe d that problem s i n discret e mathematic s wer e engagin g fo r bot h teacher s an d students an d accessibl e t o a wid e rang e o f students , i n particula r thos e wh o may hav e ha d difficult y wit h algebrai c manipulations . Student s ca n no t only wor k o n problem s i n discret e mathematics , bu t the y ca n als o pos e an d solve thei r ow n problem s tha t ar e natura l extension s o f one s provide d b y the teache r o r th e textbook . Th e natur e o f proofs , an d th e nee d t o prov e one's results , aris e naturall y i n thes e problems .

Through m y dail y interactio n wit h educatio n program s a s Divisio n Di - rector o f ESIE, 1 I a m no w eve n mor e convince d tha t discret e mathematic s opens th e doo r fo r al l student s t o discove r th e excitemen t an d versatilit y o f mathematics, th e lur e o f solving problem s bot h applie d an d theoretical , an d the pleasur e o f doin g thing s rigorously . I hav e see n student s i n classroom s throughout th e country , i n larg e urba n areas , smal l rura l schools , an d high - level magne t schools , al l workin g o n th e sam e problems , discussin g the m with thei r classmate s and , i n man y cases , convincin g thei r teacher s tha t there i s a "better " wa y t o solv e th e problem ; thes e student s don' t wan t t o leave mathematic s classe s t o g o to othe r course s becaus e the y wan t t o finish their work . I n som e cases , a s i n Philadelphia , amon g hig h schoo l student s who hav e bee n enrolle d i n mathematic s classe s wit h a heav y dosag e o f dis - crete mathematics, overal l averag e student monthl y attendanc e i n school ha s improved b y a s muc h a s 15% , and achievemen t score s fo r thes e student s i n English an d Scienc e hav e improve d significantly , a s wel l a s i n mathematics . A key component o f NSF's portfoli o o f activities to support educatio n refor m

lrThe opinion s expresse d i n thi s pape r ar e thos e o f th e author , an d no t necessaril y those o f th e Nationa l Scienc e Foundation .

A VEHICLE FO R PROBE M SOLVIN G AN D EXCITEMEN T 6 9

in the classroo m has been the development o f new mathematics instructiona l materials. Al l of the fifteen ne w mathematic s curriculu m project s funde d b y NSF, complete d o r nearin g completion , hav e heav y dose s o f discret e math - ematics; thes e spa n th e educatio n spectrum : elementary , middle , an d hig h school.

For example, th e proble m o f finding th e shortes t rout e through ke y citie s between Bosto n an d Miam i ca n b e pose d a s earl y a s fourt h grad e an d use d as a n activit y t o teac h an d reinforc e scal e measurement . I n hig h school , students ca n develo p fas t algorithm s t o comput e th e shortes t distance , bu t these algorithm s stil l requir e a knowledg e o f the intermediat e cit y distances . I have found tha t childre n of all ages like to compute shortest route s from rea l maps, rather tha n one s made up by the teacher or a textbook. An y road atla s provides plent y o f choices . Fo r example , usin g a ma p o f th e Easter n Unite d States, as k student s t o identif y th e ke y citie s betwee n Bosto n an d Miami , rather tha n tellin g the m th e one s t o use . A discussio n abou t whic h one s t o choose i s wel l wort h th e time . Olde r student s ca n us e mor e cities , younge r students ca n handl e onl y a fe w th e first time . A simpl e exampl e migh t include Boston , Ne w York , Washington , Roanoke , Columbia , Talahassee , Jacksonville, an d Miami . Thes e citie s ar e sufficien t t o provid e choices , bu t are no t s o man y a s t o b e tim e consuming .

Once th e student s ar e abl e t o wor k a fe w example s computin g shortes t routes, provid e the ma p i n the road atla s that give s times as well as distance s and hav e th e student s comput e th e shortes t tim e tri p an d compar e th e tw o answers. I hav e successfull y use d thi s exampl e i n al l grade s 4-12 ; man y o f the ne w curriculu m material s hav e suc h examples .

One of the mos t intriguin g aspect s o f specific discret e mathematics prob - lems suc h a s th e shortes t pat h proble m ar e th e man y application s o f th e same model . Fo r example , Mrs . Smit h want s t o determin e th e optima l tim e to trad e i n he r For d Taurus . Sh e ha s consulte d wit h th e For d deale r an d determined cos t an d trade-i n projection s a s indicate d i n th e followin g table :

Taurus L E year pric e ne w trade-i n pric e

1996 199 7 199 8 199 9 200 0 1995 $22,00 0 $19,00 0 $17,50 0 $16,00 0 $14,00 0 $12,00 0 1996 $24,00 0 $21,00 0 $18,50 0 $16,00 0 $14,00 0 1997 $27,00 0 $24,00 0 $21,50 0 $19,00 0 1998 $29,00 0 $26,00 0 $23,50 0 1999 $32,00 0 $29,00 0 2000 $35,00 0

T A B L E 1

The deale r assume s tha t th e averag e maintenanc e cost s ar e $50 0 i n th e second year , $100 0 i n th e thir d year , an d $200 0 i n th e fourt h an d fifth year s of ownership . I n th e yea r 2000 , Mrs . Smit h assume s sh e wil l n o longe r nee d

70 MARGARET B . COZZEN S

a ca r a s sh e wil l b e reassigne d i n Europe . Th e cos t o f keepin g th e ca r ca n then b e compute d b y th e followin g formula :

cost of keeping car between two years = cost new — trade-in price at time of sale + maintenance costs

Mrs. Smith' s proble m ca n b e solve d b y translatin g th e dat a int o a graph whos e vertice s ar e th e year s fro m 199 5 t o 199 9 an d th e edge s ar e directed fro m lowe r t o highe r year , weighte d wit h th e cos t o f keeping th e ca r during tha t year . Th e shortes t rout e betwee n 199 5 an d 200 0 i n th e grap h corresponds t o th e mos t economica l time s t o trad e i n car s fo r tota l leas t cost; th e solutio n i s t o sel l th e ca r i n 1998 , bu y a ne w one , an d sel l tha t i n 2000 (se e Figur e 1) .

F I G U R E 1 .

There ar e a number o f other problem s tha t ca n b e modele d usin g graph s and solve d usin g shortest rout e techniques. Th e shortes t rout e proble m i s an example of a problem that, unti l ten years ago, appeared i n undergraduate o r graduate course s i n mathematics , operation s research , o r compute r science , but no w appear s i n middl e schoo l an d hig h schoo l mathematic s curriculu m materials. Student s acros s th e countr y ar e inventin g ne w application s ever y day, includin g schedulin g th e activitie s i n thei r ow n classrooms .

Verifying tha t a proposed solutio n t o a problem i s indeed a solution i s a n activity tha t challenge s mathematician s o n a dail y basis . I t i s only recentl y that thi s activit y ha s bee n taugh t i n elementar y an d secondar y classrooms . We traditionall y hav e give n th e answe r t o a questio n i n th e ver y firs t para - graph o f a uni t b y statin g a theore m o r givin g a formul a an d the n merel y letting th e studen t fil l i n number s o r appl y th e resul t t o othe r situations , usually contrive d t o wor k easily . Th e correctnes s o f Dijkstra' s algorithm , the algorith m use d t o fin d shortes t distance s i n a graph , i s prove d usin g mathematical induction , a techniqu e accessibl e t o hig h schoo l students . Fo r small examples , student s ca n verif y th e correctnes s o f Dijkstra' s algorith m

A V E H I C L E F O R P R O B E M SOLVIN G AN D E X C I T E M E NT 7 1

by enumeratin g al l possibl e solutions , bu t enumeratin g al l solution s fo r a few problem s convince s student s quickl y tha t the y don' t wan t t o do that fo r all problems .

Consider anothe r exampl e o f a proble m i n discret e mathematic s tha t can b e pose d a t variou s levels , ca n b e enjoye d b y student s o f al l ages , an d has extension s tha t pos e interestin g mathematica l problem s fo r student s i n grade schoo l throug h graduat e school . Thi s exampl e i s reprinte d fro m th e new fourth grade mathematic s curriculu m material s develope d b y TERC , and publishe d b y Dal e Seymou r Publication s [2] .

Divide each small square into fourths in a different way. Use your favorite fourths, or make up new ways of dividing into fourths. Color each square's fourths right after you divide it into four parts. Use the same four colors.

F I G U R E 2 . Square s fo r a Quil t o f Fourth s

Figure 3 gives a n exampl e o f a fourth s quil t mad e b y on e fourt h grad e student. Student s ma y tak e th e quilt s hom e t o finish them . On e clas s actually too k th e best fourth s patc h fro m eac h student' s quil t an d sewe d a n actual quil t wit h 2 5 squares whic h wa s hun g a t th e entranc e o f th e school . Boys enjo y thi s exercis e a s muc h a s girls .

Students ar e aske d t o prov e tha t th e square s ar e actuall y divide d int o fourths and , durin g th e cours e o f the year , lear n ho w to verif y tha t indee d the fou r piece s o f th e subdivisio n ar e "equal " t o on e another . Sample s o f student verification s appea r i n Figure s 4 an d 5.

Even a s earl y a s fourt h grade , student s lear n th e meanin g o f proof . Their leve l o f sophisticatio n abou t th e natur e o f proo f change s ove r time , but th e notio n o f proo f i s demystifie d whe n student s hav e th e opportunit y to understan d method s o f proof, fro m th e earl y grade s throug h hig h schoo l and beyond , i n the contex t o f alread y familia r discret e topics .

72 MARGARET B . COZZEN S

F I G U R E 3 .

• c — •

• • ©

T f

/ l

/ • , / 4

• - •

• n

CD A T • / T

T • T

1 .® I ^ 3 -*-[_ ^ ^ ^[

F I G U R E 4 . A n exampl e o f a fourth grader' s proo f tha t on e of his square s i s divide d int o fourths : This works because they are all the same, so each occupies the same amount of space and there are four parts.

A B

F I G U R E 5 . A n exampl e o f a fourt h grader' s proo f tha t th e shapes i n A an d B ar e th e same : You can see these shapes are the same [in A and B], You just cut off the little triangles on top of the traingle [B] and put them to fix the squares in the line.

A VEHICLE FO R PROBE M SOLVIN G AN D EXCITEMEN T 7 3

The activit y o f designing a "fourth s quilt " i s appealing an d instructiona l for student s a t al l levels . Eve n thoug h th e activit y o f designin g a quil t ma y not b e a traditiona l applicatio n o f discret e mathematics , i t ha s prove d ver y engaging fo r students , mal e an d female . I n classroom s fro m fourt h grad e through hig h school , student s enjo y constructin g an d colorin g thei r exam - ples. I eve n trie d th e quil t exercis e a t a Christma s part y fo r th e ESI E Division suppor t an d progra m staff . Everyon e enjoye d th e fu n an d vie d fo r prizes, an d man y wer e s o captivate d wit h th e potentia l combinatoria l prob - lems tha t the y cam e u p wit h interestin g extension s o f th e origina l proble m in th e day s an d week s tha t followed . Fo r example , th e staf f cam e u p wit h the following : (Som e ar e stil l trying t o prov e their answer s to som e of them. )

1. Ho w man y distinctl y differen t fourth s squares , u p t o recoloring , ar e possible?

(a) — i f al l edge s mus t consis t o f straigh t lin e segment s (SLS) ? (b) — if all edges must b e SL S and pas s through point s on the grid ? (c) — i f al l edge s mus t b e SL S an d al l resultin g shape s convex ? (d) — i f al l edge s mus t b e SL S an d al l resultin g shape s congruent ? (e) — i f al l edge s mus t b e SL S an d th e resultin g quil t 2-colorable ? (f) — i f al l edge s mus t b e SL S an d al l piece s quadrilateral ?

2. A n an t i s placed a t eac h corner o f one square. Eac h an t walk s toward s its neighbo r a t th e sam e rate . Th e resultin g path s divid e th e squar e into fou r congruen t areas . Fin d th e boundarie s o f th e regions .

3. Wha t i s th e minimu m numbe r o f side s o f th e regula r polygo n tha t fits i n th e squar e an d ha s a n are a o f three-fourth s o f th e square ?

4. Wha t i s th e radiu s o f eac h o f th e thre e concentri c circle s tha t divid e the squar e int o fou r equa l parts ?

A thir d exampl e o f discret e mathematic s i n th e classroo m i s a n ap - plied combinatoria l situatio n fro m a n eight h grad e mathematic s curriculu m (Connected Mathematics Project [3] ) develope d a t Michiga n Stat e Univer - sity. Figur e 6 give s a diagra m fo r th e floor pla n o f th e Fail-Saf e Warehous e where Willi e ha s hi s storag e locke r fo r th e Radical Sound shop . Th e ware - house ha s tw o majo r sections , wit h large r locker s i n th e sectio n o n th e righ t of th e floor pla n an d smalle r locker s o n th e lef t o f th e floor plan . A secu - rity guar d patrol s th e warehous e a t night . Eac h patro l start s a t checkpoin t A, follow s on e aisl e o f th e warehous e t o checkpoin t B , an d anothe r aisl e t o station C . On e pat h i s shown b y th e dashe d line . T o b e sur e tha t th e entir e Fail-Safe Warehous e i s checked, th e guar d take s differen t route s on each trip . To b e sur e tha t burglar s can' t predic t time s whe n th e guar d wil l com e b y a particular point , th e guar d trie s t o tak e differen t set s o f route s eac h night .

A sampl e o f question s fo r th e student s t o answe r ar e th e following :

1. Ho w man y differen t path s ar e ther e fro m A t o C vi a B ? 2. Ho w man y differen t path s ar e ther e fro m C t o A vi a B ? 3. Ho w many differen t round-tri p path s ar e ther e fro m A to C an d bac k

to A ? I f i t take s th e guar d 2. 5 minute s t o wal k dow n on e aisl e o f

74 MARGARET B . COZZEN S

FlGURE 6 .

the warehouse , ho w lon g woul d i t tak e t o wal k al l possibl e round-tri p paths?

4. Develo p a simila r exampl e fo r th e policema n wh o patrol s you r neigh - borhood a t night . Dra w th e pictur e firs t an d the n coun t th e paths .

As thes e activitie s indicate , discret e mathematic s i n th e school s i s a vehicle t o ge t student s t o thin k mathematically , becom e proble m solvers , and becom e intereste d i n mathematics . A t th e sam e time , student s ca n work o n problem s tha t mak e sens e t o them , eithe r becaus e o f th e obviou s practical applicatio n and/o r becaus e the y ar e just plai n fu n an d challenging .

References

[1] Nationa l Counci l o f Teacher s o f Mathematics , Curriculum and Evaluation Standards for School Mathematics, NCTM , Resto n VA , 1989 .

[2] "Differen t Shapes , Equa l Areas, " Investigations in Numbers, Data, and Space, Grade 4, Dal e Seymou r Pub. , Pal o Alt o CA , 1995 .

[3] "Cleve r Counting, " Connected Mathematics, Dal e Seymou r Pub. , Pal o Alt o CA , 1995 .

DIVISION O F ELEMENTARY , SECONDARY , AN D INFORMA L E D U C A T I O N ( E S I E ) , N A -

TIONAL S C I E N C E FOUNDATION , 420 1 W I L S O N BOULEVAR D - R O O M 885 , A R L I N G T O N ,

VA 2223 0 E-mail address: mcozzens@nsf.go v

DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 3 6 , 199 7

Logic a n d Discret e M a t h e m a t i c s i n t h e School s

Susanna S . Ep p

Albert Einstei n onc e said [3] , "th e whole of science i s nothing mor e tha n a refinemen t o f ever y da y thinking. " Thi s quotatio n aptl y summarize s th e essential interdependenc e betwee n th e concret e an d commonsensica l an d th e abstract an d theoretical. Developin g students' abilitie s to shift smoothl y an d flexibly betwee n thes e tw o levels , whil e operatin g effectivel y o n eac h one , i s arguably th e centra l tas k o f mathematic s an d scienc e instruction .

In th e languag e o f th e NCT M Standards [6] , a primar y goa l o f math - ematics instructio n shoul d b e t o develo p students ' "mathematica l power, " which i s th e abilit y "t o explore , conjecture , an d reaso n logically , a s wel l a s . . . t o us e a variety o f mathematical method s t o solv e nonroutine problems. " As Ur i Treisma n an d Dic k Stanle y hav e pu t i t [9] , mathematics instructio n should "concentrat e les s on th e low-leve l us e of high-level idea s an d mor e o n the high-leve l us e o f low-leve l ideas. "

Those involve d i n th e mathematic s educatio n refor m movemen t hav e identified variou s specifi c element s tha t contribut e t o developin g students ' higher-level reasonin g skills , suc h a s experienc e workin g wit h open-ende d and slightl y ill-formed problems , opportunities fo r learnin g to perceive math - ematical issue s i n a broa d variet y o f differen t contexts , practic e i n recogniz - ing th e nee d fo r an d i n providin g justificatio n fo r mathematica l assertions , increased us e o f cooperativ e learning , an d employin g calculator s an d com - puters t o provid e answer s t o routin e part s o f problems .

But succes s i n thes e reforme d mathematica l environment s require s tha t students—whether the y kno w the y ar e doin g s o or not—correctl y appl y th e laws o f classica l logi c i n a variet y o f differen t settings . Specifically , i n orde r to b e abl e t o reaso n effectively , student s nee d t o kno w

• tha t jus t becaus e a statemen t o f th e for m "i f p the n g " i s true , on e cannot conclud e tha t "i f q then p " i s als o tru e (o r "i f no t p the n no t </", for tha t matter) ;

• tha t anothe r wa y t o phras e a statemen t o f th e for m "i f p the n g " i s "if no t q the n no t p" ;

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 , 03B65 .

https://doi.org/10.1090/dimacs/036/08

76 SUSANNA S . EP P

• wha t i t mean s fo r statement s o f the followin g for m t o b e false : "p and q" " p or g " an d "i f p the n g" ;

• tha t i f a propert y fail s t o hol d i n just on e instance , the n i t doe s no t hold universally ;

• tha t t o sho w a propert y hold s universally , on e show s tha t i t hold s i n a particula r bu t generic instance ;

• tha t certai n form s o f argumen t ar e inherentl y erroneou s (invalid ) whereas othe r form s ca n b e truste d t o produc e tru e conclusion s i f given tru e premises .

T h e P r o b l e m

Unfortunately, researc h b y cognitive psychologists strongly suggests tha t the vas t majorit y o f student s d o no t develo p th e reasonin g skill s describe d above durin g thei r hig h schoo l years. Moreover , althoug h a small proportio n of th e populatio n (approximatel y 4% ) appear s seemingl y spontaneousl y t o develop a goo d capabilit y fo r forma l reasoning , mos t o f the populatio n doe s not [1] . Thu s mos t students , bot h i n high school and college , need assistanc e in orde r t o improv e thei r abilit y t o thin k logically .

Perhaps i n a n idea l worl d th e fundamental s o f logica l reasonin g woul d be adequatel y conveye d t o student s i n th e contex t o f studyin g othe r topic s by teacher s wh o kno w ho w t o seiz e th e "teachabl e moment " an d wh o recog - nize th e importanc e o f instillin g genera l principle s o f reasonin g i n students ' minds. Bu t th e worl d i s not ideal . Fo r one thing, mos t o f us are less adept a t catching tha t elusiv e momen t tha n w e woul d wish . But , mor e importantly , if logica l reasonin g i s alway s presente d a s a subtext , i n a n implici t rathe r than explici t way , ho w ar e w e t o conve y th e expertis e require d t o teac h i t from on e generatio n t o th e next ?

One proble m i s tha t som e mathematic s teacher s ar e no t completel y se - cure i n thei r ow n reasonin g abilitie s whil e other s tak e correc t reasonin g s o much fo r grante d tha t the y ar e no t abl e t o communicat e effectivel y wit h students wh o d o no t thin k a s the y do . Anothe r proble m i s tha t whe n in - struction i n logica l reasonin g i s no t mad e a n explici t priority , i t i s usuall y subordinated t o othe r considerations .

An ironi c consequenc e o f the attentio n give n t o mathematic s instructio n over th e pas t thirt y year s i s that , especiall y i n algebra , cleve r teacher s an d textbook author s hav e devise d numerou s way s t o hel p student s obtai n cor - rect answer s t o problem s b y followin g certai n mechanica l procedure s rathe r than b y reasonin g the m through . Fo r instance , i t use d t o b e tha t student s were taugh t t o solv e th e proble m o f findin g al l rea l number s x suc h tha t (x + l)( x — 2 ) > 0 b y applyin g th e basi c principl e tha t a produc t o f tw o real number s i s positiv e i f an d onl y i f bot h number s hav e th e sam e sign . Use o f thi s approac h reinforce d th e notio n tha t succes s i n mathematic s re - sults fro m th e intelligen t applicatio n o f a smal l numbe r o f basi c principles , and i t taugh t severa l importan t method s o f logica l reasonin g (fo r instance , argument b y divisio n int o case s an d th e logi c o f and an d or). Nowadays ,

LOGIC AN D DISCRET E MATHEMATIC S I N TH E SCHOOL S 7 7

however, a popula r wa y t o teac h student s t o solv e suc h inequalitie s ask s them t o lear n tha t th e solutio n consist s o f certai n interval s an d tha t i f sub - stitution o f a valu e fro m on e o f thes e interval s make s th e inequalit y true , then th e interva l i s par t o f th e solution . Fo r mos t students , thi s method , while effective, serve s no larger educationa l purpos e tha n obtainin g a correc t answer t o a particula r problem .

Similarly, i n a n idea l situation , discussio n o f th e "vertica l lin e test " should hel p students deepe n thei r understandin g o f the relationshi p betwee n the analyti c an d geometri c version s o f th e definitio n o f a function . Instea d the rul e i s ofte n presente d i n suc h a wa y tha t student s lear n t o ge t th e right answe r t o th e questio n "doe s thi s grap h represen t a function? " with - out makin g an y rea l progres s towar d understandin g wha t a functio n is . I n practice, us e o f th e vertica l lin e tes t enable s student s t o avoi d dealin g wit h the linguisti c complexit y o f the functio n definitio n an d thu s fail s t o advanc e their abilit y t o understan d similarl y comple x statement s i n th e future . I n much th e sam e way, students wh o are taught t o find the invers e of a functio n / b y solvin g f(x) = y fo r x an d the n interchangin g x an d y ar e deprive d of th e opportunit y t o deepe n thei r understandin g bot h o f function s an d o f the logi c o f quantifie d statements . Student s no t taugh t thi s short-cu t ar e forced o f necessit y t o us e th e definitio n o f invers e function , learnin g t o as k and answe r th e question , "give n an y y i n th e co-domai n o f / ca n I find a n x i n th e domai n s o tha t f(x) = y? "

T h e D e v e l o p m e n t o f ou r Cours e

In 197 8 a t DePau l Universit y w e began developin g a cours e t o hel p stu - dents mak e th e transitio n fro m traditiona l computationally-oriente d math - ematics t o mor e abstrac t mathematica l thinking . A t th e outse t w e though t that i f we just gav e student s a n opportunit y t o lear n subjec t matte r — suc h as se t theory , relations , an d functio n propertie s — tha t form s th e basi s o f upper-level wor k i n mathematic s an d compute r science , the y woul d b e suc - cessful. Wha t w e discovere d wa s tha t student s ha d muc h mor e difficult y learning th e materia l tha n w e anticipated , an d tha t t o a grea t exten t thi s difficulty resulte d fro m a genera l lac k o f reasonin g skills .

For instance , man y application s involvin g one-to-on e function s us e on e form o f th e definition :

for al l x\ an d X2 in th e domai n o f / , i f f{x\) = f{x2) the n x\ — X2,

whereas othe r application s us e th e alternat e for m

for al l x\ an d X2 in th e domai n o f / , i f x\ ^ X2 then f(x\) ^ /(#2) « When w e first starte d teachin g th e course , w e merel y commente d o n th e equivalence o f th e definition s i n passing , usin g whicheve r wa s mos t conve - nient i n any particular situation . Bu t w e soon realized that wha t wa s obvious to u s (th e logica l equivalence o f the definitions ) wa s a majo r stumblin g bloc k for man y o f ou r students . Similarly , a larg e numbe r o f student s ha d diffi - culty determinin g whethe r o r no t particula r function s wer e one-to-one , no t

78 SUSANNA S . E P P

because the y didn' t understan d th e definition s o f th e give n functions , bu t because the y didn' t understan d wha t i t mean s fo r statement s o f th e for m displayed abov e t o b e false . Tha t is , the y di d no t understan d (eve n o n a n intuitive level ) tha t th e negatio n o f a universa l statemen t i s existentia l an d that th e negatio n o f "i f p the n g " i s u p an d no t #. "

After severa l year s o f experimentation , w e eventuall y settle d upo n th e method w e still us e today. Cognitiv e psychologist s hav e demonstrated fairl y conclusively tha t instructio n i n th e abstrac t principle s o f forma l logi c alon e does no t guarante e a n increas e i n students' reasonin g abilitie s [7] . Ou r expe - rience als o showe d tha t i n orde r t o significantl y affec t cognitiv e processe s a s fundamental an d broadl y applicabl e a s th e correc t us e o f th e rule s o f forma l logic, a one-sho t approac h i s no t sufficient . Jus t a s a person' s personalit y does no t chang e overnight , eve n afte r th e revelatio n an d acceptanc e b y th e person o f some profoun d persona l psychologica l truth , neithe r d o a person' s cognitive processe s underg o a n instantaneou s transformatio n eve n thoug h the perso n ma y hav e understoo d an d accepte d (a t som e level ) th e t r u t h o f certain logica l principles .

In our course , therefore , an d i n the boo k tha t ha s develope d ou t o f it [4] , we use severa l method s t o ti e forma l principle s o f logic to thei r us e i n actua l reasoning situations . (Substantia l portion s o f [8 ] reflec t a simila r approac h at th e hig h schoo l level. ) First , whe n w e introduce th e principles , w e includ e a ver y larg e numbe r o f natural-languag e examples . Thus , fo r instance , be - fore usin g trut h table s t o deriv e th e la w assertin g tha t th e negatio n o f a statement o f th e for m u p an d q" i s "no t p o r no t g, " w e giv e example s o f very simpl e and statement s an d hav e student s thin k abou t an d discus s wha t the negation s o f thes e statement s shoul d be . Then , afte r th e la w ha s bee n derived formally , th e bulk o f the exercise s ask student s t o appl y i t i n natural - language situations . Late r o n i n th e course , whe n th e la w is actually use d a s an importan t ste p i n a reasonin g process , w e point ou t it s occurrenc e t o th e students. An d whe n student s occasionall y us e th e la w incorrectl y i n thei r written work , mentionin g thei r erro r b y nam e help s the m bette r understan d what the y di d wrong .

Student difficultie s dealin g wit h negation s o f universa l an d existentia l statements ar e handle d similarly . Fo r instance , whe n w e introduc e student s to proo f b y contradictio n ( a difficul t topi c fo r mos t o f them) , w e migh t as k students t o prov e tha t th e doubl e o f an y irrationa l numbe r i s irrational . Here i s a versio n o f a commo n response :

Theorem: Th e doubl e o f an y irrationa l numbe r i s irrational . Proof (by contradiction): Suppos e i t i s not . Tha t is , suppos e the doubl e o f an y irrationa l numbe r i s rational . Bu t w e pre - viously prove d tha t \[2 i s irrationa l an d als o tha t 2\/ 2 i s ir - rational. Thes e result s contradic t ou r supposition . Henc e th e theorem i s true .

When th e clas s ha s no t previousl y discusse d ho w t o negat e quantifie d state - ments, a teacher ha s great difficult y helpin g students understan d th e erro r i n

LOGIC AN D DISCRET E MATHEMATIC S I N TH E SCHOOL S 7 9

the abov e "proof. " I n a clas s wher e ther e ha s bee n prio r experienc e workin g with suc h negations , th e erro r i s les s common . An d whe n i t doe s occur , th e teacher ca n recal l tha t th e clas s previousl y agree d tha t th e existenc e o f jus t one colleg e studen t age d 3 0 o r ove r wa s exactl y wha t wa s neede d t o falsif y the genera l statemen t "A H college students ar e unde r 30. " The n th e teache r can dra w th e parallel , pointin g ou t how , i n th e sam e way , th e existenc e o f just on e irrational numbe r whos e double is rational i s exactly wha t on e need s to negat e th e statemen t "th e doubl e o f an y irrationa l numbe r i s irrational. " So tha t i s th e suppositio n fro m whic h a contradictio n mus t b e deduce d i n order t o prov e th e theore m b y contradiction .

Contrast thi s approac h wit h a n approac h i n whic h genera l logica l prin - ciples hav e neve r bee n explicitl y discussed . I n suc h a case , th e teache r i s in th e awkwar d positio n o f havin g t o poin t ou t a n erro r an d a t th e sam e time convinc e th e studen t tha t i t reall y is an error . W e find tha t a studen t who ha s alread y though t abou t th e particula r logica l principl e i n questio n and a t leas t partiall y accepte d it s validit y i s muc h mor e read y t o integrat e an appreciatio n fo r th e ne w instanc e o f i t tha n a studen t wh o ha s neve r thought abou t th e issu e before . Thu s b y identifyin g a fe w logica l principle s and givin g the m name s earl y i n th e course , w e creat e a basi s fo r developin g a fulle r understandin g o f the m an d a mean s b y whic h t o communicat e wit h students abou t the m throughou t th e remainde r o f th e course .

This approac h i s simila r t o tha t use d i n bot h Englis h an d foreig n lan - guage courses. Englis h teachers agre e that th e most importan t par t o f teach- ing writin g i s havin g student s spen d tim e doin g it . Bu t intersperse d wit h actual writin g practic e i s a certai n amoun t o f explici t instructio n i n th e rules o f gramma r an d organization , an d a n importan t componen t o f writ - ing exercise s i s th e proces s o f correctio n an d revision . Similarl y fo r foreig n language instruction . Befor e th e ag e of about eleven , childre n ca n lear n lan - guage purel y b y osmosis . Bu t afte r th e ag e o f eleve n peopl e see m t o benefi t from som e forma l instructio n i n th e rule s o f a ne w languag e a s wel l a s fro m immersion i n it .

O u t c o m e s

On th e whole , w e an d ou r student s hav e bee n ver y please d wit h th e results o f our approach . W e do expec t a t time s t o hav e t o liste n t o an d rea d student explanation s tha t ar e quit e garble d (i n the earl y stage s o f discussin g set theor y proofs , fo r instance) . Deepl y ingraine d menta l habit s tak e tim e to change . Bu t wha t w e do se e i s significant growt h i n mos t student s a s th e course progresses .

For instance , w e wai t t o discus s equivalenc e relation s unti l lat e i n th e second quarter , havin g intersperse d th e mor e theoretica l cours e topic s wit h more straightforwar d topic s an d application s earlie r on . Th e advantag e i s that b y th e tim e w e reac h thi s topic , th e larg e majorit y o f student s reall y understand wha t i t mean s fo r a binar y relatio n t o b e o r no t t o b e reflexive , symmetric, an d transitiv e (whic h require s a well-developed sens e of the logi c

80 SUSANNA S . EP P

of quantifie d statements , if-then, and, an d or). Th e observatio n tha t a certain relatio n is , say , transitiv e "b y default " i s typicall y mad e wit h relis h by severa l student s simultaneously . An d whe n w e discus s th e proo f tha t an equivalenc e relatio n define d o n a se t partition s th e se t int o a unio n o f disjoint subsets , virtuall y th e whol e clas s participate s i n it s development .

Similarly b y th e tim e w e discus s th e fac t tha t an y tre e wit h n vertice s has n — 1 edge s (se e Figur e 1) , w e fin d tha t th e majorit y o f ou r student s have sufficien t familiarit y wit h th e logi c of if-then an d quantifie d statement s to comprehen d th e subtlet y o f th e proo f b y mathematica l induction . Th e difficulty i n the proo f comes in understanding wh y th e proo f o f the inductiv e step proceed s a s i t does . I n ou r course , th e structur e o f th e proo f i s see n as a natura l consequenc e o f th e genera l logica l principl e tha t t o prov e a statement o f th e for m

for al l element s i n a set , i f (hypothesis) the n (conclusion),

one assumes tha t on e has a (particula r bu t arbitraril y chosen ) elemen t o f th e set whic h make s the hypothesi s true , an d on e show s that thi s elemen t make s the conclusio n tru e also . Tha t i s why i n th e proo f o f th e inductiv e ste p on e assumes tha t k i s an y positiv e intege r fo r whic h propert y P(k) hold s (tha t is, on e assume s tha t an y tre e wit h k vertice s ha s k — 1 edges), an d the n on e shows tha t P(k + 1 ) mus t als o hol d (tha t is , on e show s tha t an y tre e wit h k + 1 vertice s ha s k edges) . Moreover , t o sho w tha t an y tre e wit h k + 1 vertices ha s k edges , applicatio n o f th e sam e logica l principl e lead s on e first to suppos e tha t T i s an y (particula r bu t arbitraril y chosen ) tre e wit h k + 1 vertices an d the n t o sho w tha t (thi s particular ) T ha s k edges .

Even afte r s o man y year s o f intimat e connectio n wit h thi s course , I a m still amazed tha t student s who are clearly bright b y many measures and hav e done extremely wel l in preceding parts o f the cours e nonetheless nee d to tak e their tim e an d fee l thei r wa y wit h eac h ne w topic . Give n encouragement , however, an d th e opportunit y t o explore , discuss , an d mak e mistakes , suc h students no t onl y succee d bu t the y als o thoroughly enjo y thei r success . Th e point i s tha t th e abilit y t o reaso n wit h mathematics , t o deduce , t o justify , and t o switc h bac k an d fort h betwee n abstrac t definition s an d theorem s an d concrete an d applie d situations , i s no t somethin g tha t student s eithe r d o or d o no t possess . No r i s i t necessaril y o r primaril y innate . Rathe r i t i s a conglomerate o f knowledge , attitudes , an d tendencie s whos e cultivatio n i s the greates t challeng e tha t mathematic s educator s ca n address .

Connection w i t h Discret e M a t h e m a t i c s

The primar y reaso n fo r th e curren t interes t i n discret e mathematic s i s that i t provide s th e theoretica l foundatio n fo r th e technolog y o f th e infor - mation age . Th e abilit y t o reaso n logicall y i n abstrac t setting s i s essentia l for succes s i n compute r scienc e course s a t al l level s o f th e undergraduat e curriculum. Moreover , knowledg e o f particula r topic s i n forma l logi c i s in - dispensible fo r understandin g th e desig n o f digita l circuit s an d automata ,

LOGIC AN D DISCRET E MATHEMATIC S I N TH E SCHOOL S 8 1

Lemma: An y tre e wit h mor e tha n on e verte x ha s a verte x o f degre e 1 . Proof: Le t T b e an y tre e wit h mor e tha n on e vertex . Pic k a verte x v at rando m an d searc h outwar d fro m v o n a pat h alon g edge s fro m on e vertex t o anothe r lookin g fo r a verte x o f degre e 1 . A s eac h ne w verte x is reached, chec k whethe r i t ha s degre e 1 . I f so , a verte x o f degre e 1 has been found . I f not , i t i s possibl e t o exi t fro m th e ne w verte x alon g a different edg e fro m tha t use d t o reac h th e vertex . Becaus e T i s a tree, i t is circuit-free, an d s o the pat h neve r return s t o a previously use d vertex . Since the numbe r o f vertices o f T i s finite, th e proces s o f building a pat h must eventuall y terminate . Whe n tha t happens , th e fina l verte x o f th e path mus t hav e degre e 1 .

Theorem: Fo r an y positiv e integer n , an y tree wit h n vertice s ha s n — 1 edges. Proof: Le t P(n) b e th e propert y

any tre e wit h n vertice s ha s n — 1 edges We us e mathematica l inductio n t o sho w tha t thi s propert y hold s fo r al l integers n > 1 .

Basis Step : Le t T b e an y tre e wit h on e vertex . The n T ha s zer o edges (becaus e i t contain s n o loops) . Sinc e 0=1-1 , th e propert y hold s for n = 1 .

Inductive Step : W e mus t sho w tha t fo r an y positiv e intege r fc, if th e property hold s fo r k the n i t hold s fo r k + 1 . Le t fcbea positiv e intege r and suppos e th e inductiv e hypothesis : tha t an y tre e wit h k vertice s ha s k — 1 edges. W e must sho w that an y tre e wit h k +1 vertice s ha s k edges . Let T b e an y tre e wit h k + 1 vertices . Sinc e k i s a positiv e integer , k + 1 > 2 , an d s o T ha s mor e tha n on e vertex . Henc e b y th e lemma , T ha s a verte x v o f degre e 1 . Als o sinc e T ha s mor e tha n on e vertex , there i s a t leas t on e othe r verte x i n T beside s v. Thu s ther e i s a n edge e connectin g v t o th e res t o f T . Le t T r b e th e subgrap h o f T consisting of all the vertices of T excep t v an d al l the edges of T excep t e .

Then T' ha s k vertices , an d T f i s circuit-fre e (sinc e T i s circuit-fre e and removin g a n edg e an d a verte x canno t creat e a circuit ) an d T' i s connected (sinc e T i s connecte d an d removin g a verte x o f degre e 1 an d its adjacen t edg e fro m a grap h doe s no t disconnec t th e graph) . Henc e T' i s a tre e wit h k vertices , an d s o T' ha s k — 1 edge s b y inductiv e hypothesis. Bu t then , sinc e T ha s on e mor e edg e tha n T" , T ha s k edges.

F I G U R E 1 . Purchased from American Mathematical Society for the exclusive use of Dipesh Bhattarai (bhdpxc) Copyright 1997 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.

82 SUSANNA S . EP P

relational databas e theory , programmin g languages , an d knowledge-base d systems. Becaus e o f it s importanc e a s a topi c i n compute r scienc e a s wel l a s its centra l rol e i n th e kin d o f critica l thinkin g i n whic h compute r scientist s must routinel y engage , logi c is now a standard topi c o f introductory discret e mathematics course s a t th e colleg e level .

Implications fo r t h e K-1 2 Curriculu m

The majorit y o f th e reasonin g skill s emphasize d i n course s suc h a s our s should no t hav e t o b e taugh t fo r th e firs t tim e a t th e colleg e level . B y th e time student s reac h us , w e hav e t o expen d a s muc h effor t helpin g the m un - learn th e incorrec t mode s of thought t o whic h the y hav e become accustome d as w e d o teachin g the m th e correc t though t processe s o n whic h mathemat - ics i s based . T o achiev e th e loft y goal s o f th e NCT M Standards, instructio n in a fe w basi c logica l principle s shoul d b e wove n throughou t th e K-1 2 cur - riculum. Kindergarte n i s no t to o earl y fo r teacher s t o begi n explorin g th e precise us e o f languag e wit h children . Indeed , eve n ver y youn g childre n ca n become sensitiv e t o an d enjo y makin g subtl e linguisti c distinctions . Fo r ex - ample, startin g i n th e primar y grades , th e Russia n mathematic s curriculu m translated a s par t o f th e Universit y o f Chicag o Schoo l Mathematic s Projec t includes exercise s specificall y designe d t o develo p children' s logica l sense [5]. In Prance , excellen t material s hav e bee n develope d fo r grade s 6-1 0 fo r help - ing student s mak e a transitio n t o abstrac t mathematica l thinking . (See , fo r instance, [2]. )

In grade s K-1 2 i n th e Unite d States , however , explici t attentio n t o th e development o f logica l reasonin g skill s ha s bee n minima l o r nonexistent . Our experienc e a s describe d abov e ha s show n tha t logi c ca n b e taugh t ex - plicitly an d successfull y withi n discret e mathematics . Includin g logi c a s a n official topi c o f discret e mathematic s throughou t th e K-1 2 year s wil l no t only provid e a basi s fo r mor e advance d stud y a t th e colleg e level , bu t wil l help insur e tha t th e principle s o f forma l reasonin g ar e n o longe r overlooked . While ther e i s a dange r tha t logi c wil l b e taugh t i n isolation , thi s ca n b e avoided b y well-constructe d curricula r materials .

The historica l rational e fo r requirin g th e stud y o f mathematics wa s tha t it sharpene d th e mind . Ove r th e year s thi s rational e ha s bee n deemphasize d and greate r attentio n ha s bee n give n t o th e goa l o f acquirin g specifi c com - putational skill s an d technique s though t t o b e neede d i n futur e course s o r in th e "rea l world. " Bu t th e compute r technolog y o f toda y render s man y o f these computationa l skill s les s important . Usin g calculator s an d computer s effectively require s genera l menta l powers , flexibilit y o f mind , an d a n un - derstanding o f concepts . Ou r primar y goa l a s teacher s shoul d b e t o develo p these abilitie s i n ou r students .

References

[1] Anderson , J . R. , Cognitive Psychology and Its Implications, 3 d ed. , W . H . Freeman , New York , 1990 .

LOGIC AN D DISCRET E MATHEMATIC S I N TH E SCHOOL S 83

[2] Arsac , G. , G . Chapiro n e t al , Initiation au Raisonnement Deductif au College, Presse s Universitaires, Lyon , Prance , 1992 .

[3] Einstein , A. , "Physic s an d Reality. " Out of my Later Years, Revise d Reprin t ed. , Bonanza Books , Ne w York , 1956 , p . 59 .

[4] Epp , S. , Discrete Mathematics with Applications, Wadsworth , Belmon t CA , 1990 . [5] Moro , Bantov a e t al. , Russian Grade 1 Mathematics, Russian Grade 2 Mathematics,

Russian Grade 3 Mathematics, Universit y o f Chicag o Schoo l Mathematic s Projec t Translation, UCSMP , Chicago , 1992 .

[6] Nationa l Counci l o f Teacher s o f Mathematics , Curriculum and Evaluation Standards for School Mathematics, NCTM , Resto n VA , 1989 .

[7] Nisbett , R . E. , G . T . Fong , D . R . Lehman , an d P . W . Cheng , "Teachin g Reasoning, " Science (238) , pp . 625-631 , 1987 .

[8] Peressini , A. , S . Ep p e t al. , Precalculus and Discrete Mathematics, Universit y o f Chicago Schoo l Mathematic s Project , Scot t Foresma n Publishin g Company , Glenco e IL, 1992 .

[9] "Th e Democratizatio n o f Undergraduat e Mathematic s Education. " CMS-M A A In - vited Address , Join t Mathematic s Meetings , Vancouver , B . C , Canada , Augus t 16 , 1993.

D E P A R T M E N T O F MATHEMATICA L SCIENCES , D E P A U L U N I V E R S I T Y , C H I C A G O , I L

60614 E-mail address: seppQcondor.depaul.ed u

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Writing Discrete(ly )

Rochelle Leibowit z

Discrete mathematic s serve s a s a n excellen t vehicl e fo r teachin g math - ematical writing . First , ver y littl e conten t i s neede d a s a prerequisite , s o discrete mat h problem s ca n b e introduce d a t an y age . Fo r example , th e 'lockers problem' , state d below 1, wa s give n t o fourt h graders , hig h schoo l students, an d colleg e mat h majors .

There are 1000 lockers, numbered from 1 to 1000, lining the hallways of a school. Early one Monday morning, the first stu- dent to arrive at school opens all the lockers. The second stu- dent to arrive at school that morning closes every other locker (that is, lockers numbered 2, 4, 6, ...). The third student to arrive approaches every third locker (lockers numbered 3, 6, 9, 12, ...) and closes the locker if it was open or opens the locker if it was closed. The fourth student approaches every fourth locker (lockers numbered 4, 8, 12, 16, ...) and closes the locker if it was open or opens the locker if it was closed. This process continues through the 1000th student who arrives that morn- ing. After the 1000th student is done with the lockers, which lockers are open?

Second, discret e mathematic s offer s man y open-ended , real-worl d modelin g problems, i n whic h ther e i s mor e tha n on e possibl e approach . On e suc h problem, discusse d later , i s th e proble m o f schedulin g final exam s a t col - lege. Third , ther e i s no t ye t a rigi d vocabular y an d symbolis m i n plac e for solvin g discret e mat h problems , s o writin g i n English , accompanie d b y good pictures , i s ofte n require d t o communicat e th e solution . Fourth , whe n introducing student s t o proo f techniques , whic h i s on e typ e o f mathemati - cal writing , discussin g discret e mat h theorem s tha t ar e intuitivel y obvious , for example , eve n intege r + od d intege r = od d integer , help s u s focu s o n

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 . 1 Editors' note : Se e als o th e equivalen t proble m i n Pete r B . Henderson' s articl e i n thi s

volume

https://doi.org/10.1090/dimacs/036/09

86 R O C H E L L E L E I B O W I T Z

the proo f techniqu e use d rathe r tha n th e materia l neede d t o understan d th e theorem.

One cours e tha t tie s th e conten t o f discret e mathematic s wit h th e pro - cess o f mathematica l writin g i s Discret e Mathematic s a t Wheato n College , a sophomore-leve l cours e fo r mathematic s an d compute r scienc e major s an d minors. Thi s course' s purpos e i s t o serv e a s a bridg e betwee n computa - tional mathematic s an d compute r scienc e (Calculu s an d C + + ) o n th e on e side and theoretica l mathematic s an d compute r scienc e (Linea r Algebr a an d Data Structures ) o n the other side . Consequently , th e emphasi s i s on writin g algorithms an d mathematica l proofs . W e spen d th e first five week s o f th e semester coverin g algorithms , logic , an d proo f techniques , includin g math - ematical induction . Th e remainde r o f th e semeste r i s spen t coverin g othe r discrete mathematic s topics , alway s keepin g i n min d th e primar y goa l o f improving mathematica l writin g skills .

The forma l (graded ) writin g fo r th e semester-lon g cours e consist s o f five problem set s an d a take-hom e final exam . Eac h proble m se t consist s o f six or seve n in-dept h mathematica l questions , whic h sometime s presen t ne w material. Th e proble m set s an d final exa m ar e ope n text , ope n notes , ope n library books . I provid e individualize d response s t o students ' writin g b y making comments , corrections , an d suggestion s o n thei r writin g styl e a s well a s o n th e mathematica l conten t o f thei r answers .

Informal (ungraded ) writin g consist s o f homewor k problem s pu t o n th e blackboard b y student s eac h da y a t th e beginnin g o f class . Student s ar e required t o pu t a certai n numbe r o f problems o n th e boar d fo r th e semester . The student s ar e encourage d t o wor k togethe r o n th e homewor k bu t onl y one studen t get s credi t fo r puttin g th e proble m o n th e board . The y ar e also encourage d t o pu t no t onl y correc t solutions , bu t als o partia l and/o r incorrect attempt s o n th e board . Livel y clas s discussion s aris e fro m th e students' boar d work , settin g th e ton e fo r th e res t o f th e clas s period .

Another exampl e o f a n informa l writin g exercis e underscore s th e im - portance o f precisio n i n technica l writing . O n on e o f th e las t day s o f th e semester, I divid e th e clas s i n two ; hal f g o int o anothe r roo m t o fill ou t th e course evaluation form s an d th e student s remainin g i n the classroo m becom e the Writers . Eac h Write r i s give n a Leg o mode l (al l model s ar e identical ) and aske d t o writ e dow n instructions , n o picture s o r diagram s allowed , o n how on e woul d buil d tha t mode l fro m a se t o f disassemble d pieces . Afte r 2 5 minutes, th e student s switc h places . Th e Writer s g o int o th e othe r roo m t o fill ou t th e cours e evaluatio n form s an d th e student s fro m th e othe r roo m come int o th e classroo m an d becom e th e Doers . Eac h Doe r i s given instruc - tions writte n b y a Write r an d a se t o f disassemble d pieces . Th e Doer s hav e 25 minute s t o complet e thei r models . Tr y it ! It' s harde r tha n yo u thin k and load s o f fun . I strongl y recommen d i t bot h fo r yoursel f an d you r class . Students an d teacher s lear n tha t technica l writin g i s no t eas y an d tha t th e technical write r need s t o b e ver y precise , t o defin e al l terms , an d no t t o assume tha t what' s intuitiv e t o hi m o r he r i s intuitiv e t o th e audience .

WRITING DISCRETE(LY ) 87

One advantag e o f teaching a writing intensiv e versio n of Discrete Mathe - matics is the unexpected turn s that com e with allowin g students th e freedo m to write an d explor e mathematics. W e navigate tw o such example s of classe s following uncharte d bu t rewardin g paths .

For th e firs t example , w e look a t th e followin g homewor k problem . "De - velop a n algorith m i n Englis h t o measur e exactl y 4 liter s o f wate r usin g a 3-lite r containe r an d a 5-lite r container , an d a n unlimite d suppl y o f wa - ter. Th e container s hav e n o marking s o n them. " Before reading on, try the problem for yourself! On e possibl e algorith m is :

1. Fil l th e 5-lite r containe r wit h water . 2. Fil l th e 3-lite r containe r wit h wate r fro m th e 5-lite r container . ( 2

liters remai n i n th e 5-lite r container. ) 3. Pou r ou t th e wate r fro m th e 3-lite r container , leavin g i t empty . 4. Pou r th e 2 liter s fro m th e 5-lite r containe r int o th e 3-lite r container .

(Now th e 3-lite r containe r ha s 2 liter s an d th e 5-lite r containe r i s empty.)

5. Fil l th e 5-lite r container . 6. Pou r som e o f th e wate r fro m th e 5-lite r containe r int o th e 3-lite r

container t o fil l up the 3-lite r container . (Th e 3-lite r containe r neede d only on e mor e lite r t o fil l i t up , s o th e 5-lite r containe r end s u p wit h 5 - 1 = 4 liters. )

Did yo u ge t th e sam e answer ? Di d yo u ge t a simila r answer , tha t is , a n algorithm tha t add s an d subtract s amount s o f wate r betwee n th e container s until th e 5-lite r containe r hold s exactl y 4 liters ? Thi s pas t year , a studen t gave a thir d typ e o f answer ; thi s answe r add s th e operatio n o f tiltin g th e containers.

1. Fil l th e 5-lite r containe r an d th e 3-lite r container . 2. Slowl y pou r ou t som e o f th e wate r fro m th e 5-lite r containe r unti l

the wate r surface , whe n th e containe r i s tilted , i s tangen t t o bot h the to p ri m an d th e botto m ri m o f th e container . (Assumin g tha t the containe r i s cylindrical, th e 5-lite r containe r no w hold s exactl y 2\ liters.)

3. Perfor m ste p 2 with th e 3-lite r container . (Assumin g tha t th e 3-lite r container i s cylindrical , i t no w hold s exactl y \\ liters. )

4. Pou r th e wate r fro m th e 3-lite r containe r int o th e 5-lite r container . Now th e 5-lite r containe r hold s 2\ + 1 ^ = 4 liters. )

Although unplanned , th e presentatio n o f thes e tw o differen t algorithm s provided m e with the perfect opportunit y t o discuss versatility of algorithms. We looked a t eac h algorith m separatel y an d derive d th e generalize d proble m that eac h on e solved. Fo r the genera l problem , suppos e containe r A ca n hol d x liter s o f water an d containe r B ca n hol d y liter s o f water wit h x > y. Wit h the restrictio n tha t x < 2y, th e firs t algorith m wil l terminate wit h containe r A holdin g 2(x — y) liter s o f water . Wit h th e restrictio n tha t bot h container s are cylindrical, th e secon d algorith m wil l terminate wit h containe r A holdin g

88 ROCHELLE LEIBOWIT Z

7}(x + y) liter s o f water . (Th e origina l homewor k proble m ha s x = 5 an d y = 3 , therefor e th e coincidenc e 2{x — y) = ±(x + y) = 4. ) W e continue d t o discuss th e versatilit y o f thes e tw o algorithms . W e als o talke d briefl y abou t the efficienc y o f algorithms . Th e clas s wen t s o wel l tha t I wil l incorporat e both "water " algorithm s i n th e future .

The secon d exampl e involve s grap h theory . I ofte n as k student s a s par t of thei r dail y homewor k assignmen t t o pos e a questio n abou t th e materia l we just covere d i n class . Th e da y afte r I presented grap h colorin g an d appli - cations t o scheduling , a studen t aske d abou t schedulin g final exam s whic h are assigne d 3-hou r tim e slots . Tha t lea d t o a discussio n o f interva l graph s and a writing assignmen t o n uni t interva l graphs . Becaus e th e student s wer e unhappy wit h thei r final exa m schedule , w e discusse d no t onl y th e mathe - matics o f uni t interva l graphs , bu t th e problem s o f modelin g thi s rea l worl d situation. Afte r th e discussion , th e student s wer e no t an y happie r abou t their final exa m schedul e bu t a t leas t the y understoo d th e difficult y i n try - ing t o pleas e everyon e an d th e necessit y o f goo d writte n communicatio n between mathematicia n (i n thi s case , th e Registrar ) an d client s (student s and faculty) .

By reinforcing writin g throughout th e semester, student s lear n tha t writ - ing an d doin g mathematic s ar e on e an d th e same . The y com e t o appreciat e that writin g mathematic s i s a n essentia l surviva l skil l fo r an y mathemati - cian. Currently , fo r mos t student s tha t I teach , thi s cours e i s a first ste p in learnin g thi s skill . However , th e connectio n betwee n doin g an d writin g mathematics ca n an d shoul d b e emphasize d muc h earlier . I t i s no t onl y the mathematicia n wh o need s t o communicat e mathematically , man y peo - ple need this skill . A lawyer need s to discus s probability whe n writin g a brief concerning DN A analysis , a buildin g contracto r need s t o understan d man - agement scienc e techniques when writing a schedule of task assignment s wit h time constraints , a conservationis t need s t o explai n function s an d graph s when writin g a repor t o n long-ter m effect s o f a huntin g ba n o n dee r popu - lation o n a n island , a homeowne r need s t o writ e trave l direction s t o his/he r house. Al l thes e an d muc h mor e involv e writin g mathematically . Teachin g this importan t skil l shoul d no t b e lef t solel y i n th e hand s o f colleg e instruc - tors. Elementar y school , middl e school , an d hig h schoo l teacher s ca n an d should devot e tim e t o teachin g thi s skill . They , a s wel l a s colleg e teachers , can us e discret e mathematic s fo r thi s purpose . I n th e proces s o f learnin g t o write mathematically , student s strengthe n thei r abilit y t o writ e clearl y an d logically, attribute s desire d o f al l writing . Discret e mathematic s thu s serve s as a n idea l too l fo r teachin g mathematica l writin g fo r th e lifelon g writer .

D E P A R T M E N T O F M A T H E M A T I C S , W H E A T O N C O L L E G E , N O R T O N M A

E-mail address: Rochelle-LeibowitzQwheatonma.ed u

DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 3 6 , 199 7

Discrete M a t h e m a t i c s an d Publi c Perception s o f M a t h e m a t i c s

Joseph Malkevitc h

1. Introductio n

A fe w year s ag o th e Matte l Corporatio n markete d a talkin g Barbi e doll , one o f whos e message s wa s "Mat h clas s i s tough. " Althoug h thi s messag e from a talkin g dol l wa s correctl y greete d wit h grea t outrag e b y variou s sec - tors o f th e mathematic s communit y becaus e i t conveye d a sexis t message , perhaps mos t member s o f th e genera l publi c woul d probabl y hav e agree d with Barbie . Fo r thes e people , no t onl y wa s mat h clas s tough , bu t math - ematics itsel f wa s tough . Man y peopl e perceiv e mathematic s t o b e toug h because o f wha t i s commonly taugh t a s mathematic s i n hig h school . Her e i s a list o f the kinds of problems typically taugh t an d teste d fo r o n standardize d tests tha t attemp t t o measur e succes s wit h hig h schoo l mathematics :

P r o b l e m Se t 1 .

1. Factor :

x3 + 5x 2 + 6x

2. Simplify :

(-2xy2z*f

3. Solv e fo r x:

3(x - 4 ) + 2(x - 3 ) = x + 2

4. Add :

x+2 x-S

x — 6 x — 4 5. Fin d th e valu e o f th e expressio n whe n x = 3 an d y = — 2:

(x*)2y-(xy)2

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 .

https://doi.org/10.1090/dimacs/036/10

90 JOSEPH MALKEVITC H

6. Prov e tha t th e line s throug h th e vertice s o f a triangl e tha t bisec t it s perimeter pas s throug h a singl e point . I s the sam e statemen t tru e fo r the are a bisector s tha t pas s throug h th e vertice s o f th e triangle ?

Compare thi s proble m se t wit h th e followin g lis t o f problems :

P r o b l e m Se t 2 . A: Desig n a n efficien t rout e fo r a pot-hol e inspectio n truck , whic h mus t

inspect ever y stretc h o f stree t i n th e stree t networ k i n Figur e 1 a t least once , an d whic h start s an d end s it s tou r a t th e locatio n marke d A. (Yo u ma y assum e tha t th e street s ar e two-way. )

F I G U R E 1

B : A smal l airpor t ha s thre e airline s tha t shar e th e us e o f th e runwa y at th e airport . I t ha s bee n decide d tha t anothe r runwa y mus t b e constructed. Wha t woul d b e a fai r syste m o f allocatin g th e cos t of th e ne w construction ? (Yo u ma y assum e tha t i t wil l b e possibl e to obtai n informatio n suc h a s numbe r o f flights pe r week , numbe r o f passengers serve d pe r wee k by thes e flights, a s well as other passenge r service an d economi c informatio n concernin g th e thre e airlines. )

C: Wha t woul d b e a fair wa y fo r a divorcing coupl e to agre e who shoul d be give n a boo k collection , a summe r home , an d som e jewelry, othe r than sellin g th e item s an d dividin g th e mone y equally ?

D : A compan y wishe s t o creat e a decima l digi t codin g syste m fo r th e products whic h i t sell s vi a a mai l orde r catalogue . Th e cod e fo r eac h item i s t o consis t o f 9 informatio n digit s an d a chec k digit . Wha t are som e o f th e consideration s whic h migh t g o int o th e desig n o f th e system?

E: Th e 5 5 ballots i n Figur e 2 have bee n collecte d fo r rankin g 5 plays fo r a dram a critic s award . I n thi s "preferenc e schedule" , th e "18 " a t th e bottom o f th e lef t colum n signifie s tha t o n 1 8 ballot s th e rankin g o f the five plays , fro m bes t t o worst , wa s 1,4,5,3,2 . Whic h pla y shoul d be designate d pla y o f th e year ?

DISCRETE MATHEMATIC S AN D PUBLI C PERCEPTION S O F MATHEMATIC S 9 1

4 - A „

4 - A .

- 4 - A ,

4 - A ,

-4- A , -4 - A.

— — **•') — \ ~ * * 1 — — **• 1 — — *"•

4 - A 0

4 - A ,

4 - A A

4 - A ,

4 - A A

18 12 10 9

F I G U R E 2

: Figur e 3 belo w show s th e 1 2 course s tha t ar e bein g ru n b y a smal l college during its summer session . A n x mean s that th e classes in tha t row and colum n hav e some students i n common. I f final examination s can b e arrange d i n 4 tim e slot s pe r day , i s i t possibl e t o schedul e al l the final examination s i n on e da y s o tha t ther e i s n o conflic t fo r an y students amon g th e time s schedule d fo r th e examination s fo r thei r courses?

1 2 3 4 5 6 7 8 9

10 11 12

1 - X

2 X

4 X

6 X

8 X

— X

10 X

—

11 X

— X

12 X

F I G U R E 3

These tw o proble m list s ar e world s apart . Th e first lis t require s suc - cessful solver s o f th e problem s t o b e comfortabl e wit h th e manipulatio n o f symbols, an d eac h o f th e problem s (othe r tha n th e geometr y problem ) ha s (essentially) on e correc t answer . Thi s lis t als o gives no hin t withi n th e prob - lems themselve s o f th e way s i n whic h mathematic s influence s dail y life . B y contrast, th e secon d lis t (certainl y i n th e statemen t o f th e problems ) down - plays th e direc t rol e symbol s play , an d th e problem s themselve s poin t t o

92 J O S E P H M A L K E V I T C H

areas o f applicability . Perhap s th e genera l publi c i s justifie d i n no t seein g the tota l pictur e abou t mathematics , whe n overwhelmingl y th e exposur e they hav e t o mathematic s consist s o f problem s o f th e kin d i n Proble m Se t 1.

The curren t hig h schoo l curriculu m i s the cause , i n m y opinion , o f muc h of th e negativ e imag e tha t th e genera l publi c attache s t o mathematics . Th e current curriculu m i s surprisingl y ofte n concerne d wit h th e kin d o f mathe - matics displaye d i n Proble m Se t 1 . Thi s i s tru e despit e th e fac t tha t thi s type o f mathematic s stray s fa r fro m achievin g man y o f th e goal s tha t th e mathematics communit y an d societ y i n genera l hop e ca n b e achieve d b y teaching mathematics . Thes e goal s includ e th e developmen t o f thinkin g skills, understandin g o f spatial concepts , an d trainin g fo r th e workplac e (se e below fo r a large r list) . I n wha t follow s I will try t o explai n wha t feature s o f "discrete mathematics" , an d th e wa y tha t i t ca n b e taught , mak e i t a usefu l tool fo r changin g th e widesprea d negativ e perception s abou t mathematic s and fo r achievin g society' s goal s fo r teachin g mathematics .

2. W h a t i s Discret e Mathematics ?

In thi s essa y I a m usin g th e phras e discret e mathematic s i n a specia l way. Here , discret e mathematics wil l mean tha t collectio n o f non-continuou s mathematical idea s tha t hav e explode d i n interes t an d stud y sinc e Worl d War II . I n man y case s thes e mathematica l idea s ha d root s i n muc h earlie r times (e.g. , grap h theor y wa s invente d b y Eule r i n 1736) , bu t th e inventio n of th e digita l compute r serve d a s a catalys t fo r th e flowering o f thes e ideas . Examples o f mathematica l tool s fallin g withi n th e rubri c o f discret e math - ematics are : matrices , graph s an d digraphs , differenc e equations , codes , and countin g techniques . Area s o f mathematic s whic h fal l primaril y withi n the domai n o f discret e mathematic s ar e rankin g system s an d socia l choice , graph theory , Marko v chains , discrete optimization, combinatorics , an d (dis - crete) probability . Jus t a s for continuou s mathematics , th e stud y o f discret e mathematics ca n b e pursue d fo r it s ow n intellectua l conten t o r fo r specifi c applications. However , a s w e shal l see , discret e mathematic s lend s itsel f t o achieving som e of the goal s for mathematic s educatio n mor e effectivel y tha n what i s currentl y taught .

3. W h y s t u d y mathematics ?

Mathematics differ s fro m othe r area s o f knowledg e i n tha t societ y ha s a vested interes t i n havin g th e publi c hav e a breadt h o f mathematica l skills . More tha n histor y o r anthropology , fo r example , mathematic s fulfill s specia l needs o f larg e sector s o f America n businesse s a s a knowledg e bas e fo r thei r employees. Obviously , societ y ha s man y interest s t o b e serve d i n promotin g the teachin g o f mathematics . Her e i s a lis t o f som e o f th e man y reason s offered fo r th e importanc e an d valu e of mathematics (i n no particular order) :

1. Promotin g skill s fo r enlightene d citizen s i n a democracy .

DISCRETE MATHEMATIC S AN D PUBLI C PERCEPTION S O F MATHEMATIC S 9 3

2. Providin g skill s fo r worker s i n a n increasingl y technologica l society . 3. Providin g understandin g o f th e physica l spac e i n whic h w e live . 4. Teachin g logica l thinkin g an d analysis . 5. Servin g a s th e languag e o f scienc e an d engineering . 6. Encouragin g flexibl e thinkin g whe n expose d t o ne w situations .

What mathematic s d o w e teac h i n hig h school s tha t i s designe d t o con - vey t o America n thes e importan t aspect s o f mathematics ? Currently , th e content o f hig h schoo l mathematic s ca n loosel y b e describe d a s follows :

Grade 9 : Algebr a Grade 10 : Geometr y Grade 11 : Algebr a an d Trigonometr y Grade 12 : Precalculus ; Calculu s

In th e contex t o f thi s over-simplifie d account , yo u ma y wis h t o tak e a sec - ond loo k a t th e problem s i n se t 1 above . Th e reaso n fo r thi s conten t i n grades 9-12 , whil e i n man y way s promotin g th e goal s mentione d above , lie s greatly i n society' s desir e t o allo w student s wh o ar e intereste d i n pursu - ing career s i n mathematics , compute r science , science , an d engineerin g t o have th e prope r skill s t o begi n colleg e leve l wor k i n thes e subjects . Th e en - try cours e i n colleg e fo r th e technologically-base d professions , mathematics , and scienc e i s Calculus . Succes s i n Calculu s i s tied t o knowledg e o f algebra , trigonometry, an d a subtle arra y o f skills wit h function s an d geometry . Thi s fact, couple d wit h a traditio n o f teachin g deductiv e geometr y (transferre d to Americ a fro m England ) an d traditio n i n general , ha s give n ris e t o th e current curriculum . However , a fe w moments ' thought , an d a loo k a t dat a concerning th e portio n o f colleg e graduate s wh o pursu e career s i n scienc e and mathematics , sho w tha t a hig h pric e i s bein g pai d fo r th e curren t cur - riculum. Althoug h th e curren t curriculu m i s generally successfu l i n locatin g the scientificall y inclined , i t result s i n vas t number s o f othe r student s wh o are "a t sea " wit h th e mathematic s the y ar e expose d to .

The botto m lin e fo r man y student s i s tha t despit e bein g expose d t o mathematics continuousl y fro m Kindergarte n throug h 10 t/l o r l l < / l grade , the typica l hig h schoo l graduat e ca n no t connec t th e valu e o f th e stud y of mathematic s wit h wha t mathematician s reall y do . Pu t differently , stu - dents hav e learne d whe n t o "call " o r hir e a doctor , electrician , geologist , o r plumber, bu t no t whe n t o "call " o r hire a mathematician . Fo r example , ho w many hig h schoo l graduate s kno w tha t mathematician s stud y optimizatio n problems (i.e . finding th e bes t o r mos t efficien t wa y o f doin g something ) and fairnes s questions ?

Another majo r failin g o f th e curren t curriculum , fro m society' s poin t o f view, i s tha t i t doe s no t sho w th e dramati c wa y tha t mathematic s ha s bee n involved in the developmen t o f new technologies. I t i s fair t o say that withou t 20t/l centur y mathematic s i t woul d hav e bee n impossibl e t o accomplis h th e following dramati c achievement s o f scienc e an d engineering :

1. Landin g a ma n o n th e moon .

94 JOSEPH MALKEVITC H

2. Developin g supersoni c planes . 3. Developin g mor e fuel-efficien t cars . 4. Makin g CAT , PET, an d MR I scans commonplace (i.e. , breakthrough s

in medica l imaging) . 5. Creatin g greate r efficienc y i n America n busines s operation s (e.g. ,

through th e us e o f linea r an d intege r programmin g models) .

Although man y peopl e ca n i n a genera l wa y se e th e connectio n betwee n more fuel-efficien t aircraf t an d mathematics , i t woul d no t b e possibl e fo r these peopl e t o writ e dow n th e mathematic s involved , eve n i n simplifie d terms. Th e reaso n fo r thi s i s tha t man y o f th e application s tha t peopl e point t o fo r demonstratin g th e importanc e o f mathematic s fo r technolog y involves th e solutio n o f differentia l an d partia l differentia l equations . Thi s mathematics i s no t reasonabl y accessibl e fo r a hig h schoo l graduat e o r eve n a colleg e graduat e (i n area s outsid e o f thos e wit h a scientific/mathematica l focus). Thi s contrast s sharpl y wit h th e situatio n fo r discret e mathematics . Research problem s i n discret e mathematic s ar e no t likel y t o b e resolve d b y typical hig h schoo l students . However , fo r discret e mathematica l problems , seeing th e ger m o f th e technica l idea s o f th e mathematic s an d takin g a fe w primitive step s wit h th e mathematica l idea s i s possible wit h a muc h smalle r knowledge bas e tha n woul d b e th e cas e fo r continuou s mathematics . (Fo r example, wit h n o knowledg e o f algebr a whatsoeve r on e ca n g o a lon g wa y in explorin g grap h theor y an d it s applications. ) Thus , alterin g th e curren t curriculum t o giv e a specia l rol e fo r idea s i n discret e mathematic s ha s muc h to recommen d it .

Furthermore, man y o f th e area s i n whic h discret e mathematic s i s be - ing applied , suc h a s operation s research , economics , an d biology , ar e area s where averag e student s hav e a riche r backgroun d knowledg e tha n fo r th e fields wher e continuou s mathematic s i s finding application s (i.e. , physics , engineering, an d chemistry) .

4. Discret e M a t h e m a t i c s i n ou r School s

Mathematics shoul d pla y a n importan t rol e i n ou r schools . Increasingly , knowledge o f th e rol e o f mathematic s i n ou r technologica l societ y wil l b e premium knowledge . Thi s raise s th e issu e abou t wha t concept s an d idea s should b e pursue d a s importan t one s i n grade s K-1 2 befor e differentiatio n of trainin g occur s a s par t o f caree r goals . M y answe r t o thi s questio n i s that w e shoul d mak e student s awar e tha t mathematic s i s involve d wit h th e following ke y area s an d issues :

Optimization: Wha t i s th e cheapest , fastest , bes t wa y o f achievin g a goal?

• Wha t i s th e optimu m blen d o f meat s fo r a make r o f col d cut s (i.e., salam i o r bologna ) t o pu t int o th e product , base d o n th e costs o f acquirin g th e meat s tha t mak e u p th e mixture ?

DISCRETE MATHEMATIC S AN D PUBLI C PERCEPTION S O F MATHEMATIC S 9 5

• Wha t mixtur e o f blend s o f gasolin e shoul d a compan y manu - facture t o optimiz e it s profit ?

• Afte r a large storm, whic h forces the cancellation of many flight s in a certain region of the country , wha t reshufflin g o f passengers, planes, an d plan e crew s wil l restor e th e syste m t o normalc y quickly an d cheaply ?

Fairness: • Ho w ca n on e fairl y divid e a n estate ? • Ho w ca n on e fairly divid e propert y betwee n a divorcin g couple ? • Wha t woul d constitut e a fai r wa y t o fun d schools ? • Ho w ca n America n electio n procedure s b e mad e mor e demo -

cratic? • I s weighte d votin g a fai r wa y t o represen t communitie s i n a

county legislature ? • Wha t make s a gam e fair ? • Wha t three-dimensiona l shape s ar e suitabl e fo r fai r dice ? • Ho w ca n tw o communitie s fairl y divid e th e cos t o f constructin g

a wate r treatmen t plan t tha t wil l benefi t bot h communities ?

Information: • Wha t code s woul d mak e i t eas y fo r businesse s t o transac t thei r

financial dealing s cheaply , safely , an d securely ? • Ho w ca n error s i n dat a transmissio n fro m oute r spac e b e cor -

rected s o that accurat e image s o f planets an d stella r object s ar e possible, eve n thoug h th e image s ar e bein g sen t wit h lo w powe r or unreliabl e transmitters ?

• Ho w ca n companie s minimiz e th e storag e spac e the y requir e fo r their records ?

• I s i t feasibl e t o sen d high-definitio n televisio n picture s alon g existing telephon e wires ?

• Ca n bar-cod e system s b e designe d tha t woul d spee d th e track - ing o f peopl e o r object s i n a transportatio n system ?

Risk: • I s it safe r t o ea t a vitamin tha t use s a non-natural color , t o tak e

a ca r ride , o r t o tak e a n airplan e ride ? • Ho w likel y i s i t tha t I wil l wi n a priz e i n a stat e lottery ? • Ho w risk y i s i t t o gamble ? • Wha t i s the risk of using milk from cow s that wer e fed genetically -

engineered feeds ? • Ho w dangerou s ar e ol d nuclea r powe r plants ?

Growth an d change : • I f curren t fishing pattern s ar e continued , wil l th e stoc k o f a

certain fish i n th e ocea n b e exhausted ? • Wha t wil l th e populatio n o f th e worl d b e i n 5 0 years i f curren t

trends continue ?

96 JOSEPH MALKEVITC H

• Ho w doe s a n epidemi c sprea d throug h a population ? • Wha t patter n o f marke t penetratio n shoul d a compan y intro -

ducing a ne w produc t expect ? • Ho w shoul d a fores t whic h contain s tree s tha t gro w a t differen t

rates b e managed ?

Unintuitive behavio r o f c o m p l e x systems : • I f weight s i n a votin g gam e ar e proportiona l t o population , i s

the power o f the legislators proportional t o the populations the y represent?

• Ca n addin g mor e processor s t o th e schedulin g o f a collectio n o f tasks increas e th e tim e t o ge t th e jo b done ?

• I f a n additiona l roa d i s buil t t o reliev e congestion , migh t con - gestion gro w worse ?

• Ca n on e batter d o better tha n anothe r i n each hal f o f a basebal l season, bu t d o wors e fo r th e seaso n overall ?

Since discret e mathematic s i s a ver y broa d are a withi n mathematics , many mor e area s an d applicatio n example s coul d b e listed .

5. A Futur e Directio n fo r M a t h e m a t i c s i n ou r School s

In ligh t o f th e ver y negativ e vie w tha t peopl e generall y (an d Barbi e i n particular) hav e o f mathematics , i t i s highl y desirabl e tha t action s b e take n that woul d chang e thes e perception s whil e a t th e sam e tim e providin g stu - dents wh o ar e mathematicall y incline d wit h th e stimulatio n tha t wil l allo w that inclinatio n to continue and flower . Discret e mathematics i s a very fertil e field t o conduc t experiment s concerne d wit h achievin g thi s goal . Alread y at th e colleg e level , th e so-calle d libera l art s mathematic s course , histori - cally taugh t wit h littl e regar d t o applications , ha s undergon e a renaissanc e with th e introductio n o f a ne w styl e o f cours e base d o n a n applie d discret e mathematics curriculu m (se e [1]) . Ther e i s thu s reaso n t o believ e tha t a n emphasis o n discret e mathematics , delivere d wit h teachin g method s tha t keep th e NCT M standard s squarel y i n view , ca n transfor m th e perceptio n that mathematic s student s ge t i n primar y an d secondar y schools , whil e a t the sam e tim e providin g a stead y strea m o f student s t o pursu e career s i n mathematics an d science .

Acknowledgment Many usefu l suggestion s fro m th e reviewer s ar e gratefull y appreciated .

References

[1] COMAP , For All Practical Purposes: Introduction to Contemporary Mathematics, 3r d ed., W . H . Freeman , Ne w York , 1994 .

[2] Hirsch , Christia n R. , an d Margare t J . Kenney , eds . Discrete Mathematics Across the Curriculum, K-12, Yearboo k o f th e Nationa l Counci l o f Teacher s o f Mathematics , Reston VA , 1991 .

[3] Malkevitch , J. , "Mathematics ' Imag e Problem" , 198 9 (Preprin t availabl e fro m th e author).

DISCRETE MATHEMATIC S AN D PUBLI C PERCEPTION S O F MATHEMATIC S 9 7

[4] (ed.) , Geometry's Future, COMAP , Lexingto n MA , 1991 . [5] Nationa l Counci l o f Teacher s o f Mathematics , Curriculum and Evaluation Standards

for School Mathematics, Resto n VA , 1989 .

M A T H E M A T I C S / C O M P U T E R S C I E N C E D E P A R T M E N T , Y O R K C O L L E G E ( C U N Y ) , J A -

MAICA, N E W Y O R K 1145 1

E-mail address: joeycQcunyvm.ciiny.ed u

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

M a t h e m a t i c a l Modelin g an d Discret e M a t h e m a t i c s

Henry 0 . Polla k

1. W h a t i s M a t h e m a t i c a l Modeling ?

When peopl e tal k abou t th e connectio n o f mathematic s wit h th e res t of th e world , the y us e a numbe r o f phrase s suc h a s "applie d mathemat - ics", "proble m solving" , "wor d problems" , an d "mathematica l modeling" , to nam e jus t a few . I n orde r t o defin e thes e mor e precisely , an d t o dif - ferentiate amon g them , I shoul d lik e t o begi n b y describin g th e serie s o f activities whic h see m t o tak e plac e whe n w e tr y t o us e mathematic s t o ex - amine somethin g i n the res t o f the world . Som e situation s involvin g discret e mathematics t o whic h thi s analysi s applie s wil l b e give n later .

(1) Th e proces s begin s wit h somethin g outsid e o f mathematic s whic h you woul d lik e t o kno w o r t o d o o r t o understand .

• Th e resul t i s a questio n i n th e rea l world , well-define d enoug h that yo u ca n recogniz e whe n yo u hav e mad e progres s o n it .

(2) Yo u nex t selec t som e importan t object s i n thi s situatio n outsid e o f mathematics, an d relationship s amon g them .

• Th e resul t i s th e identificatio n o f som e ke y concept s i n th e situation yo u wan t t o study .

(3) Yo u decid e wha t t o kee p an d wha t t o ignor e i n you r knowledg e o f the object s an d thei r interrelationships .

• Th e resul t i s a n idealize d versio n o f th e question . (4) Yo u translate th e idealize d versio n of the question int o mathematica l

terms. • Th e resul t i s a mathematica l versio n o f the idealize d question .

(5) Yo u identif y th e field o f mathematic s yo u thin k you'r e in . • Yo u brin g int o th e forefron t o f you r consciousnes s you r in -

stincts an d knowledg e abou t thi s field. (6) Yo u d o mathematics .

• Th e resul t i s solutions , theorems , specia l cases , algorithms , estimates, ope n problems .

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100 HENRY O . POLLA K

(7) Yo u no w translat e bac k int o th e settin g o f th e origina l problem . • Yo u no w hav e a theor y o f the idealize d versio n o f th e questio n

which yo u foun d i n (3 ) above . (8) Yo u confron t realit y i n th e for m o f th e origina l situatio n a s repre -

sented b y (1) . D o yo u believ e wha t i s bein g sai d i n (7) ? I n othe r words, d o you r results , whe n translate d bac k t o th e origina l situa - tion, fi t th e rea l world ?

• I f yes , yo u hav e succeeded . Yo u tel l you r friends , writ e i t up , publish som e papers , ge t a raise , ge t promoted , o r whatever .

• I f no , g o back t o th e beginning . Di d yo u pic k th e righ t object s and relationship s amon g them ? D o you r choice s o f wha t t o keep an d wha t t o ignor e nee d t o b e revisited ? Th e wa y i n which you r theor y o f the idealize d proble m fail s t o satisf y yo u should provid e som e hint s o f wher e ther e ar e difficulties .

An exampl e i n whic h thi s proces s ca n b e followe d i n detai l woul d tak e us to o fa r afiel d i n th e presen t context . Th e author' s forthcomin g pape r [7 ] contains a detaile d histor y o f suc h a problem , th e modelin g steps , an d th e repeated modelin g cycle .

2. Applie d M a t h e m a t i c s , Wor d P r o b l e m s , an d M o d e l i n g

What hav e I jus t give n i s a brie f outlin e o f "mathematica l modeling" . When I us e tha t ter m henceforth , thi s i s wha t I mean . No w wha t i s "ap - plied mathematics" ? Th e wa y th e ter m i s usually used , i t begin s wit h som e idealized version of reality, translates i t int o mathematics, doe s a lot o f math- ematics, and , a t it s best, translate s back ; i n other words , (4)-(7) . Course s i n "methods o f applie d mathematics " concentrat e o n th e mathematica l meth - ods tha t ten d t o com e u p i n (6 ) whe n yo u star t wit h question s i n physics .

What i s a "wor d problem" ? Typically , a wor d proble m begin s wit h a few word s fro m outsid e mathematic s t o provid e a semblanc e o f (4) , occur s in th e textboo k i n a plac e wher e (5 ) i s obvious , an d concentrate s o n (6) .

There i s ver y littl e agreemen t o n th e meanin g o f "proble m solving" . It ca n b e take n t o mea n doin g a wor d problem , o r applie d mathematics , or modelin g fro m beginnin g t o end . Sometimes , "proble m solving " refer s to th e proces s o f solvin g mathematica l problem s wit h n o referenc e t o a n external situatio n a t all . Whe n proble m solvin g refer s t o wor d problem s o r to applie d mathematics , i.e. , beginnin g wit h (4) , th e earlie r stage s (l)-(3 ) are sometime s referre d t o a s "proble m finding", o r "proble m formulation" .

Word problem s hav e a histor y o f bein g unrealistic , an d th e persistenc e of particula r type s lend s itsel f t o eas y caricature . W e hav e pipe s o f variou s capacities whic h ca n fill and empt y bathtub s an d John' s ag e when Susi e was twice a s ol d a s Sally . W e hav e learne d t o sa y "cente r o f mass" , "momen t of inertia " an d "pendulum " wit h a straigh t face , bu t w e g o directl y t o th e formulas withou t an y though t i n between . W e make n o attemp t t o se e if our answers mak e an y sens e i n th e origina l situatio n becaus e w e ha d n o origina l situation t o begi n with ! That' s ver y typica l o f man y wor d problems , I' m

MATHEMATICAL MODELIN G AN D DISCRET E MATHEMATIC S 10 1

afraid. O n th e othe r hand , wha t modelin g require s i s understandin g o f th e original situation , a n argumen t tha t th e idealizatio n make s sense , an d th e check tha t th e result s o f th e mathematica l wor k carr y meanin g outsid e o f mathematics.

3. Discret e M a t h e m a t i c s an d t h e Teachin g o f Modelin g

I believe that relatin g mathematic s t o th e res t o f the world i s an essentia l part o f mathematic s education . W e hav e no t don e ou r jo b i f thi s aspec t i s not included . W e ough t t o hav e wor d problems , traditiona l applie d mathe - matics, an d mathematica l modeling—al l three . Why ? I f modelin g i s wha t actually happen s whe n yo u appl y mathematic s i n th e rea l world , wh y don' t you jus t teac h that ? Ther e ar e thre e mai n difficultie s tha t I wil l discuss : mathematical modelin g take s a lo t o f time, i t require s a lo t o f knowledge o n the teacher' s part , an d ther e i s a lac k o f certaint y i n th e result s which , i n the eye s o f th e public , i s quit e uncharacteristi c o f mathematics .

It i s withou t doub t tru e tha t modelin g i s tim e consuming . S o le t u s agree tha t no t ever y proble m wit h a n applie d flavor wil l g o through th e ful l (l)-(8) above . Bu t i n term s o f a typica l wor d problem , ho w d o yo u tel l a good proble m fro m a ba d one ?

My answe r depend s o n whether th e proble m could be the middl e portio n of a genuin e model . Wha t d o I mean ? Her e i s a sampl e wor d problem : "A n electric fa n i s advertise d a s movin g 337 5 cubi c fee t o f ai r pe r minute . Ho w long wil l i t tak e th e fa n t o chang e th e ai r i n a roo m 2 7 ft . b y 2 5 ft . b y 1 0 ft.?" No w yo u al l kno w wha t yo u ar e suppose d t o do : multipl y 2 7 by 2 5 b y 10 an d divid e th e resul t int o 3375 . Bu t th e assumptio n behin d thi s i s tha t the roo m i s hermeticall y seale d an d tha t th e fa n evacuate s al l th e ai r befor e any ne w air come s in! Absurd ! Thi s i s not of f b y a little bit, it' s off by mayb e an orde r o f magnitude . Yo u could do a sensibl e discret e approximatio n t o this b y evacuatin g 10 % o f th e air , replacin g i t wit h replacin g i t wit h fres h air an d thereb y dilutin g th e ol d air , an d repeatin g thi s proces s unti l th e ol d air i s n o longe r noticeable . That' s a mode l tha t woul d mak e mor e sense . You obtai n a linea r recursio n fo r th e amoun t o f "old " ai r tha t i s lef t afte r k evacuations, an d yo u as k ho w lon g i t wil l b e unti l th e ol d ai r ca n n o longe r be perceived . Thi s i s a reasonabl e mathematica l model ; b y m y definition , the origina l wor d proble m wa s no t a goo d one .

A wor d o f caution : ther e ar e wor d problem s whic h wer e neve r mean t t o be take n seriously . Th e contex t i s deliberatel y whimsical , an d i s intende d to ad d lightnes s an d humo r t o a heav y lesson . Fo r example , Kolmogoro v i n 1966 gav e th e proble m o f a be e an d a lum p o f suga r a t tw o distinc t point s inside a triangle . Th e be e wishe s t o fly a minimu m lengt h pat h t o th e lum p of sugar , unde r th e conditio n tha t sh e mus t touc h al l thre e side s o f th e triangle alon g th e way . I hav e n o objectio n t o suc h a problem—i n fact , it' s lovely! Bu t nobod y pretend s it' s abou t actua l bees ! Wha t I objec t t o ar e problems tha t preten d t o b e rea l bu t couldn' t be .

102 HENRY O . POLLA K

Our secon d objectio n i s tha t rea l modelin g require s a lo t o f knowledg e on th e teacher' s part , knowledg e o f a lo t o f fields outsid e o f mathemat - ics! That' s true , bu t need s t o b e examine d ver y carefully . Mathematic s gets applie d i n al l aspect s o f everyda y life , intelligen t citizenship , an d othe r disciplines an d occupations . Furthermore , mos t branche s o f mathematics , certainly al l at th e schoo l an d undergraduat e level , have significan t practica l applications. I n fact , ther e ar e unexpecte d an d rathe r interestin g connec - tions betwee n thes e tw o observations . Whe n w e worr y tha t teachers , an d students, ma y no t kno w certai n fields t o whic h mathematic s i s applied , w e often hav e i n min d th e field o f physics . Wha t mathematic s i s mos t applie d to physics ? Classical , continuous , analysis . Discret e mathematic s i s just a s important fo r application s a s continuou s mathematics , an d ther e ten d t o b e many mor e application s t o everyda y life , operation s analysis , an d th e socia l sciences, wher e th e natura l experience s o f bot h teacher s an d student s ca n give a grea t dea l o f guidanc e an d insight . Thu s discret e mathematic s i s a n arena wher e w e ca n bridg e th e ga p betwee n mathematica l modelin g i n th e classroom an d mathematica l modelin g i n th e res t o f the worl d wit h unusua l effectiveness. Wha t w e ar e sayin g i s tha t mathematica l modelin g ca n b e particularly accessibl e whe n th e resultin g mathematica l field a t th e hear t of th e developmen t i s i n th e are a o f discret e mathematics . Votin g an d fai r division an d th e cleanin g o f streets ar e just a s interestin g mathematicall y a s moments o f inertia , an d the y us e a lo t o f availabl e intuitio n an d experience .

Here ar e partia l description s o f som e o f m y favorit e modelin g situation s which lea d t o discret e mathematic s an d ca n b e mad e accessibl e t o hig h school students .

(a) Traditiona l privat e lin e pricin g i n th e telephon e busines s lead s t o minimal spannin g trees , Cayley' s theore m a s wel l a s Prim' s an d Kruskal's algorithms , Shamos ' shortcuts , th e Steine r networ k prob - lem, an d NP-completeness . Th e ke y modelin g question : wha t i s meant b y "fair " pricing ? Thi s questio n drov e muc h o f th e histori - cal development . A discussio n o f th e private-lin e pricin g proble m i s given i n [7] .

(b) Buildin g a countin g circui t i n a compute r lead s t o th e proble m o f enumerating Hamiltonia n cycle s fo r th e grap h whic h i s th e verte x and edg e structur e o f a n n-dimensiona l cube . I t i s eas y t o giv e a n example o f a singl e suc h Hamiltonia n cycle , bu t ho w man y differen t cycles ar e there ? Th e grap h theor y soo n become s mixe d wit h grou p theory. Th e ke y modeling question : whe n ar e two cycles "different" ? It turn s ou t tha t fo r engineerin g purposes—an d thi s i s where model - ing i s especially important—yo u wan t tw o Hamiltonia n cycle s t o b e not differen t (i.e. , equivalent ) i f on e ca n b e obtaine d fro m th e othe r by a symmetr y o f th e n-dimensiona l cube . Ho w man y equivalenc e classes of Hamiltonian cycle s ar e possibl e o n a n n-dimensiona l cube ? The answe r appear s t o b e unknow n fo r dimension s n > 6 .

MATHEMATICAL MODELIN G AN D D I S C R E T E MATHEMATIC S 10 3

The mathematica l formulatio n o f this problem , an d th e complet e discussion fo r fou r dimension s (i.e. , countin g fro m 0 t o 15 ) ma y b e found i n [3] . Thi s pape r als o relate s th e countin g proble m t o th e earlier Gra y Cod e wor k durin g Worl d Wa r II , whic h wa s essentiall y a proble m o f analog-to-digita l conversion . Marti n Gardne r refer s t o the answe r fo r n = 5 in [2] .

(c) I n baseball , som e o f th e modelin g ha s bee n don e fo r us , a s i n th e definition o f battin g averages . I f a n additiona l hi t take s a player' s average fro m .29 9 t o .306 , ho w man y at-bat s an d ho w man y hit s has tha t playe r had ? Thi s turn s int o a wonderfu l numbe r theor y problem, an d involve s Fare y Serie s an d continue d fraction s i f w e so choose . I t i s mathematica l detectiv e work : ho w d o yo u tur n a decimal int o a fraction ? W e traditionall y teac h thi s fo r terminat - ing decimal s an d repeatin g decimals , bu t no t fo r arbitrar y decimal s known t o a certai n numbe r o f places—lik e battin g averages .

The basebal l example as such has not appeare d i n print; i t i s part of th e author' s lectur e "Som e Mathematic s o f Baseball " [6] , whic h is one o f the America n Mathematica l Society' s videotape d "Selecte d Lectures i n Mathematics". Th e sam e proble m arise s with free-thro w percentages i n basketball , an d ma y b e foun d i n [5] .

(d) E d Gilber t a t AT& T Bel l Labs , wh o wa s involve d i n th e researc h o f (a) an d (b) , i s th e originato r o f th e followin g problem : ho w d o yo u build a perfect box ? I f you have six rectangular piece s of wood, wha t patterns o f on e piec e coverin g anothe r a t a n edg e an d a t a corne r are possible ? Ther e i s som e simpl e topolog y i n this , an d th e Eule r characteristic give s a lo t o f insight . Ca n yo u buil d a perfec t bo x from si x identica l piece s o f wood ? Th e answe r i s "no t i n general" , although i t i s possible i f the dimension s o f the block s o f wood satisf y certain conditions . Gilbert' s articl e o n thi s subjec t i s [4] .

(e) Ther e ar e man y well-know n an d mor e traditional proble m area s tha t meet ou r requirements . I shal l mentio n jus t one , tha t o f codin g the - ory. Noiseles s coding , suc h a s Huffma n Codes , an d grou p code s fo r the binar y symmetri c nois y channel, ar e two very accessibl e subjects . The combinatio n o f geometry, beginnin g grou p theory , an d linea r al - gebra a t th e beginnin g o f grou p code s i s especiall y appealing .

A nice exposition o f the basic s of group code s fro m jus t th e poin t of view recommended i n the previou s paragrap h ma y b e found i n [9]. Huffman code s at a level appropriate fo r high-schoo l students ma y b e found i n [8] ; the proof s relate d t o Huffma n code s specificall y bu t no t to noiseles s codin g mor e generally , ma y b e foun d i n th e appendices . A mor e nearl y complet e expositio n o f noiseles s codin g appears , fo r example, a s chapte r 2 of [1] .

Let u s clos e wit h th e thir d objectio n t o mathematica l modeling , namel y the los s of certainty. Ther e i s personal judgment i n th e proble m formulatio n

104 HENRY O . P O L L A K

parts (l)-(3) , whic h i s especiall y noticeabl e when , i n (8) , th e result s don' t fit reality . Wors e tha n that , ther e ar e hones t difference s o f opinion ; fo r example, i f a proble m concern s fai r division , o r a n optimu m location , wha t to on e perso n look s fai r ma y no t see m fai r t o another . Or , t o giv e anothe r example, whe n competin g criteri a i n a n optimizatio n proble m ar e naturall y measured i n different units , suc h as lives and dollars , then there i s no obviou s way t o equat e them , an d disagreemen t i s inevitable . Thi s contradict s th e myth, hel d b y man y student s and , alas , som e teachers , tha t mathematic s i s a fiel d o f single righ t methods , singl e righ t answers , an d unambiguou s truth . This i s actuall y no t tru e o f pur e mathematic s either , bu t i t isn' t eve n clos e when you appl y mathematic s t o the res t o f the world. W e have to admi t tha t this observatio n ma y be especiall y distressin g t o thos e wh o like mathematic s primarily becaus e i t i s a wa y o f makin g a reasonabl e livin g an d a t th e sam e time minimizin g an y dange r o f involvemen t wit h th e rea l world . Fo r suc h people, wor d problem s ar e survivable , becaus e o f thei r degre e o f unreality , but mathematica l modelin g ma y caus e grea t unhappiness . Thei r respons e may b e t o den y tha t modelin g ha s a plac e i n th e mathematic s curriculum . Now discret e mathematic s i s especiall y usefu l i n applyin g mathematic s i n relatively controversia l areas . I s thi s on e o f th e reason s wh y it s plac e i n th e curriculum ha s bee n har d t o secure ?

References

[1] Ash , R. , Information Theory, Dove r Publications , 1990 . [2] Gardner , M. , Knotted Doughnuts and other Mathematical Entertainments, W . H . Free -

man an d Co. , Ne w York , 1986 , chapte r 2 . [3] Gilbert , E . N. , "Gra y Code s an d Path s o n th e n-Cube" , Bell System Technical Journal,

v. 37 , Ma y 1958 , pp . 81 5 - 826 . [4] "Th e Way s to Buil d a Box", Mathematics Teacher, v . 64, Dec 1971 , pp. 689-695. [5] Nort h Carolin a Schoo l o f Scienc e an d Mathematic s (G . Barret t e t al.) , Contemporary

Precalculus through applications, Janso n Publications , Providenc e RI , 1991 , pp . 170 - 172.

[6] Pollak , H . O. , "Som e Mathematic s o f Baseball" , videotape d "Selecte d Lecture s i n Mathematics", America n Mathematica l Society .

[7] "Som e Thought s o n Real-Worl d Proble m Solving" , Quantitative Literacy, Th e College Board , Ne w York , 1997 .

[8] Sacco , Copes , Sloyer , an d Stark , Information Theory, Saving Bits, Janso n Publica - tions, Providenc e RI , 1988 .

[9] Slepian , D. , "Codin g Theory" , Nuovo Cimento, N . 2 de l supplement o a l v . 13 , seri e X , 1959 , pp . 37 3 - 378 .

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E-mail address: 6182700Qmcimail.co m

DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

T h e Rol e o f Application s i n Teachin g Discret e M a t h e m a t i c s

Fred S . Robert s

1. U s i n g Application s Effectivel y i n t h e Classroo m

One o f th e majo r reason s fo r th e grea t increas e i n interes t i n discret e mathematics i s it s importanc e i n solvin g practica l problems . Conversely , practical problem s hav e stimulate d th e developmen t o f discret e mathemat - ics. Application s — discret e o r no t — shoul d pla y a majo r rol e i n th e mathematics classroom . The y mak e th e subjec t relevant . The y underscor e a reaso n fo r studyin g it . The y ar e interesting .

With regar d t o th e rol e of applications i n teaching discret e mathematics , I have developed som e rules of thumb ove r the years, based o n my experienc e with what student s respon d t o an d o n the philosophy I have developed abou t the rol e of applications i n mathematics . I n m y opinion, thes e rule s o f thum b are appropriat e a t al l grad e levels , thoug h mos t o f m y experienc e wit h the m has bee n a t th e colleg e level .

Rules o f T h u m b

1. Th e Relevanc e Rule : Choos e application s tha t ar e relevant . Ther e are plent y o f them .

2. Th e T w o Ar e B e t t e r T h a n On e Rule : Neve r settl e fo r on e appli - cation whe n tw o ar e available .

3. Th e W h y D o Thing s Twic e Rule : Stres s th e fac t tha t abstrac t methods develope d fo r dealin g wit h on e applicatio n ar e ofte n usefu l for another .

4. Th e Ge t Rea l Rule : Mentio n rea l use s o f mathematic s wheneve r possible.

5. Th e Frontier s Rule : Sho w th e frontier s o f th e subject . 6. Th e M a t h I s Aliv e Rule : Us e application s t o sho w tha t mathe -

matics i s a liv e subject , don e b y rea l people .

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106 F R E D S . R O B E R T S

7. Th e Motivat e Rule : Le t application s motivat e theory . The n appl y theory t o applie d problems .

8. Th e Don' t B e Scare d Of f Rule : Don' t hesitat e t o tal k abou t a n application becaus e you don't hav e a background i n the subject. Mos t applications ca n b e explaine d fro m genera l knowledge .

9. Th e Modelin g Rule : Choos e application s tha t involv e mode l build - ing. Illustrat e th e simplifyin g assumption s i n th e mode l an d iterat e to mor e complicate d (an d mor e realistic ) models .

In thi s paper , I wil l illustrat e thes e rule s o f thum b wit h thre e exam - ples. I n eac h case , I tak e on e simpl e mathematica l concep t an d giv e lot s of application s o f it . I hav e use d thes e an d simila r example s i n m y college - level courses , bu t hav e als o use d the m a t al l grad e levels , includin g primar y grades. Th e thre e example s I shal l discus s are :

a: Th e travelin g salesma n problem . b : Grap h coloring . c: Euleria n chain s an d paths . Almost al l of the application s I mention her e ar e discusse d i n more detai l

in m y book , Robert s [31] . Fo r som e o f them , I wil l provid e additiona l references, thoug h man y o f thes e reference s ar e t o article s tha t ar e mor e technical i n nature .

2. T h e Travelin g Salesma n P r o b l e m

The traveling salesman problem (TSP), i n it s traditiona l formulation, 1

is th e following : Ther e ar e n locations . A salesperso n mus t visi t al l o f them, i n som e order . Ther e i s a cos t o f travelin g fro m locatio n i t o locatio n j . Wha t i s th e cheapes t route ? Mos t o f thos e wh o hav e bee n expose d to discret e mathematic s hav e see n thi s problem . The y kno w i t i s difficult : No on e ha s foun d a good TSP algorithm, tha t is , a compute r algorith m fo r solving th e TS P whic h i s practica l fo r ver y larg e n, an d ther e i s stron g evidence tha t ther e i s none . (Th e proble m belong s t o th e clas s o f problem s that theoretica l compute r scientist s cal l NP-complete. ) Mos t peopl e wh o teach discret e mathematic s mentio n th e TSP . Bu t yo u ca n us e i t muc h more effectivel y b y goin g t o th e nex t step : Sho w ho w thi s proble m arise s i n practice i n man y othe r forms .

Let m e mentio n som e o f thes e othe r forms .

T h e A u t o m a t e d Telle r Machin e P r o b l e m . You r ban k has man y AT M machines . Eac h day , a courie r goe s fro m machine t o machin e t o mak e collections , gathe r compute r in - formation, an d s o on . I n wha t orde r shoul d th e machine s be visited ? Thi s proble m arise s i n practic e a t man y banks . One o f th e earlies t bank s t o us e a TS P algorith m t o solv e it , in th e earl y day s o f ATM's , wa s Shawmu t Ban k i n Boston .

xNote t h a t th e T S P i s nowaday s frequentl y referre d t o a s th e "travelin g salesperso n problem". I hav e chose n t o us e th e historica l name .

T H E ROL E O F A P P L I C A T I O N S I N T E A C H I N G D I S C R E T E MATHEMATIC S 10 7

(This exampl e i s fro m Margare t Cozzen s (persona l commu - nication), wh o first develope d i t a s a n assignmen t fo r he r un - dergraduate operation s researc h clas s a t Northeaster n Uni - versity, an d assigne d student s t o stud y th e Shawmu t Ban k ATM problem , wit h considerabl e success. )

T h e P h o n e B o o t h P r o b l e m . Onc e a week , eac h phon e booth i n a regio n mus t b e visited , an d th e coin s collected . I n what orde r shoul d tha t b e done ?

T h e P r o b l e m o f R o b o t s i n a n A u t o m a t e d Warehouse . The warehouse of the futur e wil l have orders filled by a robot . Imagine a pharmaceutical warehous e wit h stack s of goods ar - ranged i n row s an d columns . A n orde r come s i n fo r te n case s of Tylenol , si x case s o f shampoo , eigh t case s o f bandaids , etc. Eac h i s locate d b y row , column , an d height . I n wha t order shoul d th e robo t fill th e order ? Th e robo t need s t o be programme d t o solv e a TSP . I n ou r program s i n discret e mathematics fo r hig h schoo l an d middl e schoo l teacher s an d for hig h schoo l students a t DIMAC S (th e Cente r fo r Discret e Mathematics an d Theoretica l Compute r Science) , w e some - times take the student s t o see a Rutgers Universit y Industria l Engineering robot , whic h ca n b e use d t o d o exactl y this . Se e [8, 9] .

A P r o b l e m o f X - R a y Crystallography . I n x-ra y crystal - lography, w e must mov e a diffractomete r throug h a sequenc e of prescribed angles . Ther e i s a cos t i n terms o f time an d set - up fo r doin g on e mov e afte r another . Ho w d o w e minimiz e this cost ? Se e [4] .

Manufacturing. I n man y factories , ther e ar e a numbe r o f jobs tha t mus t b e performe d o r processe s tha t mus t b e run . After runnin g proces s i , a certain setu p cos t i s inferred befor e we ca n ru n proces s j , a cos t i n term s o f tim e o r mone y o r labor o f preparing th e machiner y fo r th e nex t process . Some - times thi s cos t i s minimal , fo r exampl e simpl y amountin g t o making mino r adjustments , an d sometime s i t i s major , fo r example requirin g complet e cleanin g o f equipmen t o r instal - lation o f ne w equipment . I n wha t orde r shoul d th e processe s be run ?

These application s illustrat e som e o f m y rule s o f thumb . The y al l illus - trate th e Relevanc e Rul e ( # 1 ) an d th e T w o Ar e B e t t e r T h a n On e R u l e ( # 2 ) . The y als o illustrat e th e Don' t B e Scare d Of f R u l e ( # 8 ) . You don' t hav e t o kno w anythin g abou t x-ra y crystallograph y t o tal k abou t that application . Yet , I kno w teacher s wh o ar e embarrasse d t o brin g i n

108 FRED S . ROBERT S

applications lik e thi s becaus e the y don' t kno w wha t som e word s mea n o r can't pronounc e th e words ! Wha t i s a diffractometer ? On e o f your student s might know , o r b e willin g t o find out . Th e entir e pape r wil l illustrat e thes e three rule s o f thumb , s o I wil l usuall y no t explicitl y mentio n thos e again .

These examples als o illustrate th e Ge t Rea l R u l e ( # 4 ) - i t i s especially nice t o b e abl e t o mentio n rea l companie s (suc h a s Shawmu t Bank ) tha t use mathematica l methods . I shoul d als o not e tha t al l o f thes e problem s are, i n th e abstract , th e identica l proble m w e hav e formulate d fo r th e TSP . Once w e have develope d mathematica l tool s fo r dealin g wit h th e TSP , thes e same tool s ca n b e applie d t o al l o f thes e othe r practica l problems . Thi s illustrates th e W h y D o Thing s Twic e R u l e ( # 3 ) . Ther e ar e tw o way s I illustrat e thi s rule . Sometimes , I formulat e on e versio n o f a problem , translate i t int o mathematica l languag e (wit h th e students ' help) , an d the n develop mathematica l method s neede d fo r dealin g wit h th e problem . I the n formulate anothe r practica l problem , sho w how , i n th e abstract , i t i s th e same as the first, an d then point ou t tha t littl e extra mathematical analysi s is needed. A t othe r times , I will formulate a large number o f practical problem s first, an d le t th e student s observ e ho w the y ar e relate d b y formulatin g the m all in the same abstract language , or by guessing why or how they ar e related .

I shoul d poin t ou t tha t man y o f thes e problem s i n thei r curren t formu- lation involv e simplifyin g assumptions . Fo r example , i n th e phon e boot h problem, som e telephon e booth s nee d t o b e visite d mor e ofte n tha n others , since the y fill u p faster ; an d i n th e manufacturin g problem , som e processe s cannot b e ru n befor e other s ar e completed . I n th e first roun d o f modeling , these complication s ar e ignored . Th e nex t roun d o f modelin g shoul d tr y to handl e them . Thi s i s a n illustratio n o f th e M o d e l i n g Rul e ( # 9 ) . B y discussing simplifyin g assumptions , w e teac h ou r student s t o questio n as - sumptions an d hypotheses , trai n the m t o b e mor e skeptica l abou t technica l presentations, an d ultimatel y prepar e the m t o b e bette r decisio n makers . I always tr y t o involv e m y student s i n pinpointin g oversimplification s i n a n initial mode l fo r a problem . I also involv e the m i n suggestin g ho w to modif y an abstrac t mode l t o tak e accoun t o f possibl e complications .

Recently, a grou p o f researchers a t fou r institutions , Rutger s University , AT&T Bel l Labs , Bellcore , an d Ric e University , solve d th e larges t TS P eve r solved (u p t o tha t time) . I t ha d 303 8 citie s an d aros e fro m a practica l problem involvin g th e mos t efficien t orde r i n whic h t o dril l 303 8 hole s t o make a circui t boar d (anothe r TS P application) . (Fo r informatio n abou t this, se e [1 , 41]. ) I lik e t o mentio n thi s achievement , an d tel l m y student s how a real problem wa s solved by real peopl e who are a t th e sam e institutio n as I am . Thi s illustrate s th e Ge t Rea l R u l e ( # 4 ) , th e Frontier s Rul e ( # 5 ) , an d th e M a t h I s Aliv e R u l e ( # 6 ) : I t involve s a rea l application , i t is righ t a t th e frontier s o f moder n research , an d i t wa s don e b y rea l people . Students muc h prefe r t o se e a real-worl d applicatio n t o a make-believ e on e using "widgets. " The y ge t turne d o n b y realizin g tha t the y ca n ge t t o th e frontiers o f knowledge. The y als o pay more attentio n t o things tha t ar e don e

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by rea l people . I kno w on e teache r wh o believe s s o strongly i n th e M a t h I s Alive R u l e (#6 ) tha t h e brings in slides showing pictures of mathematician s whose result s h e i s talking about . Onc e yo u hav e see n a pictur e o f a person , you someho w pa y mor e attentio n t o tha t person' s results , an d remembe r them bette r b y associatin g the m wit h th e picture .

I ofte n us e th e TS P t o introduc e th e ide a o f complexit y o f computa - tion an d t o motivat e a n interes t i n countin g an d combinatorics . I t i s a good exampl e t o illustrat e wh y on e need s t o coun t th e numbe r o f step s i n a computation befor e implementin g it . (Conside r th e brut e forc e approac h o f trying al l possibl e order s o f th e citie s i n a TS P with , say , 2 6 cities . Eve n o n a compute r tha t coul d chec k on e billio n order s pe r second , i t woul d tak e u s almost hal f a billio n year s t o loo k a t al l possibl e orders. ) Onc e I'v e intro - duced th e ide a o f countin g th e numbe r o f step s i n a computation , I find i t much easie r t o interes t student s i n method s o f countin g an d combinatorics , which I then relat e bac k t o complexit y o f computation. Al l of this illustrate s the Motivat e Rul e ( # 7 ) . Student s ar e muc h mor e intereste d i n th e rule s of countin g i f the y se e a rea l applicatio n tha t require s the m t o b e abl e t o count.

3. Grap h Colorin g

A graph consists o f a se t o f point s o r vertices, som e o f whic h ar e joine d by line s o r edges. A ver y ol d ide a i s t o color th e vertice s o f a grap h s o that i f tw o vertice s ar e joine d b y a n edge , the y ge t differen t colors . A larg e number o f thos e wh o teac h discret e mathematic s tal k abou t grap h coloring . Some mentio n on e applicatio n o f grap h coloring , th e historicall y importan t application o f m a p coloring , wher e th e goa l i s t o colo r th e ma p wit h a s few color s a s possible , s o lon g a s countrie s sharin g a borde r hav e differen t colors. W e mode l th e countrie s o f a ma p b y vertice s o f a grap h an d joi n two vertice s b y a n edg e i f thei r countrie s hav e a commo n boundary . Th e problem o f coloring a ma p s o tha t countrie s wit h a commo n boundar y mus t get differen t color s i s the sam e a s th e proble m o f colorin g th e correspondin g graph. Thi s i s a ver y importan t historica l example . I lik e bringin g histor y into m y classes , especiall y whe n ther e i s a ver y interestin g histor y o f ove r 100 years tha t als o involves important contribution s b y non-mathematician s and a historically importan t us e of computers - th e first solutio n t o th e map - coloring proble m use d 120 0 hour s o f compute r time ! Tha t i s wh y I lik e t o use th e ma p colorin g example . (Fo r mor e o n it s history , se e [2]. ) However , there i s much more to be said here , because graph colorin g has many moder n applications whic h student s find bot h interestin g an d exciting . I star t wit h some o f these example s befor e goin g bac k an d givin g th e historica l example . I find tha t student s per k u p an d tak e notic e fro m moder n an d relevan t examples. Her e ar e som e applications , almos t al l o f whic h ar e expande d upon i n m y paper s [33 , 34] .

110 F R E D S . R O B E R T S

Scheduling M e e t i n g s o f C o m m i t t e e s i n a S t a t e Leg - islature. Th e proble m i s t o assig n meetin g time s s o tha t i f two committee s hav e a membe r i n common , the y ge t differ - ent meetin g times . Th e solutio n i s t o colo r a n appropriat e graph. T o defin e a graph , w e mus t sa y wha t it s vertice s an d edges are . I n thi s case , th e vertice s ar e th e committee s an d there i s a n edg e betwee n tw o vertice s i f thei r correspondin g committees hav e a common member . The n th e color s ar e th e meeting times . I t shoul d b e note d tha t thi s proble m arise s i n many places . On e particula r plac e o f not e i s th e Ne w Yor k State Assembly . (Se e [5 ] an d [31 ] fo r mor e details. ) Thi s illustrates th e G e t Rea l R u l e ( # 4 ) .

Similar schedulin g problem s involv e assigning fina l exa m time s — classes with a commo n studen t mus t ge t differen t exa m times . Similarly , i n a n ide - alized school , student s firs t sig n u p fo r classe s an d the n classe s ar e assigne d meeting time s s o tha t classe s wit h a commo n studen t ge t differen t meetin g times. (Thi s actuall y happen s i n som e universities , a t leas t fo r th e schedul - ing o f graduat e course s i n smal l departments. ) Bot h o f thes e problem s are , in th e abstract , th e identica l proble m tha t w e hav e jus t formulate d fo r th e state legislativ e committees . A s wit h th e TSP , onc e w e hav e formulate d th e first schedulin g proble m a s a n abstrac t mathematica l proble m an d devel - oped tool s fo r dealin g wit h tha t problem , w e ca n no w "reduce " thes e ne w scheduling problem s t o th e ol d one , i n the sens e tha t i n the abstrac t version , they ar e th e sam e proble m an d s o ar e amenabl e t o solutio n usin g th e sam e tools. Thi s agai n illustrate s th e W h y D o Thing s T w i c e Rul e ( # 3 ) .

I usually giv e simple scheduling problem s a s examples, hav e the student s translate the m int o grap h problems , an d hav e the m tr y t o find grap h col - orings. W e usuall y en d u p usin g a greedy algorithm fo r doin g thi s — colo r the vertice s on e a t a time , usin g a ne w colo r onl y i f n o previousl y use d colo r can b e used . W e the n as k whethe r o r no t w e hav e foun d a colorin g wit h the fewes t numbe r o f colors . I t i s no t har d t o giv e example s wher e suc h a greed y approac h doe s no t work . I poin t ou t tha t grap h colorin g i s agai n known t o b e a difficult problem—a s wit h th e TSP , ther e i s no known "good " algorithm fo r finding a grap h colorin g wit h th e smalles t numbe r o f colors , and i t i s unlikel y tha t ther e wil l eve r b e one . Thi s i s a plac e wher e on e can introduc e differen t grap h colorin g algorithm s an d us e the m o n practi - cal problems . Th e abstrac t method s develope d fo r grap h colorin g problem s that aris e fro m on e proble m ar e usefu l fo r others . Th e W h y D o Thing s Twice R u l e ( # 3 ) ha s agai n bee n illustrated . I t i s no surpris e tha t softwar e designed t o solv e scheduling problem s i s sometimes base d o n grap h coloring . It migh t mak e a goo d exercis e t o hav e you r student s explor e th e softwar e that i s use d i n you r school , o r t o tr y t o writ e thei r ow n programs .

Practical schedulin g problem s involv e man y furthe r complications , suc h as individuals ' preference s fo r whe n the y ar e t o b e scheduled , o r certai n committees bein g require d t o mee t afte r certai n others . Also , ther e ha s

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been littl e mentio n s o fa r o f wha t make s on e schedul e (on e grap h coloring ) better tha n another . Thi s need s t o b e discusse d a s well . I s th e goa l onl y to us e th e smalles t numbe r o f colors ? O r i s i t sometime s goo d t o hav e a reasonable distributio n o f colors , i.e. , t o us e eac h colo r approximatel y th e same numbe r o f times ? Al l o f thi s illustrate s th e Modelin g R u l e ( # 9 ) . There i s a larg e literatur e o n schedulin g theory : severa l goo d reference s o n the subjec t ar e th e book s [3 , 27 , 36] .

T h e Channe l Assignmen t P r o b l e m . Th e proble m i s t o assign channel s t o radi o an d televisio n transmitters ; trans - mitters tha t interfer e mus t ge t differen t channels . Th e so - lution i s t o colo r a n appropriat e graph . Th e grap h ca n b e defined b y lettin g th e vertice s b e th e transmitter s an d let - ting a n edg e correspon d t o interference . Then , th e color s are th e channels . Grap h colorin g method s fo r solvin g th e channel assignmen t proble m ar e widel y use d a t suc h agen - cies a s th e Federa l Communication s Commission , th e Na - tional Telecommunication s an d Informatio n Administration , and NAT O (th e Ge t Rea l R u l e ( # 4 ) ) . Se e [12 , 6 , 34 ]

I usually formulate on e or two practical problem s a s graph colorin g prob- lems — explainin g wha t t o us e fo r vertice s an d edge s an d wha t correspond s to colors . Afte r a n exampl e o r two , however , I as k th e student s t o help , and the y willingl y chim e in . Afte r hearin g abou t schedulin g problems , the y can readil y translat e th e channe l assignmen t proble m int o a grap h colorin g problem. Indeed , the y ar e eage r t o thin k o f othe r problem s familia r t o the m that ca n b e formulate d a s grap h colorin g problems .

It i s wort h mentionin g tha t practica l channe l assignmen t problem s a s well a s othe r applie d problem s hav e give n ris e t o a variet y o f interestin g variations o f th e ordinar y concept s o f grap h coloring . S o far , w e hav e no t considered wha t make s one channel assignment bette r tha n another . I t i s not necessarily just tha t on e uses fewer channel s tha n th e other ; i t migh t b e tha t one ha s a smalle r separatio n betwee n larges t an d smalles t channe l use d an d thus use s les s o f th e availabl e "spectrum. " W e hav e no t considere d th e fac t that w e migh t hav e furthe r restriction s o n channel s tha t ar e close r togethe r than o n channel s tha t ar e furthe r apar t bu t stil l interfere , o r mor e generally , that w e migh t hav e differen t level s o f interference . W e hav e no t considere d the possibilit y tha t transmitter s migh t b e assigne d mor e tha n on e possibl e channel ove r whic h t o transmit , a s i s th e cas e fo r mobil e radi o telephone s in cars . Th e remova l o f eac h o f thes e simplifyin g assumption s lead s t o a n interesting generalizatio n o f grap h coloring . Som e o f the m ar e calle d T- colorings, n-tuple colorings , an d interva l coloring s [33] . Suc h generalization s are not difficul t t o explain t o students, an d man y o f them ar e at th e forefron t of moder n researc h i n grap h theory . Thi s agai n illustrate s th e M o d e l i n g R u l e (#9 ) an d th e Frontier s R u l e ( # 5 ) .

112 FRED S . ROBERT S

There i s anothe r importan t point . Real-worl d channe l assignmen t prob - lems use graphs wit h thousand s o f vertices. I t i s very har d t o find th e "best " solution unde r an y o f a numbe r o f definition s o f best. Sometimes , w e shoul d settle fo r a solution tha t ca n b e found i n a reasonable amoun t o f time, even if it i s not th e best . Thi s i s a goo d plac e to brin g i n the ide a o f approximation , and perhap s t o mentio n "heuristic " algorithm s tha t hav e bee n develope d b y real people at rea l places such as at NAT O (th e M a th I s Aliv e R u l e (#6)) .

Garbage Collectio n P r o b l e m . Garbag e truck s follo w cer - tain route s i n collectin g garbage . Th e proble m i s t o assig n each garbag e truc k rout e t o a da y o f th e wee k s o tha t i f tw o routes visi t a commo n site , the y ar e schedule d fo r differen t days. Th e solutio n i s t o colo r a n appropriat e graph . Th e vertices o f tha t grap h ar e th e route s i n questio n an d a n edg e between tw o route s mean s tha t the y visi t a commo n site . The color s ar e th e days . Thi s particula r proble m aros e fro m a more complicate d garbag e truc k routin g proble m pose d b y the Ne w Yor k Cit y Departmen t o f Sanitation . Tha t prob - lem involve s choice s o f route s a s well . Thi s illustrate s th e Get Rea l R u l e ( # 4 ) an d th e M o d e l i n g R u l e ( # 9 ) . Se e [28, 29 , 37] .

Traffic Ligh t Phasin g P r o b l e m . W e ar e puttin g i n a ne w traffic ligh t a t a traffic intersection . W e need to assig n a green light tim e t o eac h strea m o f traffi c throug h th e intersectio n so that tw o streams o f traffic tha t interfer e ge t differen t gree n light times . Th e solutio n i s t o colo r a n appropriat e graph . The vertice s o f tha t grap h ar e th e traffi c streams , a n edg e means inteference , an d th e color s ar e th e gree n ligh t times . The ide a of using graph colorin g for phasin g ne w traffic light s was firs t propose d i n a n articl e i n a transportatio n journal , Transportation Science [35] . (Se e als o [28]. )

In dealin g wit h th e Traffi c Ligh t Phasin g Problem , w e hav e omitte d any discussio n o f wha t make s on e gree n ligh t assignmen t bette r tha n an - other. Also , w e ar e no t payin g attentio n t o th e duratio n o f th e gree n ligh t times, an d th e fac t tha t on e traffi c strea m migh t requir e a longe r gree n ligh t time tha n another . Thes e complication s lea d t o generalization s o f ordinar y graph coloring , an d i n particula r th e generalizatio n know n a s interva l color - ing, whic h i s o f curren t researc h interest . So , w e hav e agai n illustrate d th e Modeling R u l e (#9 ) an d th e Frontier s R u l e ( # 5 ) . Staffers ' algorith m for traffi c ligh t phasing , an d late r one s b y Opsu t an d Robert s [20 , 21 ] an d Raychaudhuri [24 , 25] , ar e base d o n linea r programmin g method s t o fin d the bes t (interval ) grap h coloring s wit h durations . I lik e t o teac h thes e al - gorithms t o m y students , the n le t the m fin d som e loca l traffi c intersection s to appl y th e algorithm s to . Margare t Cozzen s (persona l communication ) reports tha t whe n he r student s applie d th e algorithm s t o intersection s nea r

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the campu s o f Northeaster n University , the y foun d muc h bette r traffi c ligh t phasings tha n thos e actuall y i n use . T o complet e thi s reall y practica l expe - rience, the y wen t an d convince d th e Bosto n departmen t o f transportatio n to implemen t thei r solutions ! Thi s i s a wonderfu l exampl e o f rul e o f thum b # 4 , th e Ge t Rea l Rule .

It shoul d b e note d tha t th e sam e proble m arise s i n schedulin g othe r facilities, suc h a s a classroom , computer , etc . Ther e ar e differen t user s an d some o f the m interfere . W e wis h t o assig n gree n ligh t time s (permission-to - use times ) s o tha t interferin g user s ge t differen t gree n ligh t times . I n th e abstract, thi s i s th e identica l proble m tha t w e hav e alread y analyzed , a n illustration o f th e W h y D o Thing s Twic e R u l e ( # 3 ) . I n addition , th e same problem arise s in task assignmen t problem s i n the workplace. Differen t tasks nee d t o b e assigne d times , bu t som e o f the m interfer e becaus e the y use th e sam e worker s o r tool s o r resources , anothe r illustratio n o f th e W h y D o Thing s T w i c e R u l e ( # 3 ) .

Fleet Maintenanc e P r o b l e m . Vehicle s (cars , planes, ships ) are comin g int o a facilit y fo r regula r maintenanc e accordin g to a fixe d schedule . W e wis h t o assig n a spac e t o eac h vehi - cle. I f tw o vehicle s ar e ther e a t th e sam e time , the y mus t ge t different spaces . Th e solutio n t o thi s proble m i s t o colo r a n appropriate graph . It s vertice s ar e th e vehicle s an d ther e i s an edg e betwee n tw o vehicle s i f they ar e i n th e facilit y a t th e same time . Th e color s ar e th e spaces . (Thi s i s th e firs t ex - ample I hav e give n wher e th e color s ar e no t time s o r day s o r something lik e that. Student s usuall y se e this fairl y quickly. ) It shoul d b e remarke d tha t thi s proble m wa s firs t worke d o n at IB M fo r shi p maintenanc e (th e Ge t R e a l R u l e ( # 4 ) ) . See [11 , 19 , 30] .

The Flee t Maintenanc e Proble m agai n ha s it s complications : Wha t makes on e assignmen t bette r tha n another ? Wha t i f on e vehicl e require s more spac e tha n another ? Physically , d o th e space s correspon d t o point s o r to rectangle s o r t o circles ? Eac h complicatio n lead s t o a ne w variatio n o f graph coloring , muc h a s i n th e channe l assignmen t problem . Her e agai n w e illustrate th e M o d e l i n g Rul e (#9 ) an d th e Frontier s Rul e ( # 5 ) .

I have often give n a talk t o high school audiences that describe s the man y applications I hav e give n i n thi s section . (I t certainl y i s a n illustratio n o f the T w o Ar e B e t t e r T h a n On e Rul e (#2)! ) On e o f thes e talk s le d t o a very excitin g questio n b y on e o f th e students : "Ar e ther e career s i n grap h coloring?" I thin k I mad e m y poin t tha t day ! (No , ther e ar e no t career s i n graph coloring . Yes , ther e ar e career s i n applyin g mathematics. )

4. Euleria n Chain s

Given a graph , a chain (o r wal k o r path , dependin g o n wha t terminol - ogy yo u use ) arise s i f w e follo w th e edge s fro m verte x t o vertex ; a n eulerian

114 F R E D S . R O B E R T S

chain i s a chai n tha t use s ever y edg e exactl y once . A n eulerian closed chain is a n euleria n chai n tha t begin s an d end s i n th e sam e place . Man y o f thos e who teac h discret e mathematic s mentio n th e problem s o f finding euleria n chains an d euleria n close d chains . Som e peopl e tal k abou t thei r history , by describin g th e famou s proble m o f th e Konigsber g bridges , whic h wa s solved b y th e mathematicia n Leonhar d Eule r i n 173 6 an d gav e ris e t o th e subject o f graph theory . (Se e [2] , and se e the articl e b y Newma n [18 ] i n Sci- entific American, an d th e accompanyin g translatio n o f th e origina l memoi r by Eule r [10]. ) Som e peopl e eve n g o beyon d this , t o describ e th e followin g problem (thoug h no t alway s connectin g i t t o euleria n chains) .

T h e "Chines e P o s t m a n Problem" . A mai l carrie r walk - ing a route mus t hi t ever y stree t i n the neighborhoo d an d us e the smalles t amoun t o f time . Wha t rout e shoul d th e carrie r take? Thi s proble m wa s first analyze d usin g grap h theoret - ical method s b y a rea l postma n i n China , Gua n Meig u (th e Get Rea l Rul e (#4 ) an d th e M a t h I s Aliv e R u l e (#6). ) It i s not exactl y th e sam e proble m a s that o f finding a n euler - ian close d chain, sinc e the mai l carrier ca n walk down a stree t a secon d time . However , th e euleria n chai n proble m enter s i n a critica l wa y int o th e solution : I f ther e i s a n euleria n close d chain, thi s give s th e solution . I f not , w e simpl y hav e t o find the smalles t numbe r o f edge s t o cop y s o that i n th e resultin g graph ther e i s a n euleria n close d chain . Se e [15 , 17 , 31] .

While some people teaching discrete mathematics g o as far a s mentionin g the Chines e Postman Problem , i t i s so much better t o go further, fo r instanc e by notin g tha t th e exac t sam e proble m arise s i n street sweeping an d i n snow removal. Certai n street s i n a cit y hav e t o b e swep t o r cleared , an d w e wis h to d o thi s i n th e leas t amoun t o f tim e [38 , 16 , 29 , 31] . Again , w e hav e an illustratio n o f th e W h y D o Thing s Twic e R u l e ( # 3 ) . A s i t turn s out, thes e problem s hav e interestin g complications : Onl y som e street s nee d to b e swep t ever y day ; ther e ar e one-wa y streets ; i t take s muc h longe r t o go dow n a stree t whil e sweepin g i t tha n i t doe s t o g o dow n i t whe n on e i s just passin g through . Thes e complication s ca n b e handled , an d the y lea d to interestin g variation s o f th e Chines e Postma n Proble m an d wonderfu l exercises fo r student s [38] . Her e again , w e hav e illustrate d th e M o d e l i n g Rule ( # 9 ) .

Here i s anothe r proble m tha t i s reall y th e same :

A u t o m a t e d Grap h P l o t t i n g b y C o m p u t e r . W e wis h t o draw a grap h (wit h pre-specifie d verte x locations ) b y com - puter. Whe n w e repea t a n edge , w e nee d t o paus e th e com - puter an d rais e the plotte r pe n of f th e paper . W e draw lot s of copies of th e sam e grap h an d s o would lik e to desig n a wa y of drawing i t whic h use s a s littl e tim e a s possible . Thi s i s agai n

T H E ROL E O F A P P L I C A T I O N S I N T E A C H I N G D I S C R E T E MATHEMATIC S 11 5

the Chines e Postma n Problem . I t ha s moder n practica l ap - plications i n chi p desig n a t IBM , drawin g circui t diagrams , electrical an d wate r network s fo r citie s (i t ha s bee n widel y used i n Bonn , Germany , fo r example) , contro l o f machine s for producin g lithographi c masks , an d s o on . Se e [13 , 26] . Again, w e hav e illustrate d th e W h y D o Thing s Twic e Rule ( # 3 ) an d th e Ge t Rea l R u l e (#4). )

There ar e other , mor e subtl e application s o f euleria n chains . Fo r in - stance, euleria n chain s aris e i n a telecommunication s proble m whic h i s con - cerned wit h ho w to tel l th e positio n o f a rotating roo f antenn a withou t goin g to the roof. Th e solutio n involve s finding so-calle d deBruij n diagrams , whic h also can b e connecte d t o th e desig n o f computing machine s throug h th e the - ory o f shif t registe r sequences . Se e [31 ] fo r a discussion .

How man y peopl e kno w tha t euleria n chain s hav e playe d a crucia l rol e in th e histor y o f molecula r biology ? The y wer e use d i n earl y algorithm s for finding a n RN A chai n give n fragment s o f i t tha t wer e produce d fro m decomposition b y variou s enzymes . Th e first RN A chai n wa s determine d i n 1965 b y R.W . Holle y an d hi s co-worker s a t Cornell , usin g a metho d tha t soon wa s improve d usin g euleria n chain s an d paths . Th e specifi c us e o f eulerian chain s i s a bi t complicated . However , I ca n buil d u p t o i t i n severa l class periods , whic h involv e som e rathe r simpl e bu t beautifu l application s of th e basi c countin g rule s o f combinatorics . (Se e [31] , Section s 2.1 3 an d 11.4.4.)

After I describe , o r a t leas t mention , th e us e o f eulerian chain s i n molec - ular biology , I usuall y lea d int o a discussio n o f th e man y application s o f discrete mathematic s t o moder n molecula r biology . I n particular , I mentio n the importanc e o f grap h theor y an d combinatoric s i n th e Huma n Genom e Project, th e projec t o f mappin g an d sequencin g th e entir e huma n genome . For mor e o n thi s subject , se e fo r exampl e [7 , 14 , 23 , 32 , 39 , 40] . I lik e to giv e specific problem s her e tha t hav e com e ou t o f recen t researc h i n com - putational biology . Som e o f thes e involv e euleria n chain s an d path s (a s fo r example i n connectio n wit h th e "doubl e diges t problem " i n DN A physica l mapping [22]) . Other s involv e a variet y o f question s i n grap h theor y an d combinatorics. Thi s illustrates , once again, th e Frontier s R u l e ( # 5 ) . I also mention th e increasin g collaboratio n betwee n th e biologica l sciences commu - nity an d th e mathematica l science s communit y an d th e realizatio n o n th e part o f biologica l scientist s tha t man y o f thei r problem s ar e basicall y prob - lems that ar e amenabl e t o formulatio n usin g discrete mathematics . I give ex- amples o f th e man y collaboration s betwee n loca l mathematicians/compute r scientists an d loca l biologica l scientist s tha t hav e com e about i n recent year s (the M a t h I s Aliv e R u l e (#6)) .

5. Concludin g Remar k

The mai n messag e o f thi s paper , an d th e mai n reaso n tha t w e wis h t o use application s i n ou r courses , ca n b e summe d u p a s follows . Ther e ar e s o

116 FRED S . ROBERT S

many exciting , relevan t application s o f discret e mathematic s tha t i f yo u ar e a goo d teacher , non e o f you r student s shoul d eve r agai n hav e t o ask : What is mathematics good for?

References

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, an d Stern , R.G. , "Computerize d Algorithm s fo r Orde r Processin g i n Auto - mated Warehousin g Systems, " Int. J. Prod. Res., 2 1 (1983) , 579-586 . Euler, L. , "Th e Konigsber g Bridges, " Sci. Amer., 18 9 (1953) , 66-70 . (Translatio n from 18 t h centur y article. ) Golumbic, M.C. , Algorithmic Graph Theory and Perfect Graphs, Academi c Press , New York , 1980 . Hale, W.K. , "Frequenc y Assignment : Theor y an d Applications, " Proc. IEEE, 6 8 (1980), 1497-1514 . Korte, B. , "Application s o f Combinatoria l Optimization, " i n M . Ir i an d K . Tanab e (eds.), Mathematical Programming: Recent Developments and Applications, K T K Scientific Publishing , Tokyo , an d Kluwe r Academi c Publishers , Dordrecht , 1989 , pp . 1-55. Lander, E.S. , an d Waterman , M.S . (eds.) , Calculating the Secrets of Life: Applica- tions of the Mathematical Sciences in Molecular Biology, Nationa l Academ y Press , Washington, DC , 1995 . Lawler, E.L. , Combinatorial Optimization: Networks and Matroids, Holt , Rinehar t and Winston , Ne w York , 1976 . Liebling, T.M. , Graphentheorie in Planungs-und Tourenproblemen, Lectur e Note s i n Operations Researc h an d Mathematica l System s No . 21 , Springer-Verlag, Ne w York , 1970. Minieka, E. , Optimization Algorithms for Networks and Graphs, Dekker , Ne w York , 1978. Newman, J.R. , "Leonhar d Eule r an d th e Konigsber g Bridges, " Sci. Amer., 18 9 (1953), 66 . Opsut, R.J. , an d Roberts , F.S. , "O n th e Flee t Maintenance , Mobil e Radi o Frequency , Task Assignment , an d Traffi c Phasin g Problems, " i n G . Chartrand , e t al . (eds.) , The Theory and Applications of Graphs, Wiley , Ne w York , 1981 , 479-492.

, "I-Colorings , I-Phasings , an d I-Intersectio n Assignment s fo r Graphs , an d their Applications, " Networks, 1 3 (1983) , 327-345 .

THE ROL E O F APPLICATIONS I N TEACHING DISCRET E MATHEMATIC S 11 7

[21] , "Optima l I-Intersectio n Assignment s fo r Graphs : A Linea r Programmin g Approach," Networks, 1 3 (1983), 317-326 .

[22] Pevzner , P.A. , "DNA Physica l Mappin g an d Alternating Euleria n Cycle s i n Colore d Graphs," Algorithmica, 1 3 (1995), 77-105 .

[23] Pieper , G.W. , "Compute r Scientist s Joi n Biologist s in Genome Project, " SI AM News, January 1989 , 18.

[24] Raychaudhuri , A. , "Optima l Schedulin g o f Subtasks unde r Compatibilit y an d Prece- dence Constraints, " Congr. Numer., 7 3 (1990), 223-234 .

[25] , "Optima l Multipl e Interva l Assignment s i n Frequency Assignmen t an d Traf- fic Phasing," Discr. AppL Math., 4 0 (1992), 319-332 .

[26] Reingold , E.M. , an d Tarjan, R.E. , "On a Greed y Heuristi c fo r Complet e Matching, " SIAM J. Comput., 1 0 (1981), 676-681 .

[27] Rinnoo y Kan , A.H.G. , Machine Scheduling Problems: Classification, Complexity and Computation, Nijhof , Th e Hague, 1976,

[28] Roberts , F.S. , Discrete Mathematical Models, with Applications to Social, Biological, and Environmental Problems, Prentice-Hall , Englewoo d Cliffs , NJ , 1976 .

[29] , Graph Theory and its Applications to Problems of Society, NSF-CBM S Monograph No . 29, Societ y fo r Industria l an d Applie d Mathematics , Philadelphia , 1978.

[30] , "O n the Mobile Radi o Frequenc y Assignmen t Proble m an d the Traffic Ligh t Phasing Problem, " Annals NY Acad. Sci, 319 (1979), 466-483 .

[31] , Applied Combinatorics, Prentice-Hall , Englewoo d Cliffs , NJ , 1984 . [32] (ed.) , Applications of Combinatorics and Graph Theory to the Biological and

Social Sciences, IM A Volume s in Mathematics an d its Applications, Vol . 17 , Springer - Verlag, Ne w York, 1989.

[33] , "Fro m Garbag e t o Rainbows : Generalization s o f Grap h Colorin g an d thei r Applications," i n Y. Alavi , G . Chartrand, O.R . Oellermann, an d A.J. Schwen k (eds.) , Graph Theory, Combinatorics, and Applications, Vol . 2, Wiley, Ne w York, 1991 , pp . 1031-1052.

[34] , "T-Coloring s o f Graphs: Recen t Result s an d Open Problems, " Discr. Math., 93 (1991) , 229-245 .

[35] Stoffers , K.E. , "Schedulin g of Traffic Light s - A New Approach, " Transportation Res., 2 (1968) , 199-234 .

[36] Slowinski , R. , and Weglarz, J . (eds.) , Advances in Project Scheduling, Elsevier , Am- sterdam, 1989.

[37] Tucker , A.C. , "Perfec t Graph s an d an Application t o Optimizing Municipa l Services, " SIAM Rev., 1 5 (1973), 585-590 .

[38] , an d Bodin , L. , " A Mode l fo r Municipa l Street-Sweepin g Operations, " i n W.F. Lucas , F.S . Roberts, an d R.M. Thral l (eds.) , Discrete and System Models, Vol. 3 o f Modules in Applied Mathematics, Springer-Verlag , Ne w York, 1983 , pp. 76-111 .

[39] Waterman , M.S . (ed.) , Mathematical Methods for DNA Sequences, CR C Press, Boc a Raton, FL , 1989 .

[40] Waterman , M.S. , Introduction to Computational Biology: Maps, Sequences, and Genomes, Chapma n an d Hall, 1995.

[41] Zimmer , C , "An d One for the Road," Discover, Januar y 1993 , 91-92 .

D E P A R T M E N T O F M A T H E M A T I C S, C E N T E R FO R O P E R A T I O NS R E S E A R C H ( R U T C O R ) ,

AND C E N T E R F O R D I S C R E TE MATHEMATIC S AN D T H E O R E T I C AL C O M P U T E R SCIENC E (DI -

MACS), R U T G E R S U N I V E R S I T Y , N E W BRUNSWICK , N J 0890 3

E-mail address: f r o b e r t s Q d i m a c s . r u t g e r s . e d u

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Section 3

W h a t I s Discret e M a t h e m a t i c s : Two Perspective s

What I s Discret e Mathematics ? Th e Man y Answer s S T E P H E N B . M A U R E R

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A Comprehensiv e Vie w o f Discret e Mathematics : Chapter 1 4 of th e Ne w Jerse y Mathematic s Curriculu m Framewor k

J O S E P H G . ROSENSTEI N

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

W h a t i s Discret e M a t h e m a t i c s ? T h e M a n y Answer s

Stephen B . Maure r

1. Introductio n

We advocate s o f discret e mathematic s hav e a problem : ther e i s n o agreed-on definitio n o f ou r field! W e ar e eve n wors e of f tha n Suprem e Court justice s debatin g pornography : w e don' t eve n agre e whe n w e se e it! (Ar e fractal s discret e mathematics ? Matrices ? Statistics ? Numbe r the - ory? Proofs ? Real-worl d application s o f hig h schoo l algebra ? Pattern s an d tiling? Constructio n algorithm s i n Euclidea n geometry? ) W e ar e lik e th e blind me n feelin g th e elephant ; eac h describe s hi s ow n beast .

This situatio n i s not necessaril y bad . Whe n a field is not wel l defined , i t can blosso m i n man y directions . Bu t thi s lac k o f definitio n i s differen t fro m the usua l situatio n i n mos t area s o f mathematics . Mos t mathematician s have a prett y clea r ide a wha t algebr a is , o r calculus . (Well , the y use d t o have a clea r ide a abou t calculus! ) An d thoug h on e woul d b e har d pu t t o define mathematic s generally , ther e isn' t to o muc h doub t whe n on e see s it .

However, ther e i s a big difference betwee n definin g discret e mathematic s as a field an d a s a course . Whe n a mathematicia n invent s som e ne w math - ematics, i t i s almos t irrelevan t wha t rubri c w e us e t o classif y it . Th e issu e is whethe r i t i s interestin g an d useful . Bu t course s requir e decision s — i s a particular topi c goin g t o b e include d o r not ? A t th e colleg e level , discret e mathematics course s hav e bee n aroun d fo r 2 0 year s now , an d syllab i hav e tended t o settl e int o a fe w patterns . A t th e schoo l level , discret e mathemat - ics i s stil l quit e ne w an d ther e i s littl e agreemen t o n content .

Thus, t o hel p thin k abou t th e K-1 2 curriculum , i t woul d b e usefu l t o have a singl e definitio n o r descriptio n o f discret e mathematics . Alas , I can' t provide one . Instead , i n th e first par t o f thi s pape r I offe r severa l propose d definitions an d descriptions , an d sho w shortcoming s fo r eac h one .

Having severa l definition s ca n eve n b e helpful , fo r th e followin g reason : when listenin g t o othe r advocate s o f discret e mathematics , i t i s importan t

1991 Mathematics Subject Classification. Primar y 00A35 .

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122 S T E P H E N B . M A U R E R

to catc h o n quickl y t o wha t versio n o f discrete mathematic s the y ar e talkin g about an d t o wha t end s the y ar e promotin g it . Similarly , whe n w e advocat e discrete mathematic s t o others , i t i s importan t t o mak e clea r quickl y wha t version o f discret e mathematic s we ar e talkin g abou t an d t o wha t end s we are promotin g it . Thi s pape r ca n hel p u s identif y th e differen t version s an d goals, an d giv e u s terminolog y t o tal k abou t them .

In recognizin g our differences , w e may recogniz e what i s common a s well. So, i n the final par t o f the paper , I make some suggestions a s to wha t I thin k we migh t agre e shoul d b e par t o f discret e mathematic s i n th e schools , an d what w e migh t agre e t o exclude .

2. Definin g Discret e M a t h e m a t i c s

There ar e two standard approache s t o definin g a branch o f mathematics : specifying propertie s o f th e branc h an d givin g a lis t o f topics . (Mathemati - cians usuall y star t wit h th e forme r approac h bu t ofte n en d u p wit h th e latter.) Let' s explor e bot h approaches .

A t t e m p t s t o defin e discret e m a t h e m a t i c s b y specifyin g proper - ties. Her e ar e severa l "definitions, " eac h followe d b y on e o r tw o difficulties .

Definition 1 : Discret e mathematic s i s finite mathematics , tha t is , th e mathematics o f situation s tha t ca n b e describe d b y finite sets .

This definitio n exclude s al l sorts o f importan t discret e topic s tha t requir e a t least th e se t o f al l natura l numbers : induction , differenc e equations , infinit e graphs, an d forma l languages .

Definition 2 : Discret e mathematics i s the mathematics o f discrete sets, that i s set s whic h hav e hole s betwee n an y tw o elements , a s d o th e natural number s an d th e rationa l numbers .

Many discret e topic s regularl y us e rea l numbers , suc h a s sequences , linea r programming, weighte d graphs , o r gam e theory . Whil e man y o f thes e area s could b e carrie d ou t ove r th e rationa l number s Q (fo r instance , th e theor y of linear programmin g i s unchanged ove r Q) , non e o f them are carried ou t ove r Q, an d som e o f the m canno t b e carrie d ou t a s wel l (e.g. , linea r differenc e equations wouldn' t alway s hav e closed-for m solutions) .

Definition 3 : Discret e mathematic s i s an y mathematic s tha t doesn' t involve limits .

This clai m certainl y ha s som e merit , becaus e continuou s mathematic s cer - tainly doe s involv e limits . Bu t a s a definition thi s clai m i s both to o exclusiv e and to o inclusive . To o exclusive : d o w e refus e t o discus s limit s o f sequence s in a discret e mathematic s course ? D o w e expunge fractals ? D o w e refuse t o mention tha t th e (discrete ) Poisso n distributio n i n probabilit y i s the limi t o f binomial distributions ? To o inclusive : d o w e clai m tha t al l o f abstrac t alge - bra i s part o f discrete mathematics , al l of logic, or fo r tha t matter , almos t al l of schoo l mathematics , sinc e limit s don' t appea r unti l a t leas t pre-calculus ?

W H A T I S D I S C R E T E MATHEMATICS ? T H E MAN Y ANSWER S 12 3

Definition 4 : Discret e mathematic s i s whateve r mathematic s ca n b e done i n a finite numbe r o f steps .

I mus t confes s tha t I use d t o offe r thi s definition , becaus e i t emphasize d that discret e mathematic s i s abou t algorithms . Bu t lif e i s finite, an d s o al l mathematics i s don e i n a finite numbe r o f steps . I n short , thi s i s th e defi - nition a discret e mathematicia n shoul d us e wh o want s t o b e a n intellectua l imperialist an d tak e ove r everyon e else' s field!

Definition 4 ca n b e improve d b y sayin g tha t discret e mathematic s i s mathematics wher e th e objec t o f study , rathe r tha n th e proces s o f studyin g it, i s a n algorith m tha t take s a finite numbe r o f steps . Bu t eve n i f w e coul d make thi s distinctio n precis e (betwee n object s an d th e stud y o f objects) , Definition 4 woul d no t b e goo d enough . Fo r example , bisectio n algorithms , in principle, migh t ru n forever . Shoul d the y b e excluded fro m discret e math ? Typically the y ar e not .

A t t e m p t s t o t o defin e discret e m a t h e m a t i c s b y list s o f topics . There ar e abou t a s man y propose d definin g list s a s ther e ar e discret e math - ematics textbooks . Tabl e 1 give s five lists . Lis t A i s typica l fo r a discret e structures cours e aime d a t compute r scienc e major s i n college . Lis t C i s fo r a finite mathematic s cours e aime d a t colleg e student s intereste d i n socia l science an d business . Course s correspondin g t o thes e list s hav e bee n aroun d for 2 0 years . Lis t B i s fo r a n algorithms-oriente d colleg e cours e o f mor e re - cent vintage . List s D an d E ar e fro m book s fo r hig h schoo l course s [2 , 11] . Lists A an d B ar e likel y t o b e fo r one-yea r courses , list s C , D an d E fo r a semester o f material . Additiona l book s ofte n use d fo r discret e mathematic s in school s ar e liste d i n th e reference s [4 , 5 , 3 , 9 , 10] .

These list s ar e no t all-encompassing . Topic s tha t ar e o n som e othe r list s include fractals, numbe r system s an d numbe r theory , theor y o f computation , simulation, bloc k designs , an d Poly a countin g theory . A numbe r o f term s on man y list s ar e subtopic s o f one s alread y listed . Fo r example , vector s ar e subsumed unde r linea r algebra . Similarly , trees , network s an d networ k algo - rithms com e unde r grap h theory ; semigroup s com e unde r abstrac t algebra ; coding theor y come s unde r combinatoric s and/o r abstrac t algebra .

What positiv e conclusion s ca n b e draw n fro m thes e definition s an d lists ? Anything involvin g finite set s o r abou t finite algorithm s applie d t o discret e sets, an d no t traditionall y covere d i n th e curriculum , i s probabl y discret e mathematics. Anythin g abou t grap h theory , counting , recurrence s o r ele - mentary logi c i s probabl y discret e mathematic s also .

3. Distinguishin g Approache s t o Discret e M a t h e m a t i c s

If ther e ar e s o man y variant s o f discret e mathematics , ca n w e a t leas t group th e variant s i n usefu l ways ? A s w e wil l no w show , on e wa y i s b y emphases, anothe r i s b y goals .

Grouping discret e m a t h e m a t i c s approache s b y e m p h a s e s . T o make wha t w e mea n clearer , w e grou p emphase s i n contrastin g pairs .

124 S T E P H E N B . M A U R E R

Discrete structures vs. problem-solving methodologies. A structure s course emphasize s theorem s abou t propertie s o f variou s constructs . Fo r instance, a structure s cours e migh t emphasiz e tha t al l Euleria n graph s ar e connected an d hav e al l vertice s o f eve n degree . A problem-solvin g cours e emphasizes ho w discrete mathematic s give s concepts an d technique s t o solv e problems. Fo r instance , suc h a cours e migh t emphasiz e ho w t o tel l i f a problem shoul d b e modele d b y a graph , ho w t o tel l i f that proble m i s solve d if th e grap h i s Eulerian, an d finally , ho w to tes t i f a give n grap h i s Eulerian .

This distinctio n i s similar t o th e on e betwee n rea l analysi s an d calculus . In th e former , yo u emphasiz e th e structur e of , say , th e se t o f differentiabl e functions (e.g. , i t i s close d unde r addition) , wherea s i n calculu s yo u stud y the derivativ e an d ho w i t ca n hel p yo u solv e problems .

Narrow clientele vs. broad clientele. I f a cours e i s offered a s a servic e fo r a particular group , for example, if most o f the students ar e planning to majo r in compute r science , the n typicall y th e cours e wil l emphasiz e application s of interes t t o tha t group . O n th e othe r hand , i f th e student s hav e a variet y of interests , th e cours e shoul d offe r a variet y o f applications . On e ca n argu e

T A B L E 1 . Fiv e list s o f topic s fo r discret e mathematic s course s

List A Logic an d circuit s Sets, relations , function s Induction Counting

(combinatorics recurrences generating functions )

Graph theor y Boolean algebr a Automata Abstract algebr a (intro ) Partially ordere d set s

List B Algorithms an d

algorithmic languag e Induction, iteration , recursio n Graph theor y Difference equation s Probability Logic Linear algebr a Analysis an d verificatio n

of algorithm s Sequences an d limit s Numerical Analysi s

List C Logic Counting (elementary ) Finite probabilit y Linear programmin g

and game s Statistics Social scienc e an d

business application s Modeling

List D Election theor y Fair divisio n Matrices Graphs Counting Probability Recursion

List E Logic Integers an d

polynomials Recursion an d

induction Combinatorics Graphs an d

circuits Vectors

WHAT I S DISCRETE MATHEMATICS ? TH E MAN Y ANSWER S 12 5

that eve n i f the clientel e is narrow, a broad cours e should b e given ; th e need s of a clien t grou p toda y ma y no t b e thei r need s tomorrow. I n an y event, bot h types o f discret e mathematic s course s exist , a s indicate d b y th e topi c list s in Tabl e 1 .

Structural vs. algorithmic. Th e first discret e mathematic s course s wer e about structure . Fo r instance , plana r graph s wer e characterize d a s thos e that contai n n o "homeomorph " o f eithe r K§, th e complet e grap h o n 5 ver - tices, o r if3,3 , th e "utilit y graph. " Ther e wa s littl e discussio n o f whethe r there ar e efficient way s to check if a graph meet s such a characterization, tha t is, whethe r ther e ar e goo d verificatio n algorithms . Indeed , algorithm s wer e simply no t a n objec t o f stud y i n th e course . Thi s i s a bi t odd , sinc e thes e early course s wer e give n mostl y fo r compute r scienc e students , fo r who m algorithms ar e the objec t o f study . Perhap s th e feelin g wa s tha t compute r science student s go t enoug h stud y o f algorithm s i n thei r othe r courses , an d that mathematic s course s fo r compute r scienc e shoul d mee t th e approva l o f mathematicians b y stickin g t o wha t wa s perceive d a s "rea l mathematics, " that is , structure .

Pure vs. applied. Discret e mathematic s ha s man y applications . Yet , just a s i n othe r branche s o f mathematics , on e ca n giv e a course , eve n a very interestin g course , o n purel y mathematica l aspect s o f th e topic . Ther e are discret e mathematic s textbook s tha t d o this , an d other s tha t generat e everything ou t o f applications .

Grouping discret e m a t h e m a t i c s approache s b y goals . Trut h b e told, mos t o f u s promotin g discret e mathematic s hav e som e genera l goa l in min d tha t goe s beyon d th e particula r mathematics . Sometime s th e rea l agenda i s a s broa d a s revampin g wha t schoo l i s lik e o r wha t educatio n i s all about . I n short , sometime s discret e mathematic s i s th e mean s rathe r than th e end . Thi s i s no t bad , bu t i t shoul d b e acknowledged . Belo w w e state goal s tha t hav e bee n advocate d a s reason s fo r teachin g mor e discret e mathematics. W e star t wit h mathematica l goal s an d mov e t o mor e genera l educational goals .

To introduce proofs and abstraction. Mos t colleg e student s see m t o hav e poor proo f an d abstractio n skills . Wher e ca n the y lear n thes e skill s wel l — that is , wha t topic s i n mathematic s wil l convinc e student s o f th e nee d fo r proofs an d ye t hav e proofs tha t ar e not to o har d fo r beginners ? Man y peopl e feel tha t part s o f discrete mathematic s fill the bill . I n particular , elementar y number theor y an d mathematica l inductio n ar e mentioned .

To introduce algorithms and recursion. Certai n mathematica l concept s and paradigm s ar e give n shor t shrif t i n traditiona l studies , fo r instance , algorithms an d recursion . Bot h o f thes e concept s wer e aroun d lon g befor e discrete mathematics . A s fo r algorithms , student s hav e alway s ha d t o us e them (bu t rarel y thin k abou t them) . A s fo r recursion , wh o hasn' t hear d of "reduc e t o th e previou s case" ; recursio n i s a carefu l formulatio n o f thi s

126 S T E P H E N B . M A U R E R

idea. Discret e mathematic s bring s algorithm s an d recursio n t o th e fore , b y making the m object s o f stud y an d providin g precis e way s t o discus s them .

To emphasize applications. Fe w student s ar e turne d o n b y pur e mathe - matics. W e may regre t thi s — most o f us were turned o n by pur e mathemat - ics — bu t w e can' t den y it . T o tak e a mor e positiv e attitude , mathematic s has a doubl e appeal : i t i s simultaneously beautifu l and useful. I n an y event , for mos t student s t o begi n t o appreciat e mathematics , the y hav e t o se e tha t it i s useful . Teachin g discret e mathematic s ca n sho w the m this , becaus e s o many rea l application s ar e accessibl e a t a n elementar y level .

To introduce modeling. Traditionally , i n bot h schoo l an d college , doin g mathematics wa s a proces s tha t bega n wit h a mathematicall y formulate d problem an d ende d wit h th e mathematica l solutio n o f tha t proble m - eve n if th e proble m wa s applied . Modelin g emphasize s tha t thi s traditiona l vie w is bu t on e ste p (ofte n th e easiest ) o f several :

1. A proble m i s give n i n amorphou s real-worl d term s 2. Th e proble m i s idealize d int o a mathematica l form , th e initia l mode l 3. Tha t mode l i s solve d (thi s i s th e traditiona l activity ) 4. Th e result s ar e interprete d i n th e origina l contex t 5. Th e cycl e i s repeate d unti l th e solutio n i s deeme d helpful .

Having mathematic s presente d i n thi s broade r wa y make s man y mor e stu - dents valu e it . Man y mathematician s fee l discret e problem s ar e th e bes t for introducin g modeling ; modelin g i s b y natur e complicate d an d discret e models provid e som e o f th e simple r instances .

To introduce operations research. Ther e ar e man y sort s o f optimizatio n that canno t b e touche d b y calculus , fo r instance , maximizin g flo w i n a net - work, minimizin g th e numbe r o f color s neede d fo r a map , an d maximizin g a linea r functio n whe n th e domai n i s restricte d b y inequalities . Man y suc h optimization problem s ar e intimatel y tie d t o th e modelin g approac h an d ar e highly relevan t t o busines s an d management . Ye t unti l recentl y mos t stu - dents, eve n a t th e colleg e level , neve r hear d tha t mathematic s ha s anythin g to sa y abou t optimizatio n excep t fo r th e ver y specialize d sor t i n calculus . Discrete mathematic s i s wher e the y ca n lear n th e goo d news .

To entice more students into a mathematical sciences major. Man y stu - dents ente r colleg e wit h thei r mind s almos t mad e u p abou t a major . Field s not see n befor e colleg e attrac t fe w students . Therefore , no t enoug h student s will choos e mathematica l scienc e major s wit h a discret e flavor unles s the y see som e discret e mathematic s i n school .

To introduce computers into school mathematics. Ther e ar e man y con - texts fo r introducin g computer s int o mathematic s class , fo r instance , graph - ing functions , doin g algebr a manipulations , an d drill . However , discret e mathematics i s probabl y th e mos t natura l contex t fo r introducin g comput - ers, sinc e discret e mathematic s is th e mathematic s o f computation .

To give students something fresh and relevant to them. To o man y stu - dents hav e bee n turne d of f t o mathematic s a s the y kee p seein g mor e o f th e same, wher e "th e same " i s usuall y senseles s manipulatio n (o r s o i t seem s t o

WHAT I S DISCRET E MATHEMATICS ? TH E MAN Y ANSWER S 12 7

them). Fo r instance , muc h o f high schoo l mathematic s seem s t o b e repeate d algebraic computatio n concernin g rates , time , distance , area , volume , etc . If a studen t i s successfu l an d get s t o calculus , h e o r sh e doe s th e sam e cal - culations ove r again , onl y mor e o f them , t o handl e th e optimizatio n aspec t as well .

Many educator s fee l student s ar e turne d of f because the y se e s o muc h repetition, an d tha t the y won' t b e turne d of f i f instea d the y se e somethin g completely different , especiall y i f i t i s obviousl y relevant . Discret e math - ematics ca n certainl y b e completel y differen t an d relevant . Fo r instance , the mathematic s o f fai r division , apportionmen t an d electio n method s i s a n eye-opener. Her e i s somethin g importan t i n th e struggle s ove r equit y i n to - day's world , an d mos t student s woul d neve r hav e though t mathematic s ha s something t o sa y abou t socia l equity .

To give students a chance to be creative and do research. W e ar e tol d that, i n th e future , mos t employmen t wil l requir e creativ e approache s t o open-ended problems . Therefore , educatio n shoul d involv e suc h creativ e work. I n mathematic s an d science , thi s mean s research . Som e educator s g o further an d sugges t tha t becomin g activ e junio r researcher s i s th e primar y thing kid s shoul d d o i n school . I n mos t science s i t i s possibl e t o sho w students wha t researc h i s like , an d perhap s ge t the m activel y involve d i n their ow n research , earl y on . Usin g discret e topics , th e sam e ca n b e don e i n mathematics. Thi s i s because ther e ar e part s o f discret e mathematic s wher e it i s easy t o stat e problem s tha t ar e beyon d wha t th e student s hav e learne d how t o solv e (o r i n som e cases , beyon d wha t anybod y ha s solved) .

To introduce important, active areas of mathematics. Whethe r o r no t the goa l shoul d b e t o mak e kid s researchers , certainl y the y deserv e t o b e shown wha t i s goin g o n a t th e frontiers . Discret e mathematic s i s on e are a of mathematic s wher e thi s i s possible .

To promote experimental mathematics. Par t o f th e reaso n student s ca n do researc h i n scienc e muc h earlie r tha n i n traditiona l mathematic s i s be - cause yo u ca n mak e progres s i n scienc e b y experiments , eve n i f yo u hav e not develope d a theory , wherea s i n traditiona l mathematic s th e onl y wa y to mak e progres s wa s t o conceiv e an d prov e theorems . Bu t now , wit h computers, ther e i s opportunit y fo r experimenta l mathematics , especiall y within discret e mathematics . Th e intellectua l effor t tha t goe s int o creat - ing program s t o generat e mathematica l dat a i s substantial , an d fro m thi s data student s ca n mak e conjecture s tha t the y woul d no t arriv e a t otherwise . Professional mathematician s ar e makin g muc h mor e us e o f experiments . S o should students .

To promote cooperative learning and other new classroom approaches. Traditionally, schoolwor k i s don e alone . I n mathematics , ther e ha s bee n a very competitiv e aspec t t o thi s approach , a s indicate d b y th e interes t i n mathematics competition s an d th e emphasi s o n individua l scores . Jus t a s there i s no w mor e grou p wor k a t th e professiona l leve l i n mathematics , s o can ther e b e mor e grou p wor k i n school. Grou p wor k i s most appropriat e fo r

128 STEPHEN B . MAURE R

larger, open-ende d problems . Discret e mathematic s i s a n excellen t sourc e at th e schoo l leve l fo r suc h problems .

There ar e man y othe r way s in which som e classrooms toda y ar e ver y dif- ferent fro m traditiona l classrooms . Tak e assessmen t fo r example . I n som e places test s hav e largel y bee n replace d b y broade r method s suc h a s portfo - lios. Fo r a portfoli o t o asses s mor e tha n a test , th e portfoli o mus t involv e items mor e open-ende d tha n traditiona l tes t problems . Thu s onc e agai n th e opportunity discret e mathematic s provide s fo r open-ende d problem s make s it a goo d contex t i n whic h t o introduc e th e ne w approach .

To teach students to think. Traditiona l schoo l mathematic s emphasize s technique, techniqu e tha t ca n b e mastere d i n a mechanica l way . Thu s man y students hav e cope d wit h mathematic s b y learnin g ho w t o "tur n th e crank " instead o f learning ho w t o think . Mechanica l strategie s ar e no t s o successfu l in discret e mathematics , fo r ther e ar e man y fewe r part s o f discret e mathe - matics tha t ca n b e routinized . Fo r instance , ther e ar e endles s varietie s o f counting problems . Also , th e firs t tim e a studen t see s a grap h theor y prob - lem, n o previousl y learne d solutio n metho d wil l hel p directly . Thi s lac k o f standard technique s ca n hav e a downside : student s ma y ge t frustrate d an d give up. Bu t a s long a s the difficult y leve l of material i s carefully monitored , the lac k o f standar d technique s ca n mak e student s think .

Some word s o f caution : Som e o f th e goal s jus t liste d ar e contradic - tory. Fo r instance , topic s tha t ar e relevan t t o schoo l student s o r whic h pro - vide accessibl e unsolve d problem s ar e ofte n no t particularl y dee p o r activ e mathematics. Example : man y discret e course s a t th e schoo l leve l includ e substantial materia l o n th e theor y o f elections . However , thi s i s not a large , very active , o r centra l are a o f discret e mathematics ; i t doe s no t hav e man y connections t o other part s o f discrete mathematic s an d th e solutio n method s do no t generaliz e t o othe r areas . Theor y o f election s i s rarel y include d i n college discret e mathematic s courses , an d i t woul d no t b e include d i n schoo l courses i n orde r t o mee t th e previousl y describe d goals , fo r instance , o f in - troducing proof s an d abstraction , introducin g algorithm s an d recursion , o r introducing activ e area s o f mathematics .

Also, t o achiev e man y o f thes e goal s i t i s no t necessar y t o us e discret e mathematics. W e shoul d promot e an y goa l w e fee l i s important , bu t w e should no t equat e discret e mathematic s wit h thos e goals . T o d o s o onl y obscures th e issue s an d hinder s th e effort .

4. Suggestion s fo r C o m m o n Groun d

Let m e mak e som e proposals. 1 I sai d a t th e star t tha t on e purpos e o f describing th e man y meaning s o f discret e mathematic s i s s o tha t eac h o f u s can mak e ou r position s clea r t o others . S o le t m e identif y m y views . F m primarily intereste d i n discret e mathematic s becaus e o f it s content , no t a s

1 These proposal s wer e wel l receive d a t th e conference . O f course , it' s eas y t o b e wel l received i f yo u ar e sufficientl y vague !

WHAT I S DISCRET E MATHEMATICS ? TH E MAN Y ANSWER S 12 9

a mean s toward s pedagogica l goals . Th e conten t tha t interest s m e (a t leas t for introductor y courses ) i s not th e forma l structur e bu t rathe r th e concepts , the problem-solvin g paradigm s (lik e recursion) , an d th e rol e o f algorithm s (see [6 , 7]) . I hav e a broa d audienc e i n mind , no t jus t mathematic s an d computer scienc e majors .

Also, I a m uncomfortabl e wit h th e ton e o f muc h pedagogica l discussio n in th e mathematic s communit y today . I don' t thin k tha t th e sol e rol e o f mathematics educatio n i s to ge t student s t o think , o r tha t experimentatio n is central (e.g. , student s shoul d discove r al l ke y idea s fo r themselve s throug h experimentation, an d an y topi c fo r whic h experimenta l confirmatio n ca n b e obtained i s appropriat e t o study) , o r tha t ever y topi c taugh t shoul d b e on e linked t o real-worl d applications . I fee l ther e ha s bee n to o muc h bashin g o f traditional methods ; ther e i s muc h goo d i n them , a t leas t i n th e hand s o f good teachers . I do feel there i s need fo r pedagogica l change . I t i s incumben t on u s al l t o vers e ourselve s i n ne w method s an d giv e the m a fai r try ; bu t i t remains t o b e see n wha t th e righ t mi x o f ol d an d ne w wil l be .

Because I tak e thi s view , I limi t m y proposal s fo r commo n groun d t o content.

Principles fo r selectin g discret e m a t h e m a t i c s topic s fo r schools . Some discrete mathematics is appropriate in schools for each student.

This i s i n fac t a principl e o f th e NCT M Standard s [8] , wher e example s are give n o f discret e mathematic s topic s appropriat e fo r student s a t variou s grade levels .

Relevance to calculus should not be the main criterion for selecting math- ematics to teach in school. Th e traditional mean s fo r decidin g what t o put i n the school curriculum wa s "i s it goo d backgroun d fo r calculus? " T o maintai n this poin t o f vie w severel y crimp s an y change , an d besides , mos t student s are (o r wil l be) a s likel y t o tak e som e sor t o f discret e mathematic s i n colleg e as continuou s mathematics .

Concepts from some non-traditional areas of mathematics belong in the student repertoire, beginning early, sometimes in elementary school. Som e discrete topes , no t traditionall y taugh t i n schools , shoul d b e taugh t there , sometimes beginnin g i n elementar y school . Tabl e 2 give s m y proposals .

T A B L E 2 . Discret e Mat h Topic s fo r School s

"Definite" "Maybes " algorithmic languag e compute r programmin g graph theor y counting/combinatoric s probability an d statistic s logi c an d proo f concept s vectors an d matrice s modelin g

recursion an d iteratio n

For instance , grap h theor y i s a "Definite " becaus e s o many situation s ca n b e pictured wit h graph s — an y binar y relatio n (e.g. , adjacenc y o f countries) ,

130 S T E P H E N B . M A U R E R

any networ k problem , an y proble m abou t transitio n betwee n configuration s (e.g., almos t an y puzzl e wher e yo u hav e t o mov e pieces) . Thus , student s ought t o becom e familia r wit h th e concep t o f a grap h itsel f an d wit h variou s properties a grap h ca n have , suc h a s bein g connected .

The importanc e o f probability/statistic s an d matri x algebr a i s b y no w well known (se e the Standard s [8]) , but algorithmi c languag e ma y nee d som e explanation. Thi s refer s t o th e sor t o f language neede d t o discus s algorithm s in precis e ways . Compute r scientist s refe r t o suc h languag e a s pseudocode. Students nee d t o b e familia r wit h concept s lik e loo p (i.e. , for-next ) an d if - then statements , an d the y nee d t o hav e terminolog y t o us e suc h concept s carefully.

Why the n i s compute r programmin g a "Maybe" ? Certainl y student s should us e computer s — ther e i s muc h goo d mathematica l softwar e — bu t writing program s i n a compute r languag e i s anothe r matter , eve n i f thes e programs ar e merel y translation s o f idea s th e student s hav e alread y ex - pressed i n algorithmi c language . I n programmin g ther e ar e alway s s o man y technical detail s one can ge t hun g u p on. Suc h implementatio n o f algorithm s might bes t b e lef t optional , o r lef t fo r a compute r scienc e course .

As fo r counting , o f cours e student s shoul d d o some , an d wil l d o som e as par t o f probability , bu t a detaile d stud y o f forma l countin g method s i s what I classif y a s "Maybe" . Som e student s lov e t o count , other s regar d i t as borin g abstraction . Fo r mos t students , thi s i s perhaps bes t lef t t o colleg e discrete mathematics .

As fo r logi c an d proof , thes e idea s mus t appea r a t leas t informally ; th e experimental approac h shoul d no t pus h the m ou t entirely . Bu t t o presen t them explicitl y an d a t length , a s i n traditiona l Euclidea n geometr y courses , may hav e th e sam e stultifyin g effec t o n man y student s a s tha t cours e ha s had. 2

As fo r modeling , agai n m y concer n i s wit h a ful l head-o n approach . Fo r instance, ever y tim e on e turn s a n applie d proble m int o a grap h proble m on e is doin g modeling , an d I a m al l fo r this . Bu t i t i s probabl y to o muc h t o elaborate o n th e explici t stage s o f modelin g (a s describe d unde r goa l 4) , o r to dea l wit h al l th e length , detail , specia l cases , an d partia l result s o f rea l modeling o f rea l problems .

Recursion i s m y ow n hobbyhorse , bu t yo u ca n d o quit e wel l withou t it unti l th e poin t wher e yo u ar e seriou s abou t "algorithmics " - no t jus t using algorithmi c construct s bu t actuall y creating , verifyin g an d analyzin g the efficienc y o f algorithm s — an d thes e activitie s ma y no t b e appropriat e until lat e hig h schoo l o r college . T o th e exten t tha t on e i s doin g recursio n when on e devise s a recurrenc e relatio n fo r a sequenc e (say , th e formul a fn — fn-i + fn-2 fc> r th e Fibonacc i numbers) , the n o f cours e on e shoul d introduce recursio n early . Bu t a s fo r recursiv e algorithms , o r fo r proof s

2In discussion , som e conferees wer e inclined t o mov e logi c and proo f t o th e "Definites" , and t o mov e recursio n also . I n schoo l ther e ar e to o fe w place s t o practic e proofs , especiall y with th e declin e o f Euclidea n geometry .

WHAT I S DISCRETE MATHEMATICS ? TH E MAN Y ANSWER S 13 1

obtained b y firs t restatin g a proble m i n a recursiv e formulation , thi s migh t be postponed .

One migh t lis t severa l mor e "Maybes" , bu t pleas e not e th e sor t o f thing s from Tabl e 1 that I hav e lef t out . First , I hav e lef t ou t topic s o f interes t t o special group s only , e.g. , automat a (compute r scientists) , circuit s (electrica l and compute r engineers) , busines s applications . Schoo l student s haven' t o r shouldn't defin e themselve s s o narrowly, an d schoo l course s shoul d no t cate r to narro w interests . Second , I hav e lef t ou t abstrac t topics , e.g. , set s an d relations, abstrac t algebra . Third , I hav e lef t ou t mino r area s o f discret e mathematics tha t hav e fe w tie s t o majo r areas , e.g. , electio n theory , thoug h I realiz e there ar e ferven t champion s o f such area s becaus e o f their relevanc e to rea l lif e (se e goa l 8 earlier) .

W h a t genera l impression s shoul d b e fostered . Returnin g t o prin - ciples, m y las t concer n i s wit h wha t peopl e retai n afte r thei r mathematic s education ha s ended . Someon e ha s sai d tha t learnin g i s wha t remain s afte r detailed technique s hav e bee n forgotte n an d fundamenta l concept s hav e got - ten rusty . Le t u s refe r t o thi s remainde r a s general impressions. W e shoul d strive har d t o instil l ou r student s wit h certai n correc t genera l impressions , especially the majorit y o f our student s wh o will not us e mathematics directl y in thei r wor k bu t wh o wil l nee d t o hav e som e sens e o f ho w other s ar e usin g mathematics fo r them . Fo r instance , al l student s shoul d ge t th e genera l impression tha t mathematic s i s ver y useful , thoug h I fea r tha t man y leav e school wit h th e contrar y impressio n tha t mathematic s i s a useles s sorcer y with x , y an d z.

Students should leave school with several general impressions about dis- crete mathematics:

• Mathematica l model s ca n b e continuou s o r discrete . • Muc h optimizatio n doe s no t us e calculus . • Computatio n an d th e us e o f computer s involve s interestin g mathe -

matics. • A key theme i n mathematics i s the metho d o f reducing to the previou s

case (recursion) .

W h a t schoo l discret e m a t h e m a t i c s shoul d no t b e . Th e guideline s above leav e a lo t o f room , s o le t m e narro w thing s somewha t b y suggestin g some restrictions . A hig h schoo l discret e mathematic s cours e shoul d not b e

• A computer scienc e oriente d cours e — ther e i s a muc h broade r clien - tele.

• Bille d a s a n advance d placemen t cours e — ther e isn' t suc h a n ad - vanced placemen t tes t an d thoug h th e Colleg e Boar d ha s considere d it, on e isn' t planne d [1] .

• To o forma l (discret e structures ) — thi s i s no t appropriat e a t th e school level .

132 STEPHEN B . MAURE R

5. Concludin g R e m a r k s

What i s discret e mathematics ? I f yo u wante d a 30-secon d definition , I hav e lef t yo u n o bette r of f tha n befor e yo u rea d thi s article . Bu t i f yo u sought som e examples, the flavor, an d th e goals of discrete mathematics, an d you wante d t o recogniz e th e differen t varieties , the n I hop e I hav e helped . If yo u wante d som e idea s fo r wha t t o includ e fro m discret e mathematic s i n the schools , I hop e I hav e helpe d yo u a s well .

References

[1] Bailey , Harol d F. , "Th e Statu s o f Discret e Mathematic s i n th e Hig h Schools" , thi s volume.

[2] Crisler , Nancy , Patienc e Fishe r an d Gar y Froelich , Discrete Mathematics Through Applications, W . H . Freema n fo r COMA P (Consortiu m fo r Mathematic s an d it s Ap - plications), Ne w York , 1994 .

[3] COMAP , For All Practical Purposes: Introduction to Contemporary Mathematics, 3rd ed. , W . H . Freeman , Ne w York , 1994 .

[4] Cozzens , Margare t B. , an d Richar d D . Porter , Mathematics with Calculus, D. C . Heath , Lexingto n MA , 1987 .

[5] Dossey , Joh n A. , Alber t D . Otto , Lawrenc e E . Spence , an d Charle s Vande n Eynden , Discrete Mathematics, 2n d ed. , Scott , Foresman , Glenvie w IL , 1993 .

[6] Maurer , Stephe n B. , an d Anthon y Ralston , Discrete Algorithmic Mathematics, Addison-Wesley, Readin g MA , 1991 .

[7] "Algorithms : Yo u Can' t Teac h Discret e Mathematic s withou t Them" , Dis- crete Mathematics Across the Curriculum, K-12, 199 1 NCTM Yearboo k (Margare t J . Kenney an d Christia n R . Hirsch , eds.) , NCTM , Resto n VA , 1991 , pp. 195-206 .

[8] NCTM , Curriculum and Evaluation Standards for School Mathematics, NCTM , Re - ston VA , 1989 .

[9] Nort h Carolin a Schoo l o f Scienc e an d Mathematic s (G . Barret t e t al.) , Contemporary Precalculus through Applications, Janso n Publications , Providenc e RI , 1991 .

[10] Sandefur , Jame s T. , Discrete Dynamical Systems, Oxfor d Universit y Press , Ne w York , 1990.

[11] Universit y o f Chicag o Schoo l Mathematic s Projec t (A . Peressin i e t al.) , Precalculus and Discrete Mathematics, Scott-Foresman , Glenvie w IL , 1991 .

D E P A R T M E N T O F MATHEMATIC S AN D STATISTICS , SWARTHMOR E C O L L E G E , SWARTH -

MORE P A 19081-139 7 E-mail address: smaurerl@cc.swarthmore.ed u

DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

A Comprehensiv e Vie w o f Discret e M a t h e m a t i c s : C h a p t e r 1 4 o f t h e Ne w Jerse y M a t h e m a t i c s

C u r r i c u l u m Framewor k

Joseph G. Rosenstei n

Introduction

This articl e contain s th e chapte r o f th e New Jersey Mathematics Cur- riculum Framework 1 whic h deal s wit h discret e mathematics . Th e firs t thre e pages o f the articl e describe s wha t thi s documen t i s and wh y i t wa s written .

On Ma y 1 , 1996 , th e Ne w Jerse y Boar d o f Educatio n adopte d cor e cur - riculum conten t standard s i n seve n conten t areas , includin g mathematics .

These standard s describ e wha t al l Ne w Jerse y student s nee d t o kno w and b e abl e t o d o a t th e en d o f grade s 4 , 8 , an d 12 . Statewid e assessment s reflecting thes e standard s ar e bein g develope d a t thes e grad e levels , an d students wil l b e expecte d t o demonstrat e tha t the y mee t thes e standard s i n order t o graduat e fro m hig h school .

The standard s fo r mathematic s include s a discret e mathematic s stan - dard; thu s al l Ne w Jerse y student s wil l b e expecte d t o demonstrat e under - standing an d proficienc y i n discret e mathematics .

The developmen t an d adoptio n o f standard s extende d ove r a perio d o f three years , and , a s Directo r o f th e Ne w Jerse y Mathematic s Coalition , I was ver y muc h involve d a t ever y ste p alon g th e way . Th e mathematic s standards represen t wha t Ne w Jerse y mathematic s educator s believ e ar e high achievabl e standard s fo r al l student s i n th e state .

How wil l Ne w Jerse y teacher s ensur e tha t thei r student s ca n mee t thes e standards? Durin g th e pas t fou r years , th e Ne w Jerse y Mathematic s Coali - tion, workin g in collaboration wit h th e New Jersey Departmen t o f Educatio n and wit h a n Eisenhowe r gran t fro m th e Unite d State s Departmen t o f Educa - tion, ha s develope d a resourc e book , th e New Jersey Mathematics Curricu- lum Framework] thi s 688-pag e documen t wa s develope d t o assis t teacher s

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 . 1 Rosenstein, Josep h G. , Jane t H . Caldwell , an d Warre n D . Crown , New Jersey Math-

ematics Curriculum Framework, Ne w Jerse y Mathematic s Coalition , 1996 .

133

https://doi.org/10.1090/dimacs/036/14

134 JOSEPH G . ROSENSTEI N

and administrator s i n implementin g th e mathematic s standard s a t bot h th e classroom an d th e distric t level . Th e preliminar y versio n wa s publishe d i n Spring 1995 , and a revise d versio n i n Decembe r 1996 . Th e preliminar y ver - sion include d th e contribution s o f man y Ne w Jerse y educators ; th e revise d version incorporate d th e suggestion s o f man y reviewer s an d reflecte d th e standards adopte d b y th e Board .

I a m please d t o hav e spearheade d an d directe d thi s effort ; w e hav e pro - duced a valuabl e guid e fo r Ne w Jerse y teacher s and , throug h it s availabilit y on th e Worl d Wid e Web, 2 fo r thos e o f othe r states . Th e New Jersey Math- ematics Curriculum Framework i s no t intende d t o b e a curriculum ; rathe r it i s intende d t o b e a structur e (i.e. , a "framework" ) aroun d whic h a dis - trict ca n buil d it s own curriculu m (o r curricula) . Thi s particula r framework , however, provide s muc h mor e detai l abou t th e conten t o f K-1 2 mathematic s than an y othe r stat e framewor k o f whic h I a m aware ; fo r tha t reason , i t should b e a valuabl e resourc e t o al l teacher s o f mathematics .

What shoul d student s b e expecte d t o kno w an d b e abl e t o do ? Th e discrete mathematic s standard , lik e th e othe r mathematic s standard s (an d those i n othe r conten t areas) , consist s o f a genera l statemen t abou t discret e mathematics followe d b y fiv e o r si x statements , calle d "cumulativ e progres s indicators", whic h describ e wha t student s shoul d b e abl e t o d o a t eac h o f the thre e grad e levels . Th e discret e mathematic s standar d an d cumulativ e progress indicator s appea r a t th e en d o f thi s Introduction .

How wil l teacher s b e abl e t o reflec t thes e indicator s i n thei r curricula ? The discret e mathematic s chapte r o f th e Framework (lik e eac h o f th e othe r chapters) i s intende d t o respon d t o thi s question . Th e chapte r consist s o f a K-1 2 overvie w o f discret e mathematics , followe d b y section s addressin g five differen t grad e levels ; fo r eac h grad e leve l ther e i s a (self-contained ) overview o f discret e mathematic s fo r tha t grad e level , followe d b y a numbe r of classroo m activitie s tha t illustrat e ho w eac h indicato r coul d b e addresse d at tha t grad e level . Thes e material s ar e arrange d i n thi s articl e i n th e following sections :

1. Grade s K-1 2 Overvie w 2. Grade s K- 2 Overvie w 3. Grade s K- 2 Indicator s an d Activitie s 4. Grade s 3- 4 Overvie w 5. Grade s 3- 4 Indicator s an d Activitie s 6. Grade s 5- 6 Overvie w 7. Grade s 5- 6 Indicator s an d Activitie s 8. Grade s 7- 8 Overvie w 9. Grade s 7- 8 Indicator s an d Activitie s

10. Grade s 9-1 2 Overvie w 11. Grade s 9-1 2 Indicator s an d Activitie s

2 h t t p : / / d i m a c s. r u t g e r s. e d u / nj _ m a t h _ c o a l i t i o n /f ramework . h t m l/

A C O M P R E H E N S I V E VIE W O F D I S C R E T E MATHEMATIC S 13 5

Note tha t becaus e th e material s fo r eac h grad e leve l ar e self-contained , there i s considerabl e overla p betwee n th e overview s (eve n numbere d sec - tions). Not e als o tha t al l reference s fo r eac h grad e leve l ar e provide d a t th e end o f th e od d numbere d sections .

The activitie s i n thi s chapte r ar e base d o n activitie s use d b y teacher s i n the DIMACS-sponsore d an d NSF-funde d Leadershi p Progra m i n Discret e Mathematics, whic h I have directe d sinc e it s inceptio n i n 198 9 (se e article i n this volume) . Th e organizatio n o f discret e mathematic s int o fiv e area s an d the lis t o f indicators , on e fo r eac h are a a t eac h grad e level , emerge d fro m a series o f discussion s i n 199 3 b y Rutger s Universit y facult y associate d wit h DIM ACS. Althoug h I hav e bee n responsibl e fo r th e selectio n an d writin g o f the activities , a s well as the overal l organization o f the material , I would lik e to acknowledg e th e assistanc e I receive d fro m a numbe r o f people , includin g many participants i n the Leadership Program, wh o reviewed an d commente d on draft s o f thi s chapter . Th e expectatio n i s that , throug h th e wonder s o f the Web , th e entir e Framework an d thi s chapte r o n discret e mathematic s i n particular wil l continu e t o evolve .

And now , Chapte r 1 4 of the New Jersey Mathematics Curriculum Frame- work, whic h addresse s th e followin g standar d an d cumulativ e progres s indi - cators o f th e New Jersey Core Curriculum Content Standards:

All student s wil l appl y th e concept s an d method s o f discret e math - ematics t o mode l an d explor e a variet y o f practica l situations .

Cumulative Progres s Indicator s

B y t h e en d o f Grad e 4 , students : 1. Explor e a variet y o f puzzles , games , an d countin g problems . 2. Us e network s an d tre e diagram s t o represen t everyda y situations . 3. Identif y an d investigat e sequence s an d pattern s foun d i n nature , art ,

and music . 4. Investigat e ways to represent an d classif y dat a accordin g to attributes ,

such a s shape o r color , an d relationships , an d discus s th e purpos e an d usefulness o f suc h classification .

5. Follow , devise , an d describ e practica l list s o f instructions .

Building u p o n knowledg e an d skill s gaine d i n t he precedin g grades , by t h e en d o f Grad e 8 , students :

6. Us e systemati c listing , counting , an d reasonin g i n a variet y o f con - texts.

7. Recogniz e commo n discret e mathematica l models , explor e thei r prop - erties, an d desig n the m fo r specifi c situations .

8. Experimen t wit h iterativ e an d recursiv e processes , wit h th e ai d o f calculators an d computers .

9. Explor e method s fo r storing , processing , an d communicatin g infor - mation.

136 J O S E P H G . ROSENSTEI N

10. Devise , describe , an d tes t algorithm s fo r solvin g optimizatio n an d search problems .

Building upo n knowledg e an d skill s gaine d i n t h e precedin g grades , by t h e en d o f Grad e 12 , students :

11. Understan d th e basi c principlie s o f iteration , recursion , an d mathe - matical induction .

12. Us e basic principle s t o solv e combinatorial an d algorithmi c problems . 13. Us e discret e model s t o represen t an d solv e problems . 14. Analyz e iterativ e processe s with th e ai d o f calculators an d computers . 15. Appl y discret e method s t o storing , processing , an d communicatin g

information. 16. Appl y discret e method s t o problem s o f voting , apportionment , an d

allocations, an d us e fundamenta l strategie s o f optimizaio n t o solv e problems.

1. Grade s K-1 2 Overvie w

Descriptive S t a t e m e n t . Discret e mathematic s i s the branc h o f math - ematics tha t deal s wit h arrangement s o f distinc t objects . I t include s a wid e variety o f topic s an d technique s tha t aris e i n everyda y life , suc h a s ho w t o find th e bes t rout e fro m on e cit y t o another , wher e th e object s ar e citie s arranged o n a map . I t als o include s ho w t o coun t th e numbe r o f differen t combinations o f toppings fo r pizzas , ho w best t o schedul e a list o f tasks t o b e done, an d ho w computer s stor e an d retriev e arrangement s o f informatio n o n a screen . Discret e mathematic s i s the mathematic s use d b y decision-maker s in ou r society , fro m worker s i n governmen t t o thos e i n healt h care , trans - portation, an d telecommunications . It s variou s application s hel p student s see th e relevanc e o f mathematic s i n th e rea l world .

M e a n i n g an d Importance . Durin g th e pas t 3 0 years , discret e math - ematics ha s grow n rapidl y an d ha s evolve d int o a significan t are a o f mathe - matics. I t i s the language of a large body of science and provide s a framewor k for decision s tha t individual s wil l nee d t o mak e i n thei r ow n lives , i n thei r professions, an d i n thei r role s a s citizens . It s man y practica l application s can hel p student s se e th e relevanc e o f mathematic s t o th e rea l world . I t does no t hav e extensiv e prerequisites , ye t i t pose s challenge s t o al l students . It i s fu n t o do , i s ofte n geometr y based , an d ca n stimulat e a n interes t i n mathematics o n th e par t o f student s a t al l level s an d o f al l abilities .

K-12 D e v e l o p m e n t an d Emphases . Althoug h th e ter m "discret e mathematics" ma y see m unfamiliar , man y o f it s theme s ar e alread y presen t in the classroom. Wheneve r object s ar e counted, ordered , o r listed, wheneve r instructions ar e presente d an d followed , wheneve r game s ar e playe d an d analyzed, teacher s ar e introducing themes of discrete mathematics . Throug h understanding thes e themes , teacher s wil l be able to recogniz e an d introduc e them regularl y i n classroo m situations . Fo r example , whe n callin g thre e

A COMPREHENSIV E VIE W O F DISCRET E MATHEMATIC S 13 7

students t o wor k a t th e thre e segment s o f th e chalkboard , th e teache r migh t ask In how many different orders can these three students work at the board? Another versio n o f th e sam e questio n i s How many different ways, such as ABC, can you name a triangle whose vertices are labeled A, B, and C? A similar, bu t slightl y differen t questio n i s In how many different orders can three numbers be multiplied?

Two importan t resource s o n discret e mathematic s fo r teacher s a t al l levels ar e th e 199 1 NCTM Yearboo k Discrete Mathematics Across the Cur- riculum K-12 an d th e 199 7 DI M ACS Volum e Discrete Mathematics in the Schools. Th e materia l i n thi s chapte r i s draw n fro m activitie s tha t hav e been reviewe d an d classroom-teste d b y th e K-1 2 teacher s i n th e Rutger s University Leadershi p Progra m i n Discret e Mathematic s ove r th e pas t nin e years; thi s progra m i s funde d b y th e Nationa l Scienc e Foundation .

Students shoul d lear n t o recogniz e example s o f discret e mathematic s i n familiar settings , an d explor e an d solv e a variet y o f problem s fo r whic h dis - crete techniques hav e proved useful . Thes e ideas should b e pursued through - out th e schoo l years . Student s ca n star t wit h man y o f th e basi c idea s i n concrete settings , includin g game s an d genera l play , an d progressivel y de - velop thes e idea s i n mor e complicate d setting s an d mor e abstrac t forms . Five majo r theme s o f discret e mathematic s shoul d b e addresse d a t al l K-1 2 grade level s — s y s t e m a t i c listing , counting , an d reasoning ; discret e mathematical modelin g usin g graph s (networks ) an d trees ; iter - ative (tha t is , repetitive ) pattern s an d processes ; organizin g an d processing information ; an d followin g an d devisin g list s o f instruc - tions, calle d "algorithms, " an d usin g algorithm s t o find t h e bes t solution t o real-worl d problems . Thes e five themes ar e discusse d i n th e paragraphs below .

Students shoul d us e a variet y o f strategie s t o systematicall y lis t an d count th e numbe r o f way s ther e ar e t o complet e a particula r task . Fo r example, elementar y schoo l student s shoul d b e abl e t o mak e a lis t o f al l possible outcome s o f a simpl e situatio n suc h a s th e numbe r o f outfit s tha t can b e wor n usin g tw o coat s an d thre e hats . Middl e schoo l student s shoul d be abl e t o systematicall y lis t an d coun t th e numbe r o f differen t four-block - high tower s that ca n b e buil t usin g blu e an d re d block s (se e example below) , or th e numbe r o f possibl e route s fro m on e locatio n o n a ma p t o another , o r the numbe r o f differen t "words " tha t ca n b e mad e usin g five letters . Hig h school students shoul d b e abl e to determin e th e numbe r o f possible ordering s of a n arbitrar y numbe r o f object s an d t o describ e procedure s fo r listin g an d counting al l such orderings . Thes e strategie s fo r listin g an d countin g shoul d be applied b y both middl e school and hig h schoo l students t o solv e problem s in probability .

Following i s a lis t o f al l four-block-hig h tower s tha t ca n b e buil t usin g clear block s an d soli d blocks . Th e 1 6 tower s ar e presente d i n a systemati c list — th e first 8 tower s hav e a clea r bloc k a t th e botto m an d th e secon d 8 towers hav e a soli d bloc k a t th e bottom ; withi n eac h o f thes e tw o groups ,

138 J O S E P H G . ROSENSTEI N

t h e firs t 4 tower s hav e t h e secon d bloc k clear , a n d t h e secon d 4 tower s hav e t h e secon d bloc k solid ; etc .

I fill I E l l If eac h towe r i s describe d alphabeticall y a s a sequenc e o f C' s a n d S's ,

representing "clear " a n d "solid " — t h e towe r a t t h e left , fo r e x a m p l e , woul d be C-C-C-C , an d t h e t h i r d towe r fro m t h e lef t woul d b e C-C-S-C , readin g from t h e b o t t o m u p — t h e n t h e sixtee n tower s woul d b e i n a l p h a b e t i c a l order:

C-C-C-C C-S-C- C C-C-C-S C-S-C- S C-C-S-C C-S-S- C C-C-S-S C-S-S- S

s-c-c-c s-s-c- c s-c-c-s s-s-c- s s-c-s-c s-s-s- c s-c-s-s s-s-s- s

T h e r e ar e o t h e r way s o f systematicall y listin g t h e 1 6 towers; fo r example , t h e lis t coul d contai n firs t t h e on e towe r w i t h n o soli d blocks , t h e n t h e fou r towers w i t h on e soli d block , t h e n t h e si x tower s w i t h tw o soli d blocks , t h e n t h e fou r tower s wit h t h r e e soli d blocks , a n d finall y t h e on e towe r w i t h fou r solid blocks .

D i s c r e t e m a t h e m a t i c a l m o d e l s s u c h a s g r a p h s ( n e t w o r k s ) a n d t r e e s (suc h a s thos e p i c t u r e d below ) ca n b e use d t o represen t a n d solv e a variety o f problem s base d o n real-worl d s i t u a t i o n s .

Example s of graphs : A A

W . » ( i 1 Example s of

"9T\ ""' . Q>

In t h e left-mos t g r a p h o f t h e figure s above , al l seve n d o t s ar e linke d int o a networ k consistin g o f t h e si x lin e segment s emergin g fro m t h e cente r dot ; these si x lin e segment s for m t h e tre e a t t h e fa r righ t whic h i s sai d t o "span " t h e origina l g r a p h sinc e i t reache s al l o f it s p o i n t s . A n o t h e r example : i f we t h i n k o f t h e secon d g r a p h a s a stree t m a p a n d w e m a k e t h e s t r e e t s on e way, w e ca n represen t t h e s i t u a t i o n usin g a directe d g r a p h wher e t h e lin e segments a r e replace d b y arrows .

E l e m e n t a r y schoo l s t u d e n t s shoul d recogniz e t h a t a s t r e e t m a p ca n b e represented b y a g r a p h a n d t h a t route s ca n b e represente d b y p a t h s i n t h e graph; middl e schoo l s t u d e n t s shoul d b e abl e t o fin d cost-effectiv e way s o f linking site s int o a networ k usin g s p a n n i n g trees ; a n d hig h schoo l s t u d e n t s should b e abl e t o us e efficien t m e t h o d s t o organiz e t h e performanc e o f indi - vidual t a s k s i n a large r projec t usin g directe d g r a p h s .

I t e r a t i v e p a t t e r n s a n d p r o c e s s e s ar e use d b o t h fo r describin g t h e world an d i n solvin g p r o b l e m s . A n iterativ e p a t t e r n o r proces s i s on e whic h

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 13 9

involves repeatin g a singl e ste p o r sequenc e o f step s man y times . Fo r ex - ample, elementar y schoo l student s shoul d understan d tha t multiplicatio n corresponds t o repeatedl y addin g th e sam e numbe r a specifie d numbe r o f times. The y shoul d investigat e ho w decorativ e floor tiling s ca n ofte n b e described a s th e repeate d us e o f a smal l pattern , an d ho w th e pattern s o f rows i n pin e cone s follo w a simpl e mathematica l rule . Middl e schoo l stu - dents shoul d explor e ho w simpl e repetitiv e rule s ca n generat e interestin g patterns b y usin g spirolateral s o r Log o commands , o r ho w the y ca n resul t in extremel y comple x behavio r b y generatin g th e beginnin g stage s o f fracta l curves. The y shoul d investigat e th e way s tha t th e plan e ca n b e covere d b y repeating patterns , calle d tessellations . Hig h schoo l student s shoul d under - stand ho w man y processe s describin g th e chang e o f physical , biological , an d economic system s ove r tim e ca n b e modele d b y simpl e equation s applie d repetitively, an d us e thes e model s t o predic t th e long-ter m behavio r o f suc h systems.

Students shoul d explor e differen t method s o f arranging , organizing , analyzing, transforming , an d c o m m u n i c a t i n g information , an d un - derstand ho w thes e method s ar e use d i n a variet y o f settings . Elemen - tary schoo l student s shoul d investigat e way s t o represen t an d classif y dat a according t o attribute s suc h a s colo r o r shape , an d t o organiz e dat a int o structures lik e table s o r tre e diagram s o r Ven n diagrams . Middl e schoo l students shoul d b e abl e t o read , construct , an d analyz e tables , matrices , maps an d othe r dat a structures . Hig h schoo l student s shoul d understan d the applicatio n o f discret e method s t o problem s o f informatio n processin g and computin g suc h a s sorting , codes , an d erro r correction .

Students shoul d b e abl e t o follo w an d devis e list s o f instructions , called "algorithms, " an d us e t h e m t o find t h e bes t solutio n t o real-world problem s — wher e "best " ma y b e defined , fo r example , a s most cost-effective o r a s mos t equitable . Fo r example , elementar y schoo l students shoul d b e abl e t o carr y ou t instruction s fo r gettin g fro m on e loca - tion t o another , shoul d discus s differen t way s o f dividin g a pil e o f snacks , and shoul d determin e th e shortes t pat h fro m on e sit e t o anothe r o n a ma p laid ou t o n th e classroo m floor. Middl e schoo l student s shoul d b e abl e t o plan a n optima l rout e fo r a clas s tri p (se e th e vignett e i n th e Introductio n to thi s Framework entitle d Short-circuiting Trenton), writ e precis e instruc - tions fo r addin g tw o two-digi t numbers , and , pretendin g t o b e th e manage r of a fast-foo d restaurant , devis e wor k schedule s fo r employee s whic h mee t specified condition s ye t minimiz e th e cost . Hig h schoo l student s shoul d b e conversant wit h fundamenta l strategie s o f optimization , b e abl e t o us e flow charts t o describ e algorithms , an d recogniz e bot h th e powe r an d limitation s of computer s i n solvin g algorithmi c problems .

IN SUMMARY , discret e mathematic s i s a n excitin g an d appropriat e vehicle fo r workin g towar d an d achievin g th e goa l o f educatin g informe d citizens wh o ar e bette r abl e t o functio n i n ou r increasingl y technologica l so - ciety; hav e bette r reasonin g powe r an d problem-solvin g skills ; ar e awar e o f

140 JOSEPH G . ROSENSTEI N

the importanc e o f mathematic s i n ou r society ; an d ar e prepare d fo r futur e careers which wil l require ne w and mor e sophisticated analytica l an d techni - cal tools. I t i s an excellen t too l fo r improvin g reasonin g an d problem-solvin g abilities.

N o t e : Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted in the Introduction to this Framework, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences.

References.

• Kenny , M . J., Ed . Discrete Mathematics Across the Curriculum K-12. 199 1 Yearbook o f th e Nationa l Counci l o f Teacher s o f Mathematic s (NCTM) . Reston, VA , 1991.

• Rosenstein , J . G. , D . Pranzblau , an d F . Roberts , Eds . Discrete Mathemat- ics in the Schools. Proceeding s o f a 199 2 DIMACS Conferenc e o n "Discret e Mathematics i n th e Schools. " DIMAC S Serie s on Discret e Mathematic s an d Theoretical Compute r Science . Providence , RI : American Mathematica l So - ciety (AMS) , 1997 .

On-Line Resources . h t t p : / / d i m a c s. r u t g e r s. edu/nj _math_coalit ion/framework. html /

The Framework wil l be available at thi s site during Sprin g 1997 . In time , w e hop e t o pos t additiona l resource s relatin g t o thi s standard, suc h a s grade-specifi c activitie s submitte d b y Ne w Jersey teachers , an d t o provid e a foru m t o discus s th e Mathe- matics Standards.

2. Grade s K- 2 Overvie w

The five major theme s o f discret e mathematics , a s discusse d i n th e K-1. 2 Overview,3 ar e s y s t e m a t i c listing , counting , an d reasoning ; discret e mathematical modelin g usin g graph s (networks ) an d trees ; iter - ative (tha t is , repetitive ) pattern s an d processes ; organizin g an d processing information ; an d followin g an d devisin g list s o f instruc - tions, calle d "algorithms, " an d usin g t h e m t o find t h e bes t solutio n t o real-worl d problems .

Despite thei r formidabl e titles , thes e five theme s ca n b e addresse d wit h activities a t th e K- 2 grad e leve l whic h involv e purposefu l pla y an d simpl e analysis. Indeed , teacher s wil l discove r tha t man y activitie s the y alread y are usin g i n thei r classroom s reflec t thes e themes . Thes e five theme s ar e discussed i n th e paragraph s below .

3Since K- 2 grad e leve l teacher s ma y no t rea d th e K-1 2 Overview , and , mor e generally , teachers a t othe r grad e level s wil l begi n thei r revie w o f thi s chapte r o f th e Framework b y turning t o th e sectio n addressin g thei r ow n grad e levels , th e grad e leve l overview s hav e significant overlap .

A C O M P R E H E N S I V E VIE W O F D I S C R E T E MATHEMATIC S 14 1

Activities involvin g s y s t e m a t i c listing , counting , an d reasonin g can b e don e ver y concretel y a t th e K- 2 grad e level . Fo r example , dressin g cardboard tedd y bear s wit h differen t outfit s become s a mathematica l activ - ity whe n th e tas k i s t o mak e a lis t o f al l possibl e outfit s an d coun t them ; pictured belo w ar e th e si x outfit s tha t ca n b e arrange d usin g on e o f tw o types o f shirt s an d on e o f thre e type s o f shorts . Similarly , playin g an y gam e involving choice s become s a mathematica l activit y whe n childre n reflec t o n the move s the y mak e i n th e game .

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An importan t discret e mathematica l m o d e l i s that o f a networ k o r graph, whic h consist s o f dot s an d line s joining th e dots ; th e dot s ar e ofte n called vertices (vertex i s th e singular ) an d th e line s ar e ofte n calle d edges. (This i s differen t fro m othe r mathematica l use s o f th e ter m "graph." ) Th e two terms "network " an d "graph " ar e use d interchangeabl y fo r thi s concept . An exampl e o f a grap h wit h seve n vertice s an d twelv e edge s i s give n below . You ca n thin k o f the vertice s o f this grap h a s island s i n a river an d th e edge s as bridges . Yo u ca n als o thin k o f the m a s building s an d roads , o r house s and telephon e cables , o r peopl e an d handshakes ; whereve r a collectio n o f things ar e joine d b y connectors , th e mathematica l mode l use d i s tha t o f a network o r graph . A t th e K- 2 level , childre n ca n recogniz e graph s an d us e life-size model s of graphs i n various ways. Fo r example, a large version of this graph, o r an y othe r graph , ca n b e "drawn " o n th e floor usin g pape r plate s as vertice s an d maskin g tap e a s edges . Childre n migh t selec t tw o "islands " and fin d a wa y t o g o fro m on e islan d t o th e othe r islan d b y crossin g exactl y four "bridges. " (Thi s ca n b e don e fo r an y tw o island s i n thi s graph , bu t no t necessarily i n anothe r graph. )

Children ca n recogniz e an d wor k wit h repetitiv e pattern s an d pro - cesses involvin g number s an d shapes , usin g object s i n th e classroo m an d i n the worl d aroun d them . Fo r example , childre n a t th e K- 2 leve l ca n creat e (and decorate ) a patter n o f triangles o r squares (a s pictured here ) tha t cove r a sectio n o f th e floor (thi s i s calle d a "tessellation") , o r star t wit h a num - ber an d repeatedl y ad d three , o r us e clappin g an d movemen t t o simulat e rhythmic patterns .

142 J O S E P H G . ROSENSTEI N

Children a t th e K- 2 grad e level s shoul d investigat e way s o f sortin g i t e m s accordin g t o attribute s lik e color , shape , o r size , an d way s o f ar- ranging dat a int o charts , tables , an d family trees . Fo r example, the y can sort attribut e block s or stuffed animal s b y color o r kind, a s in the diagram, and ca n coun t th e number o f children wh o hav e birthday s i n each mont h by organizing themselve s int o birthday-mont h groups .

BI G ITEM S ROUN D ITEM S

• A Finally, a t the K-2 grade levels , childre n shoul d b e able t o follow an d

describe simpl e procedure s an d determin e an d discus s wha t i s t h e best solutio n t o a problem . Fo r example, the y shoul d b e abl e t o follo w a prescribe d rout e fro m th e classroo m t o anothe r roo m i n th e schoo l (a s pictured below ) an d to compare variou s alternat e routes , an d in the second grade shoul d determin e th e shortes t pat h fro m on e sit e to another o n a ma p laid ou t on the classroom floor .

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A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 14 3

3. Grade s K- 2 Indicator s an d Acti vitie s

The cumulativ e progres s indicator s fo r grad e 4 appear belo w i n boldfac e type. Eac h indicato r i s followe d b y activitie s whic h illustrat e ho w i t ca n b e addressed i n th e classroo m i n kindergarte n an d grade s 1 and 2 .

Experiences wil l b e suc h tha t al l student s i n grade s K-2 :

1. Explor e a variet y o f puzzles , games , an d countin g problems .

• Student s us e tedd y bea r cut-out s with , fo r example , shirt s o f tw o colors and short s of three colors, and decid e how many differen t outfit s can b e mad e b y makin g a lis t o f al l possibilitie s an d arrangin g the m systematically. (Se e illustratio n i n K- 2 Overview. )

• Student s us e pape r face s o r Mr . Potat o Hea d typ e model s t o creat e a "regula r face " give n a nose , mouth , an d a pai r o f eyes . The n the y use anothe r pai r o f eyes , the n anothe r nose , an d the n anothe r mout h (or othe r parts ) an d explor e an d recor d th e numbe r o f face s tha t ca n be mad e afte r eac h additiona l par t ha s bee n included .

• Student s rea d A Three Hat Day an d the n tr y t o creat e a s man y different hat s a s possibl e wit h thre e hats , a feather , a flower, an d a ribbon a s decoration . Student s coun t th e differen t hat s they'v e mad e and discus s thei r answers .

• Student s coun t th e numbe r o f square s o f eac h siz e ( l x l , 2 x 2 , 3 x 3 ) tha t the y ca n fin d o n th e squar e gri d below . The y ca n b e challenged t o fin d th e number s o f smal l square s o f eac h siz e o n a larger squar e o r rectangula r grid .

• Student s wor k i n groups t o figur e ou t th e rule s of addition an d place - ment tha t ar e use d t o pas s fro m on e ro w t o th e nex t i n th e diagra m below, an d us e thes e rule s t o fin d th e number s i n th e nex t fe w rows .

1 1 1

1 2 1 1 3 3 1

1 4 6 4 1

In thi s diagram , calle d Pascal' s triangle , eac h numbe r i s th e su m o f the tw o number s tha t ar e abov e it , t o it s lef t an d right ; th e number s on th e lef t an d righ t edge s ar e al l 1 .

• Student s cu t ou t fiv e "coins " labele d 1^ , 2yf , 4^ , 8^ , an d 16ĵ . Fo r each numbe r i n th e countin g sequenc e 1 , 2 , 3 , 4 , 5 , . . . (a s fa r a s i s appropriate fo r a particula r grou p o f students) , student s determin e

144 JOSEPH G . ROSENSTEI N

how to obtai n tha t amoun t o f mone y usin g a combinatio n o f differen t coins.

• Student s pla y simpl e game s an d discus s wh y the y mak e th e move s they do . Fo r example , tw o student s divid e a six-piec e domin o se t (with 0-0 , 0-1 , 0-2, 1-1 , 1-2 , an d 2-2 ) an d tak e turn s placin g dominoe s so that dominoe s whic h touc h hav e th e sam e number s an d s o that al l six dominoe s ar e use d i n th e chain .

2. U s e network s an d tre e diagram s t o represen t everyda y situ - ations.

• Student s find a wa y o f gettin g fro m on e islan d t o another , i n th e graph describe d i n th e K- 2 Overvie w lai d ou t o n th e classroo m floor with maskin g tape , b y crossin g exactly fou r bridges . The y mak e thei r own graphs , namin g eac h o f th e islands , an d mak e a "from-to " lis t o f islands fo r whic h the y hav e foun d a four-bridge-route . (Note : i t ma y not alway s b e possibl e t o fin d four-bridge-routes. )

• Student s coun t th e numbe r o f edge s a t eac h verte x (calle d th e de - gree o f th e vertex ) o f a networ k an d construc t graph s wher e al l ver - tices hav e th e sam e degree , o r wher e al l th e vertice s hav e on e o f tw o specified degrees .

• O n a patter n o f island s an d bridge s lai d ou t o n th e floor, student s try t o find a wa y o f visitin g eac h islan d exactl y once ; the y ca n leav e colored marker s t o kee p track o f islands alread y visited . Not e tha t fo r some pattern s thi s ma y no t b e possible ! Student s ca n b e challenge d to find a wa y o f visitin g eac h islan d exactl y onc e whic h return s the m to thei r startin g point . Simila r activitie s ca n b e foun d i n Inside, Outside, Loops, and Lines b y Herber t Kohl .

A • Student s creat e a ma p wit h make-believ e countrie s (se e exampl e below), an d colo r th e map s s o tha t countrie s whic h ar e nex t t o eac h other hav e differen t colors . How many colors were used? Could it be done with fewer colors? with four colors? with three colors? with two colors? A numbe r o f interestin g ma p colorin g idea s ca n b e foun d i n Inside, Outside, Loops and Lines b y Herber t Kohl .

A COMPREHENSIVE VIE W O F DISCRET E MATHEMATIC S 14 5

3. Identif y an d investigat e sequence s an d p a t t e r n s foun d i n na - ture, art , an d music .

• Student s us e a calculato r t o creat e a sequenc e o f te n number s start - ing wit h zero , eac h o f whic h i s thre e mor e tha n th e previou s one ; o n some calculators , thi s ca n b e don e b y pressin g 0 + 3 = = = . . . , where = i s pressed te n times . A s they proceed , the y coun t on e 3 , tw o 3s, thre e 3s , etc .

• Student s "tessellate " th e plane , b y usin g group s o f square s o r tri - angles (fo r example , fro m set s o f patter n blocks ) t o completel y cove r a shee t o f pape r withou t overlapping ; the y recor d thei r pattern s b y tracing aroun d th e block s on a sheet o f paper an d colorin g the shapes .

• Student s liste n to or read Grandfather Tang's Story b y Ann Tomper t and the n us e tangram s t o mak e th e shape-changin g fo x fairie s a s th e story progresses . Student s ar e the n encourage d t o d o a retellin g o f the stor y wit h tangram s o r t o inven t thei r ow n tangra m character s and stories .

• Student s rea d The Cat in the Hat o r Green Eggs and Ham b y Dr . Seuss an d identif y th e patter n o f event s i n th e book . Student s coul d create thei r ow n book s wit h simila r patterns .

• Student s collec t leave s an d not e th e pattern s o f th e veins . The y look a t ho w th e vein s branc h of f o n eac h sid e o f th e cente r vei n an d observe tha t thei r branche s ar e smalle r copie s o f th e origina l vei n pattern. Student s collec t feathers , ferns , Quee n Anne' s lace , broccoli , or cauliflowe r an d not e i n eac h cas e ho w th e patter n o f the origina l i s repeated i n miniatur e i n eac h o f it s branche s o r clusters .

• Student s liste n fo r rhythmi c pattern s i n musica l selection s an d us e clapping, instruments , an d movemen t t o simulat e thos e patterns .

• Student s tak e a "pattern s walk " throug h th e school , searchin g fo r patterns i n th e bricks , th e pla y equipment , th e shape s i n th e class - rooms, th e numbe r sequence s o f classrooms , th e floor s an d ceilings , etc.; th e purpos e o f thi s activit y i s t o creat e a n awarenes s o f al l th e patterns aroun d them .

4. Investigat e way s t o represen t an d classif y dat a accordin g t o attributes, suc h a s shap e o r color , an d relationships , an d discuss t h e purpos e an d usefulnes s o f suc h classification .

146 J O S E P H G . R O S E N S T E I N

• S t u d e n t s sor t themselve s b y m o n t h o f b i r t h , a n d t h e n w i t h i n eac h g r o u p b y heigh t o r b i r t h d a t e . ( O t h e r sortin g activitie s ca n b e foun d in Mathematics Their Way, b y M a r y B a r a t t a - L o r t o n . )

• E a c h s t u d e n t i s give n a car d w i t h a differen t n u m b e r o n it . S t u d e n t s line u p i n a ro w a n d p u t t h e n u m b e r s i n numerica l orde r b y exchangin g cards, on e a t a t i m e , wit h adjacen t children . (Afte r practice , t h i s ca n b e accomplishe d w i t h o u t talking. )

• S t u d e n t s dra w stic k figure s o f m e m b e r s o f t h e i r famil y a n d a r r a n g e t h e m i n orde r o f size .

• S t u d e n t s sor t stuffe d animal s i n variou s way s a n d explai n wh y t h e y sorted t h e m a s t h e y did . S t u d e n t s ca n us e Tabletop, Jr. softwar e t o sort character s accordin g t o a variet y o f a t t r i b u t e s .

• Usin g a t t r i b u t e blocks , b u t t o n s , o r o t h e r object s w i t h clearl y distin - guishable a t t r i b u t e s suc h a s color , size , a n d s h a p e , s t u d e n t s develo p a sequenc e o f object s wher e eac h differ s fro m t h e previou s on e i n onl y one a t t r i b u t e . Tabletop, Jr. softwar e ca n als o b e use d t o cr eat e suc h sequences o f objects .

• S t u d e n t s us e tw o H u l a Hoop s (o r larg e circle s d r a w n o n p a p e r s o t h a t a p a r t o f thei r interior s overlap ) t o assis t i n sortin g a t t r i b u t e block s o r o t h e r object s accordin g t o tw o characteristics . Fo r example , give n a collection o f object s o f differen t color s a n d shapes , s t u d e n t s ar e aske d t o plac e t h e m s o t h a t al l re d i t e m s g o insid e h o o p # 1 a n d al l o t h e r s go o n t h e outside , a n d s o t h a t al l s q u a r e item s g o insid e h o o p # 2 a n d al l o t h e r s g o o n t h e outside . What items should be placed in the overlap of the two hoops? What is inside only the first hoop? What is outside both hoops?

T h i s i s a n exampl e o f a Ven n d i a g r a m . S t u d e n t s ca n als o us e Ven n d i a g r a m s t o organiz e t h e similaritie s a n d difference s betwee n t h e in - formation i n tw o storie s b y placin g al l feature s o f t h e firs t stor y i n h o o p # 1 a n d al l feature s o f t h e secon d s t o r y i n h o o p # 2 , w i t h com - m o n feature s i n t h e overla p o f t h e tw o hoops . A simila r activit y ca n b e foun d i n t h e Shapetown lesso n t h a t i s describe d i n t h e F i r s t Fou r S t a n d a r d s o f t h i s Framework. Tabletop, Jr. softwar e allow s s t u d e n t s t o a r r a n g e a n d sor t d a t a , a n d t o explor e thes e concept s easily .

5. F o l l o w , d e v i s e , a n d d e s c r i b e p r a c t i c a l l i s t s o f i n s t r u c t i o n s .

• S t u d e n t s follo w direction s fo r a t r i p w i t h i n t h e classroo m — fo r example, s t u d e n t s ar e aske d wher e t h e y woul d en d u p i f t h e y s t a r t e d

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 147

at a give n spo t facin g i n a certai n direction , too k thre e step s forward , turned left , too k tw o step s forward , turne d right , an d move d forwar d three mor e steps .

• Student s follo w ora l direction s fo r goin g fro m th e classroo m t o th e lunchroom, an d represen t thes e direction s wit h a diagram . (Se e K- 2 Overview fo r a sampl e diagram. )

• Student s agre e o n a procedur e fo r filling a bo x wit h rectangula r blocks. Fo r example , a bo x wit h dimension s 4 " x 4 " x 5 " ca n b e fille d with 1 0 blocks o f dimension s 1 " x2" x4". (Linkin g cube s ca n b e use d to creat e th e rectangula r blocks. )

• Student s explor e th e questio n o f finding th e shortes t rout e fro m school t o hom e o n a diagra m lik e th e on e picture d below , lai d ou t on th e floor usin g maskin g tape , wher e student s plac e a numbe r o f counters o n each lin e segment t o represen t th e lengt h o f that segment . (The shortes t rout e wil l depen d o n th e placemen t o f th e counters ; what appear s t o b e th e mos t direc t rout e ma y no t b e th e shortest. )

• Student s find a wa y throug h a simpl e maze . The y discus s th e dif - ferent path s the y too k an d thei r reason s fo r doin g so .

• Student s us e Log o software t o giv e the turtl e precis e instruction s fo r movement i n specifie d directions .

R e f e r e n c e s .

• Baratta-Lorton , Mary . Mathematics Their Way. Menl o Park, CA : Addiso n Wesley, 1993 .

• Casey , Nancy , an d Mik e Fellows . This is MEGA-Mathematics! — Stories and Activities for Mathematical Thinking, Problem-Solving, and Commu- nication. Lo s Alamos , CA : Lo s Alamo s Nationa l Laboratories , 1993 . ( A version i s availabl e onlin e a t http://www.c3.lanl.gov/mega-math )

• Geringer , Laura . A Three Hat Day. Ne w York: Harpe r Ro w Junio r Books , 1987.

• Kenney , M . J. , Ed . Discrete Mathematics Across the Curriculum K-12. 1991 Yearbook o f the National Counci l of Teachers of Mathematics (NCTM) . Reston, VA : 1991.

• Kohl , Herbert . Insides, Outsides, Loops, and Lines. Ne w York : W . H . Freeman, 1995 .

148 J O S E P H G . R O S E N S T E I N

• Murphy , Pat . By Nature's Design. Sa n Francisco , CA : Chronicl e Books , 1993.

• Rosenstein , J . G. , D . Franzblau , an d F . Roberts , Eds . Discrete Mathemat- ics in the Schools. Proceeding s o f a 199 2 DIMAC S Conferenc e o n "Discret e Mathematics i n th e Schools. " DIMAC S Serie s on Discret e Mathematic s an d Theoretical Compute r Science . Providence , RI : American Mathematica l So - ciety (AMS) , 1997 .

• Seuss , Dr . Cat in the Hat. Boston , MA : Houghto n Mifflin , 1957 .

• Seuss , Dr . Green Eggs and Ham. Rando m House .

• Tompert , Ann . Grandfather Tang's Story. Crow n Publishing , 1990 .

Software.

• Logo. Man y version s o f Log o ar e commerciall y available .

• Tabletop, Jr. Broderbun d Software . TERC .

4. Grade s 3- 4 Overvie w

The five major theme s o f discrete mathematics , a s discusse d i n the K-1 2 Overview, ar e s y s t e m a t i c listing , counting , an d reasoning ; discret e mathematical modelin g usin g graph s (networks ) an d trees ; iter - ative (tha t is , repetitive ) pattern s an d processes ; organizin g an d processing information ; an d followin g an d devisin g list s o f instruc - tions, calle d "algorithms, " an d usin g t h e m t o find t h e bes t solutio n t o real-worl d problems .

Despite thei r formidabl e titles , thes e five theme s ca n b e addresse d wit h activities a t th e 3- 4 grad e leve l whic h involv e purposefu l pla y an d simpl e analysis. Indeed , teacher s wil l discove r tha t man y activitie s tha t the y al - ready ar e usin g i n thei r classroom s reflec t thes e themes . Thes e five theme s are discusse d i n th e paragraph s below .

The following discussion of activities at the 3-4 grade levels in discrete mathematics presupposes that corresponding activities have taken place at the K-2 grade levels. Hence 3-4 grade teachers should review the K-2 grade level discussion of discrete mathematics and might use activities similar to those described there before introducing the activities for this grade level.

Activities involvin g systemati c listing , counting , an d reasonin g should b e don e ver y concretel y a t th e 3- 4 grad e levels , buildin g o n simila r activities a t th e K- 2 grad e levels . Fo r example , th e childre n coul d systemati - cally lis t an d coun t th e tota l numbe r o f possible combination s o f dessert an d beverage tha t ca n b e selecte d fro m picture s o f those tw o type s o f food s the y have cu t ou t o f magazine s o r tha t ca n b e selecte d fro m a restauran t menu . Similarly, playin g game s lik e Nim , dot s an d boxes , an d dominoe s become s a mathematica l activit y whe n childre n systematicall y reflec t o n th e move s they mak e i n th e gam e an d us e those reflection s t o decid e o n th e nex t move .

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 149

An importan t discret e mathematica l mode l i s tha t o f a graph , which i s use d wheneve r a collectio n o f thing s ar e joine d b y connector s — such a s building s an d roads , island s an d bridges , o r house s an d telephon e cables — or , mor e abstractly , wheneve r th e object s hav e som e define d rela - tionship t o eac h other ; thi s kin d o f mode l i s described i n th e K- 2 Overview . At th e 3- 4 grad e levels , childre n ca n recogniz e an d us e model s o f graph s i n various ways, for example , b y finding a way to ge t fro m on e island t o anothe r by crossin g exactl y fou r bridges , o r b y finding a rout e fo r a cit y mai l carrie r which use s eac h stree t once , o r b y constructin g a collaboratio n grap h fo r the clas s whic h describe s wh o ha s worke d wit h who m durin g th e pas t week . A specia l kin d o f grap h i s calle d a "tree. " Thre e view s o f th e sam e tre e ar e pictured i n th e diagra m below ; th e first suggest s a famil y tree , th e secon d a tree diagram , an d th e thir d a "real " tree .

At th e 3- 4 grad e levels , student s ca n us e a tre e diagra m t o organiz e th e six way s tha t thre e peopl e ca n b e arrange d i n order . (Se e th e Grade s 3- 4 Indicators an d Activitie s fo r a n example. )

Students ca n recogniz e an d wor k wit h repetitiv e pattern s an d pro - cesses involvin g number s an d shapes , wit h classroo m object s an d i n th e world aroun d them . Childre n a t th e 3- 4 grad e level s ar e fascinate d wit h th e Fibonacci sequenc e o f number s 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21, 34, 55 , 89, . . . wher e every numbe r i s th e su m o f th e previou s tw o numbers . Thi s sequenc e o f numbers turn s u p i n petal s o f flowers, i n th e growt h o f population s (se e th e activity involvin g rabbits), i n pineapples an d pin e cones , and i n lots o f othe r places i n nature . Anothe r importan t sequenc e t o introduc e a t thi s ag e i s th e doubling sequenc e 1 , 2 , 4 , 8 , 16 , 32 , . . . an d t o discus s differen t situation s in whic h i t appears .

Students a t th e 3-4 grade levels should investigat e ways of sorting i t e m s according t o attribute s lik e colo r o r shape , o r b y quantitativ e informatio n like size , arrangin g dat a usin g tre e diagram s an d buildin g chart s an d ta - bles, an d recoverin g hidde n informatio n i n game s an d encode d mes - sages. Fo r example , the y ca n sor t letter s int o zi p cod e orde r o r sor t th e class alphabetically , creat e ba r chart s base d o n informatio n obtaine d exper - imentally (suc h a s sod a drin k preference s o f th e class) , an d pla y game s lik e hangman t o discove r hidde n messages .

Students a t th e 3- 4 grad e level s shoul d describ e an d discus s simpl e algorithmic procedure s suc h a s providin g an d followin g direction s fro m one locatio n t o another , an d shoul d i n simpl e case s determin e an d discus s what i s t h e bes t solutio n t o a problem. Fo r example , the y migh t follo w a recipe t o mak e a cak e o r t o assembl e a simpl e to y fro m it s componen t parts .

150 J O S E P H G . ROSENSTEI N

Or the y migh t find th e bes t wa y o f playin g tic-tac-to e o r th e shortes t rout e that ca n b e use d t o ge t fro m on e locatio n t o another .

5. Grade s 3- 4 Indicator s an d A c t i v i t i e s

Building upo n knowledg e an d skill s gaine d i n th e precedin g grades , ex - periences wil l b e suc h tha t al l student s i n grade s 3-4 :

1. Explor e a variet y o f puzzles , games , an d countin g problems .

• Student s rea d One Hundred Hungry Ants b y Elinor Pincze s an d the n illustrate an d writ e thei r ow n stor y book s (perhap s title d 18 Ailing Alligators o r 24 Furry Ferrets) i n a styl e simila r t o th e boo k usin g as man y differen t arrangement s o f the animal s a s possibl e i n creatin g their books . The y rea d thei r book s t o student s i n th e lowe r grades .

• Student s coun t th e numbe r o f square s o f eac h siz e ( l x l , 2x2 , 3 x 3 , 4x4, 5x5 ) tha t the y ca n find o n a geoboard , an d i n large r squar e o r rectangular grids .

• Student s determin e th e numbe r o f possibl e combination s o f desser t and beverag e that coul d be selected from picture s of those two types of foods the y hav e cu t ou t o f magazines . Subsequently , the y determin e the numbe r o f possibl e combination s o f desser t an d beverag e tha t could b e chose n fro m a restauran t menu , an d ho w man y o f thos e combinations coul d b e ordere d i f the y onl y hav e $4 .

• Student s find th e numbe r o f differen t way s t o mak e a ro w o f fou r flowers eac h o f whic h coul d b e re d o r yellow . The y ca n mode l thi s with Unfi x cube s an d explai n ho w the y kno w tha t al l combination s have bee n obtained .

• Student s determin e th e numbe r o f differen t way s an y thre e peopl e can b e arrange d i n order , an d us e a tre e diagra m t o organiz e th e information. Th e tre e diagra m belo w represent s th e si x way s tha t Barbara (B) , Maria (M) , and Tarvand a (T) , can b e arrange d i n order . The thre e branche s emergin g fro m th e "start " positio n represen t th e three peopl e who could b e first; eac h pat h fro m lef t t o righ t represent s the arrangemen t o f th e thre e peopl e liste d t o th e right .

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 15 1

• Eac h studen t use s fou r square s t o mak e design s wher e eac h squar e shares a n entir e sid e wit h a t leas t on e o f th e othe r thre e squares . Geoboards, attribut e block s o r Linke r cube s ca n b e used . How many different shapes can be made? Thes e shape s ar e calle d "tetrominoes. "

• Eac h grou p o f students receive s a ba g containin g fou r colore d beads . One grou p ma y b e give n 1 red , 1 blac k an d 2 gree n beads ; othe r groups ma y hav e th e sam e fou r bead s o r differen t ones . Student s take turn s drawin g a bea d fro m th e bag , recordin g it s color , an d replacing i t i n th e bag . Afte r 2 0 bead s ar e drawn , eac h grou p make s a ba r grap h illustratin g th e numbe r o f bead s draw n o f eac h color . They mak e anothe r ba r grap h illustratin g th e numbe r o f bead s o f each colo r actuall y i n th e bag , an d compar e th e tw o ba r graphs . A s a follow-u p activity , student s shoul d dra w 2 0 o r mor e time s fro m a bag containin g a n unknow n mixtur e o f bead s an d tr y t o guess , an d justify, ho w man y bead s o f eac h colo r ar e i n th e container .

• Student s determin e wha t amount s o f postag e ca n an d canno t b e made usin g onl y 3̂ f and 5 ^ stamps .

• Student s generat e additiona l row s of Pascal's triangl e (below) . The y color al l odd entrie s on e color an d al l even entries anothe r color . The y examine th e pattern s tha t result , an d tr y t o explai n wha t the y see . They discus s whethe r thei r conclusion s appl y t o a large r versio n o f Pascal's triangle .

1 1 1

1 2 1 1 3 3 1

1 4 6 4 1

• Student s mak e a table indicatin g whic h stamps o f the denomination s ljzf, 2JZ( , 4^, 8JZ( , 16jz(, 32jzf would b e used (wit h n o repeats) t o obtai n eac h amount o f postage from 1 ^ to 63ĵ . Fo r the table, the y lis t the availabl e denominations acros s the to p an d th e postag e amount s fro m 1$ to 63^ at th e left ; the y pu t a checkmar k i n th e appropriat e spo t i f they nee d the stam p fo r tha t amount , an d leav e i t blan k otherwise . The y tr y to find a patter n whic h coul d b e use d t o decid e whic h amount s o f postage coul d b e mad e i f additiona l stamp s (lik e 64jz f an d 128^ ) wer e used.

• Student s pla y game s lik e Ni m an d reflec t o n th e move s the y mak e in th e game . (Se e Math for Girls and Other Problem Solvers, b y D . Downie e t al. , fo r othe r game s fo r thi s grad e level. ) I n Nim , yo u star t

152 JOSEPH G . ROSENSTEI N

with a numbe r o f piles of objects — for example , yo u coul d star t wit h two piles , on e wit h five buttons , th e othe r wit h seve n buttons . Tw o students alternat e moves , and eac h move consists of taking some or all of th e button s fro m a singl e pile ; th e chil d wh o take s th e las t butto n off th e tabl e win s th e game . Onc e the y maste r thi s game , student s can tr y Ni m wit h thre e piles , startin g wit h thre e pile s whic h hav e respectively 1 , 2 , an d 3 buttons .

• Student s pla y game s lik e dots and boxes an d systematicall y thin k about th e move s the y mak e i n th e game . I n dot s an d boxes , yo u start wit h a squar e (o r rectangular ) arra y o f dots , an d tw o student s alternate drawin g a lin e whic h joins tw o adjacen t dots . Wheneve r al l four side s o f a squar e hav e bee n drawn , th e studen t put s he r o r hi s initial i n th e squar e an d draw s anothe r line ; th e perso n wit h initial s in mor e square s win s th e game .

2. U s e network s an d tre e diagram s t o represen t everyda y situ - ations.

• Student s mak e a collaboratio n grap h fo r th e member s o f th e clas s which describe s wh o ha s worke d wit h who m durin g th e pas t week .

• Student s dra w specifie d pattern s o n th e chalkboar d withou t retrac - ing, suc h a s thos e below . Alternatively , the y ma y trac e thes e pat - terns i n a smal l bo x o f sand , a s don e historicall y i n Africa n cultures . (See Ethnomathematics, Drawing Pictures With One Line, o r Insides, Outsides, Loops, and Lines.) Alternatively , o n a patter n o f island s and bridge s lai d ou t o n th e floor wit h maskin g tape , student s migh t try t o tak e a wal k whic h involve s crossin g eac h bridg e exactl y onc e (leaving colore d marker s o n bridge s alread y crossed) ; not e tha t fo r some pattern s thi s ma y no t b e possible . Th e pattern s give n her e ca n be used , bu t student s ca n develo p thei r ow n pattern s an d tr y t o tak e such a wal k fo r eac h patter n tha t the y create .

E l f ffi • Student s creat e "huma n graphs " wher e the y themselve s ar e th e ver - tices an d the y us e piece s o f yar n (severa l fee t long ) a s edges ; eac h piece o f yar n i s hel d b y tw o students , on e a t eac h end . The y migh t

A COMPREHENSIVE VIE W O F DISCRET E MATHEMATIC S 15 3

create graph s wit h specifie d properties ; fo r example , the y migh t cre - ate a huma n grap h wit h fou r vertice s o f degre e 2 , or , a s i n th e figure below, wit h si x vertice s o f whic h fou r hav e degre e 3 an d tw o hav e degree 2 . (Th e degree of a verte x i s th e tota l numbe r o f edge s tha t meet a t th e vertex. ) The y migh t coun t th e numbe r o f different shape s of huma n graph s the y ca n for m wit h fou r student s (o r five, o r six) .

• Student s us e a floor pla n o f thei r schoo l t o ma p ou t alternat e route s from thei r classroo m t o th e school' s exits , an d discus s whethe r th e fire dril l rout e i s i n fac t th e shortes t rout e t o a n exit .

• Student s dra w graph s o f their ow n neighborhoods , wit h edge s repre - senting street s an d vertice s representin g location s wher e road s meet . Can you find a route for the mail carrier in your neighborhood which enables her to walk down each street, without repeating any streets, and which ends where it begins? Can you find such a route if she needs to walk up and down each street in order to deliver mail on both sides of the street?

• Student s colo r map s (e.g. , the 2 1 counties o f New Jersey ) s o that ad - jacent countie s (o r countries ) hav e different colors , usin g a s few color s as possible. Th e clas s coul d the n shar e a N J cak e froste d accordingly . (See The Mathematician's Coloring Book.)

• Student s recogniz e an d understan d famil y tree s i n socia l an d histor - ical studies , an d i n storie s tha t the y read . Wher e appropriate , the y create thei r ow n famil y trees .

3. Identif y an d investigat e sequence s an d pattern s foun d i n na - ture, art , an d music .

• Student s rea d A Cloak for a Dreamer b y A . Friedman , an d mak e outlines o f cloak s o r coat s lik e thos e wor n b y th e son s o f th e tailo r in th e boo k b y tracin g thei r uppe r bodie s o n larg e piece s o f paper . Students coul d use pattern block s or pre-cut geometri c shapes to cover (tessellate) th e pape r cloak s wit h pattern s lik e thos e i n th e boo k o r try t o mak e thei r ow n clot h designs .

154 J O S E P H G . ROSENSTEI N

• Student s rea d Sam and the Blue Ribbon Quilt b y Lis a Ernst , an d by rotating , flipping, o r slidin g cut-ou t squares , rectangles , triangles , etc., creat e thei r ow n symmetrica l design s o n quil t square s simila r t o those foun d i n th e book . Th e design s fro m al l th e member s o f th e class ar e pu t togethe r t o mak e a patchwor k clas s quil t o r t o for m th e frame fo r a mat h bulleti n board .

• Student s tak e a "patter n walk " throug h th e neighborhood , searchin g for pattern s i n the trees , the houses, the buildings , th e manhol e cover s (by th e way , why are they always round?) , th e cars , etc. ; th e purpos e of thi s activit y i s t o creat e a n awarenes s o f th e pattern s aroun d us . By Nature's Design i s a photographi c journe y wit h a n ey e fo r man y of thes e natura l patterns .

• Student s "tessellate " th e plan e usin g squares , triangles , o r hexagon s to completel y cove r a shee t o f pape r withou t overlapping . The y als o tessellate the plan e using groups of shapes, like hexagons and triangle s as i n th e figure below .

• Student s migh t as k i f thei r parent s woul d b e willin g t o giv e the m a penny fo r th e first tim e the y d o a particular chore , tw o pennies fo r th e second tim e the y d o th e chore , fou r pennie s fo r th e thir d time , eigh t pennies fo r th e fourt h time , an d s o on . Befor e asking , the y shoul d investigate, perhap s usin g tower s o f Unifi x cube s tha t kee p doublin g in height , ho w lon g thei r parent s coul d actuall y affor d t o pa y the m for doin g th e chore .

• Student s cu t a shee t o f pape r int o tw o halves , cu t th e resultin g tw o pieces into halves, cut th e resultin g fou r piece s into halves, etc. If they do this a number of times, say 12 times, and stacked all the pieces of paper on top of each other, how high would the pile of paper be? Students estimat e th e heigh t befor e performin g an y calculations .

• Student s colo r hal f a larg e square, the n hal f o f the remainin g portio n with anothe r color , the n hal f o f th e remainin g portio n wit h a thir d color, etc . Will the entire area ever get colored? Why, or why not?

• Student s coun t th e numbe r o f rows o f bract s o n a pineappl e o r pin e cone, o r rows o f petal s o n a n artichoke , o r rows o f seed s o n a sun - flower, an d verif y tha t thes e number s al l appea r i n th e sequenc e 1 , 1 , 2 , 3 , 5 , 8 , 1 3 , 2 1 , 3 4 , . . . o f Fibonacc i numbers , wher e eac h numbe r i s

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 15 5

the su m o f th e tw o previou s number s o n th e list . Student s find othe r pictures depictin g Fibonacc i number s a s the y aris e i n nature , refer - ring, fo r example , t o Fibonacci Numbers in Nature. I n Mathematical Mystery Tour b y Mar k Wahl , a n elementar y schoo l teache r provide s a year' s wort h o f Fibonacc i exploration s an d activities .

• Usin g a larg e equilatera l triangl e provide d b y th e teacher , student s find an d connec t th e approximat e midpoint s o f th e thre e sides , an d then colo r the triangle in the middle. (Se e Stage 1 picture.) The y the n repeat thi s procedure wit h eac h of the thre e uncolore d triangle s t o ge t the Stag e 2 picture, an d the n repea t thi s procedur e agai n wit h eac h of the nin e uncolore d triangle s t o ge t th e Stag e 3 picture. Thes e ar e th e first thre e stage s o f th e Sierpinsk i triangle ; subsequen t stage s becom e increasingly intricate . How many uncolored triangles are there in the Stage 3 picture ? How many would there be in the Stage 4 picture if the procedure were repeated again?

/ \ / • \ rwv\ /fl^ & Stag e 0 Stag e 1 Stag e 2 Stag e 3

4. Investigat e way s t o represen t an d classif y d a t a accordin g t o attributes, suc h a s shap e o r color , an d relationships , an d discuss t h e purpos e an d usefulnes s o f suc h classification .

• Student s ar e provide d wit h a se t o f inde x card s o n eac h o f whic h i s written a wor d (o r a number) . Workin g i n groups , student s pu t th e cards i n alphabetica l (o r numerical ) order , explai n th e method s the y used t o d o this , an d the n compar e th e variou s method s tha t wer e used.

• Student s brin g to class names of cities and thei r zi p codes where thei r relatives an d friend s live , paste thes e a t th e appropriat e location s on a map o f th e Unite d States , an d loo k fo r pattern s whic h migh t explai n how zi p code s ar e assigned . The n the y compar e thei r conclusion s with pos t offic e informatio n t o se e whethe r the y ar e consisten t wit h the wa y tha t zi p code s actuall y ar e assigned .

• Student s sen d an d decod e message s i n whic h eac h lette r ha s bee n replaced b y th e lette r whic h follow s i t i n th e alphabe t (o r occur s tw o letters later) . Student s explor e othe r codin g system s describe d i n Let's Investigate Codes and Sequences b y Mario n Smoothey .

• Student s collec t informatio n abou t th e sof t drink s the y prefe r an d discuss variou s way s o f presentin g th e resultin g information , suc h a s tables, ba r graphs , an d pi e charts , displaye d bot h o n pape r an d o n a computer.

156 JOSEPH G . ROSENSTEI N

• Student s pla y th e gam e o f Set i n whic h participant s tr y t o identif y three card s fro m thos e o n displa y which , fo r eac h o f fou r attribute s (number, shape , color , an d shading) , al l shar e th e attribut e o r ar e al l different. Simila r idea s ca n b e explore d usin g Tabletop, Jr. software .

5. Follow , devise , an d describ e practica l list s o f instructions .

• Student s follo w a recip e t o mak e a cak e o r t o assembl e a simpl e to y from it s componen t parts , an d the n writ e thei r ow n version s o f thos e instructions.

• Student s giv e writte n an d ora l direction s fo r goin g fro m th e class - room t o anothe r roo m i n th e school , an d represen t thes e direction s with a diagra m draw n approximatel y t o scale .

• Student s rea d Anno's Mysterious Multiplying Jar b y Mitsumas a Anno. Durin g a secon d readin g the y devis e a metho d t o recor d an d keep trac k o f th e increasin g numbe r o f item s i n th e boo k an d pre - dict ho w tha t numbe r wil l continu e t o grow . Eac h grou p explain s it s method t o th e class .

• Student s writ e step-by-ste p direction s fo r a simpl e tas k lik e makin g a peanu t butte r an d jell y sandwich , an d follo w the m t o prov e tha t they work .

• Student s fin d an d describ e th e shortes t pat h fro m th e compute r t o the doo r o r fro m on e locatio n i n th e schoo l buildin g t o another .

• Student s fin d th e shortes t rout e fro m schoo l t o hom e o n a ma p (se e figure below) , wher e eac h edg e ha s a specifie d numerica l lengt h i n meters; student s modif y length s t o obtai n a differen t shortes t route .

• Student s writ e a program whic h wil l create specifie d picture s o r pat - terns, suc h a s a hous e o r a clow n fac e o r a symmetrica l design . Log o software i s well-suite d t o thi s activity . I n Turtle Math, student s us e Logo command s t o g o o n a treasur e hunt , an d loo k fo r th e shortes t route t o complet e th e search .

• Workin g i n groups , student s creat e an d explai n a fai r wa y o f sharin g a bagfu l o f simila r candie s o r cookies . (Se e als o th e vignett e enti - tled Sharing A Snack i n th e Introductio n t o thi s Framework.) Fo r example, i f th e ba g ha s 3 0 brownie s an d ther e ar e 2 0 children , the n they migh t sugges t tha t eac h chil d get s on e whol e browni e an d tha t the teache r divid e eac h o f th e remainin g brownie s i n half . O r the y might sugges t tha t eac h pai r o f childre n figur e ou t ho w t o shar e on e

A COMPREHENSIV E VIE W O F DISCRET E MATHEMATIC S 15 7

brownie. What if there were 30 hard candies instead of brownies? What if there were 25 brownies? What if there were 15 brownies and 15 chocolate chip cookies? Th e purpos e o f this activit y i s for student s to brainstor m possibl e solution s i n th e situation s wher e ther e ma y b e no solutio n tha t everyone perceive s a s fair .

• Student s devis e a strateg y fo r neve r losin g a t tic-tac-toe .

• Student s find different way s of paving just enoug h streets of a "mudd y city" (lik e the street ma p below , perhap s lai d out o n the floor) s o tha t a chil d ca n wal k fro m an y on e locatio n t o an y othe r locatio n alon g paved roadways . I n "mudd y city " non e of the road s ar e paved, s o tha t whenever i t rain s al l street s tur n t o mud . Th e mayo r ha s aske d th e class t o propos e differen t way s o f pavin g th e road s s o tha t a perso n can ge t fro m an y on e locatio n t o an y othe r locatio n o n pave d roads , but s o tha t th e fewes t numbe r o f road s possibl e ar e paved .

• Student s divid e a collectio n o f Cuisenair e rod s o f differen t length s into tw o o r thre e group s whos e tota l length s ar e equa l (o r a s clos e t o equal a s possible) .

References.

• Anno , M . Anno's Mysterious Multiplying Jar. Philome l Books , 1983 .

• Asher , M . Ethnomathematics. Brooks/Col e Publishin g Company , 1991 .

• Casey , Nancy , an d Mik e Fellows . This is MEGA-Mathematics! - Stories and Activities for Mathematical Thinking, Problem-Solving, and Commu- nication. Lo s Alamos , CA : Lo s Alamo s Nationa l Laboratories , 1993 . ( A version i s availabl e onlin e a t h t t p : / / w w w . c 3 . l a n l . g o v / m e g a - m a t h )

• Chavey , Darrah . Drawing Pictures with One Line: Exploring Graph The- ory. Consortiu m fo r Mathematic s an d It s Application s (COMAP) , Modul e # 2 1 , 1992 .

• Downie , D. , T . Slesnick , an d J . Stenmark . Math for Girls and Other Prob- lem Solvers. EQUALS . Lawrenc e Hal l o f Science , 1981 .

• Ernst , L . Sam Johnson and the Blue Ribbon Quilt. Mulberr y Paperbac k Book, 1992 .

• Francis , R . The Mathematician's Coloring Book. Consortiu m fo r Mathe - matics an d It s Application s (COMAP) , Modul e # 1 3 , 1989 .

• Fibonacci Numbers in Nature. Dal e Seymou r Publications .

158 J O S E P H G . R O S E N S T E I N

• Friedman , A . A Cloak for a Dreamer. Pengui n Books . Scholastic .

• Kenney , M . J. , Ed . Discrete Mathematics Across the Curriculum K-12. 1991 Yearbook o f the National Counci l of Teachers of Mathematics (NCTM) . Reston, VA : 1991.

• Kohl , Herbert . Insides, Outsides, Loops, and Lines. Ne w York : W . H . Freeman, 1995 .

• Murphy , P . By Nature's Design. Sa n Francisco, CA : Chronicle Books, 1993.

• Pinczes , E . J . One Hundred Hungry Ants. Houghto n Miffli n Company , 1993.

• Rosenstein , J . G. , D . Franzblau , an d F . Roberts , Eds . Discrete Mathemat- ics in the Schools. Proceeding s o f a 199 2 DIMAC S Conferenc e o n "Discret e Mathematics i n th e Schools. " DIMAC S Serie s o n Discret e Mathematic s an d Theoretical Compute r Science . Providence , RI : American Mathematica l So - ciety (AMS) , 1997 .

• Set. Se t Enterprises .

• Smoothey , Marion . Let's Investigate Codes and Sequences. Ne w York : Marshall Cavendis h Corporation , 1995 .

• Tompert , Ann . Grandfather Tang's Story. Crow n Publishing , 1990 .

• Wahl , Mark . Mathematical Mystery Tour: Higher-Thinking Math Tasks. Tucson, AZ : Zephy r Press , 1988 .

Software.

• Logo. Man y version s o f Log o ar e commerciall y available .

• Tabletop, Jr. Broderbun d Software . TERC .

• Turtle Math. LCSI .

6. Grade s 5- 6 Overvie w

The five majo r theme s o f discret e mathematics , a s discusse d i n th e K - 12 Overview , ar e systemati c listing , counting , an d reasoning ; dis - crete m a t h e m a t i c a l modelin g usin g graph s (networks ) an d trees ; iterative (tha t is , repetitive ) pattern s an d processes ; organizin g and processin g information ; an d followin g an d devisin g list s o f in - structions, calle d "algorithms, " an d usin g t h e m t o fin d t h e bes t solution t o real-worl d problems . Tw o importan t resource s o n discret e mathematics fo r teacher s a t al l levels are the 199 1 NCTM Yearboo k Discrete Mathematics Across the Curriculum K-12 an d th e 199 7 DIMAC S Volum e Discrete Mathematics in the Schools.

Despite thei r formidabl e titles , thes e five theme s ca n b e addresse d wit h activities a t th e 5- 6 grad e leve l whic h involv e bot h th e purposefu l pla y an d simple analysi s suggeste d fo r elementar y schoo l student s an d experimenta - tion an d abstractio n appropriat e a t th e middl e grades . Indeed , teacher s wil l

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 159

discover tha t man y activitie s tha t the y alread y ar e usin g i n thei r classroom s reflect thes e themes . Thes e five theme s ar e discusse d i n th e paragraph s below.

The following discussion of activities at the 5-6 grade levels in discrete mathematics presupposes that corresponding activities have taken place at the K-4 grade levels. Hence 5-6 grade teachers should review the K-2 and 3-4 grade level discussions of discrete mathematics and might use activities similar to those described there before introducing the activities for this grade level.

Activities involvin g s y s t e m a t i c listing , counting , an d reasonin g at K- 4 grad e level s ca n b e extende d t o th e 5- 6 grad e level . Fo r example , they migh t determin e th e numbe r o f possibl e licens e plate s wit h tw o letter s followed b y thre e number s followe d b y on e letter , an d decid e whethe r thi s total numbe r o f licens e plate s i s adequat e fo r al l Ne w Jerse y drivers . The y need t o becom e familia r wit h th e ide a o f permutations , tha t is , th e differen t ways i n which a grou p o f item s ca n b e arranged . Thus , fo r example , i f thre e children ar e standin g b y th e blackboard , ther e ar e altogethe r si x differen t ways, cal l permutations , i n whic h thi s ca n b e done ; fo r example , i f the thre e children ar e Am y (A) , Bethan y (B) , an d Coriande r (C) , th e si x differen t permutations ca n b e describe d a s ABC , ACB , BAC , BCA , CAB , an d CBA. Similarly , th e tota l numbe r o f differen t way s i n whic h thre e student s out o f a clas s o f thirt y ca n b e arrange d a t th e blackboar d i s altogethe r 30x29x28, o r 24,36 0 ways , a n amazin g total !

An importan t discret e mathematica l m o d e l i s tha t o f a networ k or graph , whic h consist s o f dot s an d line s joinin g th e dots ; th e dot s ar e often calle d vertices (vertex i s th e singular ) an d th e line s ar e ofte n calle d edges. (Thi s i s differen t fro m othe r mathematica l use s o f th e ter m "graph" ; the tw o term s "network " an d "graph " ar e use d interchangeabl y fo r thi s concept.) A n exampl e o f a grap h wit h 2 4 vertice s an d 3 8 edge s i s give n below. Graph s ca n b e use d t o represen t island s an d bridges , o r building s and roads , o r house s an d telephon e cables ; whereve r a collectio n o f thing s are joine d b y connectors , th e mathematica l mode l use d i s tha t o f a graph . At th e 5- 6 level , student s shoul d b e familia r wit h th e notio n o f a grap h and recogniz e situation s i n whic h graph s ca n b e a n appropriat e model . Fo r example, the y shoul d b e familia r wit h problem s involvin g route s fo r garbag e pick-ups, schoo l buses , mai l deliveries , sno w removal , etc. ; the y shoul d b e able t o mode l suc h problem s b y usin g graphs , an d b e abl e t o solv e suc h problems b y finding suitabl e path s i n thes e graphs , suc h a s i n th e tow n whose stree t ma p i s th e grap h below .

p • i

T T T T

> • • • •

160 J O S E P H G . R O S E N S T E I N

Students shoul d recogniz e an d wor k wit h repetitiv e pattern s an d processes involvin g number s an d shapes , wit h object s foun d i n th e class - room an d i n th e worl d aroun d them . Buildin g o n thes e explorations , fifth- and sixth-grader s shoul d als o recogniz e an d wor k wit h iterativ e an d re - cursive processes . The y explor e iteratio n usin g Log o software, wher e the y recreate a variet y o f interestin g pattern s (suc h a s a checkerboard ) b y iter - ating th e constructio n o f a simpl e componen t o f th e patter n (i n thi s cas e a square) . A s wit h younge r students , 5t h an d 6t h grader s ar e fascinate d with th e Fibonacc i sequenc e 1,1 , 2,3,5,8,13,21,34, 5 5 , 8 9 , . . . wher e ever y number i s th e su m o f th e previou s tw o numbers . Althoug h th e Fibonacc i sequence start s wit h smal l numbers , th e number s i n th e sequenc e becom e large very quickly . Student s ca n no w also begin t o understan d th e Fibonacc i sequence an d othe r sequence s recursively — where each term o f the sequenc e is describe d i n term s o f precedin g terms .

Students i n th e 5t h an d 6t h grad e shoul d investigat e sortin g i t e m s using Venn diagrams, an d continu e their exploration s of recovering hidde n information b y decodin g messages . The y shoul d begi n t o explor e ho w codes ar e use d t o communicat e information , b y traditiona l method s such a s Mors e cod e o r semaphor e (flag s use d fo r ship-to-shi p messages ) an d also b y curren t method s suc h a s zi p codes , whic h describ e a locatio n i n th e United State s b y a five-digit (o r nine-digit ) number . Student s shoul d als o explore modular arithmeti c throug h application s involvin g clocks, calendars , and binar y codes .

Finally, a t grade s 5-6 , student s shoul d b e abl e t o describe , devise , and tes t algorithm s fo r solvin g a variet y o f problems . Thes e includ e finding th e shortest rout e fro m on e location to another , dividin g a cake fairly , planning a tournament schedule , an d plannin g layout s fo r a class newspaper .

Two importan t resource s o n discret e mathematic s fo r teacher s a t al l levels i s th e 199 1 NCT M Yearboo k Discrete Mathematics Across the Cur- riculum K-12 an d th e 199 7 DIMAC S Volum e Discrete Mathematics in the Schools. Anothe r importan t resourc e fo r 5- 6 teacher s i s This Is MEGA- Mathematics!

7. Grade s 5- 6 Indicator s an d Activitie s

The cumulativ e progres s indicator s fo r grad e 8 appear belo w i n boldfac e type. Eac h indicato r i s followe d b y activitie s whic h illustrat e ho w i t ca n b e addressed i n th e classroo m i n grade s 5 an d 6 .

Building upo n knowledg e an d skill s gaine d i n th e precedin g grades , ex - periences wil l b e suc h tha t al l student s i n grade s 5-6 :

6. U s e s y s t e m a t i c listing , counting , an d reasonin g i n a variet y of differen t c o n t e x t s .

• Student s determin e th e numbe r o f different sandwiche s o r hamburg - ers that ca n be created a t loca l eateries usin g a combination o f specifi c ingredients.

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 161

• Student s find th e number o f different way s to make a row of flowers each o f which i s red or yellow, i f the row has 1, 2, 3, 4, or 5 flowers. Modeling thi s wit h Unifi x cubes , the y discove r tha t addin g a n addi - tional flower to the row doubles the number o f possible rows , provid e explanations fo r this, and generalize to longer rows . Simila r activitie s can be found i n the Pizza Possibilities an d Two- Toned Towers lesson s that ar e described i n the First Fou r Standard s o f this Framework.

• Student s find th e number o f ways o f asking thre e differen t student s in th e class to write thre e homewor k problem s o n the blackboard .

• Student s understan d an d us e th e concep t o f permutation . The y determine th e numbe r o f way s an y five item s ca n b e arrange d i n order, justif y thei r conclusio n usin g a tree diagram , an d use factoria l notation, 5! , to summarize th e result.

• Student s find the number of possible telephone number s wit h a given area cod e an d investigate wh y several year s ag o the telephone com - pany introduce d a ne w area cod e (908 ) in New Jersey, an d why ad - ditional are a code s ar e being introduce d i n 1997 . Is the situation the same with zip codes?

• Student s estimat e an d then calculat e th e number o f possible licens e plates wit h tw o letter s followe d b y three number s followe d b y one let - ter. The y investigat e wh y the state licens e burea u trie d t o introduc e license plate s wit h seve n character s an d why thi s attemp t migh t hav e been unsuccessful .

• Student s explor e the sequence of triangular number s 1 , 1+2, 1+2+3, 1+2+3+4, . . . whic h represen t th e number o f dots i n the triangula r arrays below , an d find the location o f the triangula r number s i n Pas- cal's triangle .

o o o

o o o o o o

o o o o o o o o o o

o o o o o o o o o o o o o o o

• Student s loo k fo r pattern s i n the various diagonal s o f Pascal' s tri - angle, an d in the differences betwee n consecutiv e term s i n thes e di- agonals. Patterns in Pascal's Triangle Poster i s a nic e resourc e fo r introducing thes e ideas .

• Student s analyz e simpl e game s lik e th e following : Bet h win s th e game wheneve r th e tw o dic e giv e a n eve n total , an d Hobar t win s whenever th e tw o dic e giv e a n od d total . The y pla y th e gam e a number o f times , an d usin g experimenta l evidence , decid e whethe r the gam e i s fair, and , if not, whic h playe r i s more likel y to win. The y then tr y t o justif y thei r conclusion s theoretically , b y countin g th e

162 J O S E P H G . R O S E N S T E I N

number o f combination s o f dic e tha t woul d resul t i n a wi n fo r eac h player.

• Student s creat e a tabl e i n th e for m o f a gri d whic h indicate s ho w many o f eac h o f th e coin s o f th e fictitious countr y "Ternamy " — i n denominations o f 1 , 3 , 9 , 27 , an d 8 1 "terns " — ar e neede d t o mak e up an y amoun t fro m 1 t o 200 . The y lis t th e denomination s i n th e columns a t th e to p o f th e tabl e an d th e amount s the y ar e tryin g t o make i n th e row s a t th e left . The y writ e th e numbe r o f eac h coi n needed t o ad d u p t o th e desire d amoun t i n th e appropriat e square s i n that row . Th e onl y "rule " t o b e followe d i s tha t th e leas t numbe r o f coins mus t b e used ; fo r example , thre e l' s shoul d alway s b e replace d by on e 3 . Thi s tabl e ca n b e use d t o introduc e bas e 3 ("ternary" ) numbers, an d the n number s i n othe r bases .

7. Recogniz e c o m m o n discret e m a t h e m a t i c a l models , explor e their properties , an d desig n t h e m fo r specifi c situations .

• Student s experimen t wit h drawin g make-believ e map s whic h ca n b e colored wit h two , three , an d fou r color s (wher e adjacen t countrie s must hav e different colors) , an d explai n wh y their fictitious maps , an d real map s lik e th e ma p o f th e 5 0 states, canno t b e colore d wit h fewe r colors. Not e tha t i t wa s prove n i n 197 6 that n o ma p ca n b e draw n o n a flat surfac e whic h require s mor e tha n fou r colors . The Mathemati- cian's Coloring Book contain s a variet y o f map-colorin g activities , a s well a s historica l backgroun d o n th e ma p colorin g problem .

• Student s pla y game s usin g graphs . Fo r example , i n th e strollin g game, tw o player s strol l togethe r o n a pat h throug h th e grap h whic h never repeat s itself ; the y alternat e i n selectin g edge s fo r th e path , and th e winne r i s th e on e wh o select s th e las t edg e o n th e path . Who wins ? I n th e gam e below , Charle s an d Dian e bot h star t a t V , Charles pick s th e first edg e (marke d 1 ) an d the y bot h strol l dow n that edge . The n Dian e pick s th e secon d edg e (marke d 2 ) an d th e game continues . Dian e ha s wo n thi s pla y o f th e gam e sinc e th e pat h cannot b e continue d afte r th e sixt h edg e withou t repeatin g itself . Does Diane have a way of always winning this game, or does Charles have a winning strategy? What if there was a different starting point? What if a different graph was used? What if the path must not cross itself (instead of requiring that it not repeat itself) ? Student s shoul d try t o explain i n each case why a certain playe r ha s a winning strategy .

• Student s find path s i n graph s whic h utiliz e eac h edg e exactly once ; a path i n a grap h i s a sequenc e o f edges eac h o f which begin s wher e th e previous on e ends. The y appl y thi s ide a by converting a street ma p t o

3 / v N

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 163

a graph wher e vertices on the graph correspon d t o intersections o n th e street map , an d b y usin g thi s grap h t o determin e whethe r a garbag e truck ca n complet e it s secto r withou t repeatin g an y streets . Se e th e segment Snowbound: Euler Circuits o n the videotap e Geometry: New Tools for New Technologies; the modul e Drawing Pictures With One Line provide s a stron g backgroun d fo r problem s o f thi s kind .

• Student s pla n emergenc y evacuatio n route s a t schoo l o r fro m hom e using graphs .

• Al l o f th e student s togethe r creat e a "huma n graph " wher e eac h child i n th e clas s i s holdin g tw o strings , on e i n eac h hand . Thi s ca n be accomplishe d b y placin g i n th e cente r o f th e roo m a numbe r o f pieces o f yar n (eac h si x fee t long ) equa l t o th e numbe r o f students , and havin g eac h studen t tak e th e end s o f tw o strings . Th e childre n are aske d t o untangl e themselves , an d discus s o r writ e abou t wha t happens.

• Student s pla y th e gam e o f Sprouts , i n which tw o student s tak e turn s in buildin g a grap h unti l on e o f the m (th e winner! ) complete s th e graph. Th e rule s are : star t th e gam e wit h tw o o r thre e vertices ; eac h person add s a n edg e (i t ca n b e a curve d line! ) joinin g tw o vertices , and the n add s a ne w verte x a t th e cente r o f tha t edge ; n o mor e tha n three edge s ca n occu r a t a vertex ; edge s ma y no t cross . I n th e sampl e game below , th e secon d playe r (B ) win s becaus e th e firs t playe r (A ) cannot dra w an edge connecting the only two vertices that hav e degre e less tha n thre e withou t crossin g a n existin g edge .

m o o

STAR T

#1 r~\ AMOVE S

#2 O B MOVE S

^ AMOVE S

^ B MOVE S

. Experimen t w i t h iterativ e an d recursiv e processes , w i t h t h e aid o f calculator s an d computers .

• Student s develo p a metho d fo r solvin g th e Towe r o f Hano i problem : There ar e thre e pegs , o n th e first o f whic h i s stacke d five disks , eac h smaller tha n th e one s underneath i t (se e diagram below) ; the proble m is t o mov e th e entir e stac k t o th e thir d peg , movin g disks , on e a t a time, fro m an y pe g t o eithe r o f th e othe r tw o pegs , wit h n o dis k eve r placed upo n a smalle r one . How many moves are required to do this?

164 J O S E P H G . R O S E N S T E I N

• Student s us e iteration i n Logo software t o draw checkerboards , stars , and othe r designs . Fo r example , the y iterat e th e constructio n o f a simple componen t o f a pattern, suc h a s a square, t o recreat e a n entir e checkerboard design .

• Student s us e pape r rabbit s (prepare d b y th e teacher ) wit h whic h to simulat e Fibonacci' s 13t h centur y investigatio n int o th e growt h o f rabbit populations : / / you start with one pair of baby rabbits, how many pairs of rabbits will there be a year later? Fibonacci' s assump - tion wa s tha t eac h pai r o f bab y rabbit s result s i n anothe r pai r o f baby rabbit s tw o month s late r — allowin g a mont h fo r maturatio n and a mont h fo r gestation . Onc e mature , eac h pai r ha s bab y rab - bits monthly . (Eac h pai r o f student s shoul d b e provide d wit h 1 8 cardboard pair s eac h o f bab y rabbits , not-yet-matur e rabbits , an d mature rabbits. ) The Fascinating Fibonaccis b y Trud i Garlan d illus - trates th e rabbit proble m an d a number o f other interestin g Fibonacc i facts. I n Mathematics Mystery Tour b y Mar k Wahl , a n elementar y school teache r provide s a year' s wort h o f Fibonacc i exploration s an d activities.

• Student s us e calculators to compar e th e growt h o f various sequences , including counting by 4's ( 4 , 8 , 1 2 , 1 6 , . . . ) , doubling (1 , 2,4, 8 , 1 6 , . . . ) , squaring (1,4,9,16 , 2 5 , . . . ), an d Fibonacc i ( 1 , 1 , 2 , 3 , 5 , 8 , 1 3 , . . . ) .

• Student s explore their surroundings t o find rectangular object s whos e ratio o f lengt h t o widt h i s th e "golde n ratio. " Sinc e th e golde n rati o can b e approximate d b y th e rati o o f tw o successiv e Fibonacc i num - bers, student s shoul d cu t a rectangular peephol e o f dimensions 21m m x 3 4 m m ou t o f a piec e o f cardboard , an d us e i t t o "frame " potentia l objects; whe n i t "fits, " th e objec t i s a golden rectangle. The y describ e these activitie s i n thei r mat h journals .

• Student s stud y th e pattern s o f patchwor k quilts , an d mak e on e o f their own . The y migh t first rea d Eight Hands Round.

• Student s mak e equilatera l triangle s whos e side s ar e 9" , 3" , an d 1 " (or othe r length s i n rati o 3:1) , an d us e the m t o construc t "Koc h snowflakes o f stage 2 " (a s show n below ) b y pastin g th e 9 " triangl e o n a larg e shee t o f paper , thre e 3 " triangle s a t th e middl e o f th e thre e sides o f the 9 " triangl e (pointin g outward) , an d twelv e 1 " triangle s a t the middl e o f th e expose d side s o f th e twelv e 3 " segment s (pointin g outward). T o ge t Koc h snowflake s o f stag e 3 , ad d forty-eigh t 1/3 " equilateral triangles . How many 1/9" equilateral triangles would be needed for the Koch snowflake of stage 4 ? Fractals for the Classroom is a valuabl e resourc e fo r thes e kind s o f activitie s an d explorations .

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 165

Stag e 0 Stag e 1 Stag e 2

• Student s mar k on e en d o f a lon g strin g an d mak e anothe r mar k midway betwee n th e two ends. The y the n continu e marking the strin g by followin g som e simpl e rul e suc h a s "mak e a ne w mar k midwa y between th e las t midwa y mar k an d th e marke d end " an d the n repea t this instruction . Student s investigat e th e relationshi p o f th e length s of the segment s betwee n marks . How many marks are possible in this process if it is assumed that the marks take up no space on the string? What happens if the rule is changed to u make a new mark midway between the last two marks?"

. Explor e m e t h o d s fo r storing , processing , an d communicatin g information.

• Afte r discussin g possible methods fo r communicating messages acros s a footbal l field, team s o f student s devis e method s fo r transmittin g a short messag e (usin g flags, flashlights, ar m signals , etc.) . Eac h tea m receives a messag e o f th e sam e lengt h an d mus t transmi t i t t o mem - bers o f the tea m a t th e othe r en d o f the field as quickly an d accuratel y as possible .

• Student s devis e rule s s o tha t arithmeti c expression s withou t paren - theses, suc h a s 5 x 8 — 2/7 , ca n b e evaluate d unambiguously . The y then experimen t wit h calculator s t o discove r th e calculators ' built-i n rules fo r evaluatin g thes e expressions .

• Student s explor e binar y arithmeti c an d arithmeti c fo r othe r base s through application s involving clocks (bas e 12) , days of the week (bas e 7), an d binar y (bas e 2 ) codes .

• Student s assig n eac h lette r i n th e alphabe t a numerica l valu e (pos - sibly negative ) an d the n loo k fo r word s wort h a specifie d numbe r o f points.

• Student s sen d an d decod e message s i n whic h letter s o f th e messag e are systematicall y replace d b y othe r letters . The Secret Code Book by Hele n Huckl e show s thes e codin g system s a s wel l a s others .

• Student s us e Venn diagrams t o sort an d the n repor t o n their findings in a survey . Fo r example , the y ca n see k response s t o th e question , When I grow up I want to be a) rich and famous, b) a parent, c) in a profession I love, where respondents ca n choos e more than on e option . The result s ca n b e sorte d int o a Ven n diagra m lik e that below , wher e

166 J O S E P H G . ROSENSTEI N

entries "m " an d "f ' ar e use d fo r mal e an d femal e students . Th e clas s can the n determin e answer s t o question s lik e Are males or females in our class more likely to have a single focus? Tabletop, Jr. softwar e can b e use d t o sor t an d explor e dat a usin g Ven n diagrams .

10. D e v i s e , describe , an d tes t algorithm s fo r solvin g optimiza - t i o n an d searc h problems .

• Student s us e a systemati c procedur e t o find th e tota l numbe r o f routes fro m on e locatio n i n thei r tow n t o another , an d th e shortes t such route . (Se e Problem Solving Using Graphs.)

• I n Turtle Math, student s us e Log o command s t o g o o n a treasur e hunt, an d loo k fo r th e shortes t rout e t o complet e th e search .

• Student s discus s an d writ e abou t variou s method s o f dividing a cak e fairly, suc h a s th e "divider/choose r method " fo r tw o peopl e (on e per - son divides , th e othe r chooses ) an d th e "lon e choose r method " fo r three peopl e (tw o peopl e divid e th e cak e usin g th e divider/choose r method, the n eac h cut s his/he r hal f int o thirds , an d the n th e thir d person take s on e piec e fro m eac h o f th e others) . Fair Division: Get- ting Your Fair Share can be used t o explore methods o f fairly dividin g a cak e o r a n estate .

• Student s conduc t a clas s surve y fo r th e to p te n song s an d discus s different way s t o us e th e informatio n t o selec t th e winners .

• Student s devis e a telephon e tre e fo r disseminatin g message s t o al l 6th grad e student s an d thei r parents .

• Student s schedul e th e matche s o f a volleybal l tournamen t i n whic h each tea m play s eac h othe r tea m once .

• Student s us e flowchart s t o represen t visuall y th e instruction s fo r carrying ou t a comple x project , suc h a s schedulin g th e productio n of th e clas s newspaper .

• Student s develo p an algorith m t o create a n efficien t layou t fo r a clas s newspaper.

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 16 7

References.

• Bennett , S. , e t al . Fair Division: Getting Your Fair Share. Consortiu m for Mathematic s an d It s Application s (COMAP) . Modul e # 9 , 1987 .

• Chavey , D . Drawing Pictures With One Line. Consortiu m fo r Mathematic s and It s Application s (COMAP) . Modul e # 2 1 , 1992 .

• Cozzens , M. , an d R . Porter . Problem Solving Using Graphs. Consortiu m for Mathematic s an d It s Application s (COMAP) . Modul e # 6 , 1987 .

• Francis , R . The Mathematician's Coloring Book. Consortiu m fo r Mathe - matics an d It s Application s (COMAP) . Modul e # 1 3 .

• Garland , Trudi . The Fascinating Fibonaccis. Pal o Alto , CA : Dal e Seymo r Publications, 1987 .

• Huckle , Helen . The Secret Code Book. Dia l Books .

• Kenney , M . J. , Ed . Discrete Mathematics Across the Curriculum K-12. 1991 Yearbook o f the National Counci l of Teachers of Mathematics (NCTM) . Reston, VA , 1991 .

• Paul , A . Eight Hands Round. Ne w York : Harpe r Collins , 1991 .

• Peitgen , Heinz-Otto , e t al . Fractals for the Classroom: Strategic Activities Volume One & Two. Reston , VA : NCT M an d Ne w York : Springer-Verlag , 1992.

© Rosenstein, J . G. , D . Pranzblau , an d F . Roberts , Eds . Discrete Mathemat- ics in the Schools. Proceeding s o f a 199 2 DIMAC S Conferenc e o n "Discret e Mathematics i n th e Schools. " DIMAC S Serie s o n Discret e Mathematic s an d Theoretical Compute r Science . Providence , RI : American Mathematica l So - ciety (AMS) , 1997 .

• Wahl , Mark . Mathematical Mystery Tour: Higher-Thinking Math Tasks. Tucson, AZ : Zephy r Press , 1988 .

Software.

• Logo. Man y version s o f Log o ar e commerciall y available .

• Tabletop, Jr. Broderbund , TERC .

• Turtle Math. LCSI .

Video.

• Geometry: New Tools for New Technologies, videotap e b y th e Consortiu m for Mathematic s an d It s Application s (COMAP) . Lexington , MA , 1992 .

168 J O S E P H G . ROSENSTEI N

8. Grade s 7- 8 Overvie w

The five major theme s o f discrete mathematics , a s discusse d i n th e K-1 2 Overview, ar e systemati c listing , counting , an d reasoning ; discret e mathematical modelin g usin g graph s (networks ) an d trees ; iter - ative (tha t is , repetitive ) pattern s an d processes ; organizin g an d processing information ; an d followin g an d devisin g list s o f instruc - tions, calle d "algorithms, " an d usin g t h e m t o fin d t h e b e s t solutio n t o real-worl d problems .

Despite thei r formidabl e titles , thes e five theme s ca n b e addresse d wit h activities a t th e 7- 8 grad e leve l whic h involv e bot h th e purposefu l pla y an d simple analysi s suggeste d fo r elementar y schoo l student s an d experimenta - tion an d abstractio n appropriat e a t th e middl e grades . Indeed , teacher s wil l discover tha t man y activitie s tha t the y alread y ar e usin g i n thei r classroom s reflect thes e themes . Thes e five theme s ar e discusse d i n th e paragraph s below.

The following discussion of activities at the 1-8 grade levels in discrete mathematics presupposes that corresponding activities have taken place at the K-6 grade levels. Hence 1-8 grade teachers should review the K-2, 3- 4, and 5-6 grade level discussions of discrete mathematics and might use activities similar to those described there before introducing the activities for this grade level.

Students i n 7t h an d 8t h grad e shoul d b e abl e t o us e p e r m u t a t i o n s and combination s an d othe r countin g strategie s i n a wid e variet y of contexts . I n additio n t o workin g wit h permutations , wher e th e orde r of th e item s i s importan t (se e Grade s 5- 6 Overvie w an d Activities) , the y should als o be abl e t o wor k wit h combinations , wher e th e orde r o f the item s is irrelevant . Fo r example , th e numbe r o f differen t thre e digi t number s tha t can b e mad e usin g thre e differen t digit s i s 1 0 x 9 x 8 because eac h differen t ordering o f th e thre e digit s result s i n a differen t number . However , th e number o f differen t pizza s tha t ca n b e mad e usin g thre e o f te n availabl e toppings i s (1 0 x 9 x 8)/( 3 x 2 x 1 ) becaus e th e order m whic h th e topping s are adde d i s irrelevant ; th e divisio n b y 3 x 2 x 1 eliminates th e duplication .

An importan t discret e mathematica l m o d e l i s tha t o f a networ k or graph , whic h consist s o f dot s an d line s joinin g th e dots ; th e dot s ar e often calle d vertices {vertex i s th e singular ) an d th e line s ar e ofte n calle d edges. (Thi s i s different fro m othe r mathematica l use s o f the ter m "graph." ) Graphs ca n b e use d t o represen t island s an d bridges , o r building s an d roads , or house s an d telephon e cables ; whereve r a collectio n o f thing s ar e joine d by connectors , th e mathematica l mode l use d i s that o f a graph . Student s i n the 7t h an d 8t h grade s shoul d b e abl e to us e graph s t o m o d e l situation s and solv e problem s usin g t h e model . Fo r example , student s shoul d b e able t o us e graph s t o schedul e a school' s extracurricula r activitie s s o that, i f at al l possible , n o on e i s excluded becaus e o f conflicts . Thi s ca n b e don e b y creating a grap h whos e vertice s ar e th e activities , wit h tw o activitie s joine d

A COMPREHENSIVE VIE W O F DISCRET E MATHEMATIC S 16 9

by a n edg e i f the y hav e a perso n i n common , s o tha t th e activitie s shoul d be schedule d fo r differen t times . Colorin g th e vertice s o f th e grap h s o tha t adjacent vertice s hav e differen t colors , usin g a minimu m numbe r o f colors , then provide s a n efficien t solutio n t o th e schedulin g proble m — a separat e time slo t i s neede d fo r eac h color , an d tw o activitie s ar e schedule d fo r th e same tim e slo t i f the y hav e th e sam e color .

Students ca n recogniz e an d wor k wit h iterativ e an d recursiv e pro - cesses, extendin g thei r earlie r exploration s o f repetitiv e pattern s an d procedures. I n th e 7t h an d 8t h grade , the y ca n combin e thei r under - standing o f exponent s an d iteratio n t o solv e problem s involvin g compoun d interest wit h a calculato r o r spreadsheet . Topic s whic h befor e wer e viewe d iteratively — arrivin g a t th e presen t situatio n b y repeatin g a procedur e n times — ca n no w b e viewe d recursivel y - arrivin g a t th e presen t situatio n by modifyin g th e previou s situation . The y ca n appl y thi s understandin g t o Fibonacci numbers , t o th e Towe r o f Hano i puzzle , t o program s i n Logo , t o permutations an d t o othe r areas .

Students i n the 7t h and 8t h grades should explor e ho w code s ar e use d t o c o m m u n i c a t e information , b y traditional method s suc h as Morse cod e or semaphor e (flag s use d fo r ship-to-shi p messages ) an d als o b y curren t methods suc h a s zi p codes . Student s shoul d investigat e an d repor t abou t various code s tha t ar e commonl y used , suc h a s binar y codes , UPC s (univer - sal produc t codes ) o n grocer y items , an d ISB N number s o n books . The y should als o explor e ho w informatio n i s processed . A usefu l metapho r is ho w a waitin g lin e o r queu e i s handle d (o r "processed" ) i n variou s sit - uations; a t a bank , fo r example , th e queu e i s usuall y processe d i n first-in - first-out (FIFO ) order , bu t i n a supermarke t o r restauran t ther e i s usually a pre-sorting int o smalle r queue s don e b y th e shopper s themselve s befor e th e FIFO proces s i s activated .

In th e 7t h an d 8t h grade , student s shoul d b e abl e t o us e algorithm s t o find t h e b e s t solutio n i n a numbe r o f situation s — includin g th e shortest rout e fro m on e cit y t o anothe r o n a map , th e cheapes t wa y o f connecting site s int o a network , th e fastes t way s o f alphabetizin g a lis t o f words, th e optima l rout e fo r a clas s tri p (se e th e Short-Circuiting Trenton lesson i n the Introductio n t o thi s Framework) , o r optima l wor k schedule s fo r employees a t a fast-foo d restaurant .

Two importan t resource s o n discret e mathematic s fo r teacher s a t al l levels ar e th e 199 1 NCTM Yearboo k Discrete Mathematics Across the Cur- riculum K-12 an d th e 199 7 DI M ACS Volum e Discrete Mathematics in the Schools. Teacher s o f grade s 7- 8 woul d als o find usefu l th e textboo k Discrete Mathematics Through Applications.

9. Grade s 7- 8 Indicator s an d A c t i v i t i e s

The cumulativ e progresse s indicator s fo r grad e 8 appea r belo w i n bold - face type . Eac h indicato r i s followe d b y activitie s whic h illustrat e ho w i t can b e addresse d i n th e classroo m i n grade s 7 an d 8 .

170 J O S E P H G . ROSENSTEI N

Building upo n knowledg e an d skill s gaine d i n th e precedin g grades , ex - periences wil l b e suc h tha t al l student s i n grade s 7-8 :

6. U s e systemati c listing , counting , an d reasonin g i n a variet y of differen t c o n t e x t s .

• Student s determin e th e numbe r o f possibl e differen t sandwiche s o r hamburgers tha t ca n b e create d a t loca l eaterie s usin g a combinatio n of specifie d ingredients . The y fin d th e numbe r o f pizza s tha t ca n b e made wit h thre e ou t o f eigh t availabl e topping s an d relat e th e resul t to th e number s i n Pascal' s triangle .

• Student s determin e th e numbe r o f dominoes i n a se t tha t goe s u p t o 6:6 o r 9:9 , th e numbe r o f candles use d throughou t Hannukah , an d th e number o f gift s give n i n th e son g "Th e Twelv e Day s o f Christmas, " and connec t th e result s throug h discussio n o f the triangula r numbers . (Note tha t i n a 6: 6 se t o f dominoe s ther e i s exactl y on e domin o wit h each combinatio n o f dot s fro m 0 t o 6. )

• Student s determin e th e numbe r o f way s o f spellin g "Pascal " i n th e array belo w b y followin g a pat h fro m to p t o botto m i n whic h eac h letter i s directl y below , an d jus t t o th e righ t o r lef t o f th e previou s letter.

P A A

s s s c c c c

A A A A A L L L L L L

• Student s desig n differen t licens e plat e system s fo r differen t popula - tion sizes ; fo r example , how large would the population be before you would run out of plates which had only three numbers, or only five numbers, or two letters followed by three numbers?

• Student s fin d th e numbe r o f different way s of making a row of six re d and yello w flowers, organiz e an d tabulat e th e possibilitie s accordin g to th e numbe r o f flowers of th e firs t color , an d explai n th e connectio n with th e number s i n th e sixt h ro w o f Pascal' s triangle . (Se e als o Visual Patterns in Pascal's Triangle.)

• Student s pos e and ac t ou t problem s involvin g the number o f differen t ways a grou p o f peopl e ca n si t aroun d a table , usin g a s motivatio n the scen e o f th e Ma d Hatte r a t th e te a party . (Se e Mathematics, a Human Endeavor, p . 394. )

• Student s coun t th e tota l numbe r o f different cube s tha t ca n b e mad e using eithe r re d o r gree n pape r fo r eac h face . (T o solv e thi s problem , they wil l hav e t o us e a "brea k u p th e proble m int o cases " strategy. )

A C O M P R E H E N S I V E VIE W O F D I S C R E T E MATHEMATIC S 17 1

• Student s determin e th e numbe r o f handshake s tha t tak e plac e i f each perso n i n a roo m shake s hand s wit h ever y othe r perso n exactl y once, an d relat e thi s tota l t o th e numbe r o f lin e segment s joinin g th e vertices i n a polygon , t o th e numbe r o f two-flavo r ice-crea m cones , and t o triangula r numbers .

• Student s coun t th e numbe r o f triangle s o r rectangle s i n a geometri c design. Fo r example , the y shoul d b e abl e t o coun t systematicall y the numbe r o f triangle s (an d trapezoids ) i n th e figure belo w t o th e left, notin g tha t ther e ar e triangle s o f thre e sizes , an d th e numbe r o f rectangles i n th e 4 x 5 gri d picture d belo w t o th e right , listin g firs t all dimension s o f rectangle s tha t ar e present .

7. Recogniz e c o m m o n discret e mathematica l m o d e l s , explor e their properties , an d desig n t h e m fo r specifi c situations .

• Student s find th e minimu m numbe r o f color s neede d t o assig n color s to al l vertices i n a graph s o that an y two adjacent vertice s ar e assigne d different color s an d justif y thei r answers . Fo r example , student s ca n explain wh y on e of the graph s belo w require s fou r color s whil e fo r th e other, thre e color s ar e sufficient .

• Student s us e graph colorin g to solve problems which involv e avoidin g conflicts suc h as : schedulin g th e school' s extr a curricula r activities ; scheduling referee s fo r socce r games ; determinin g th e minimu m num - ber o f aquarium s neede d fo r a specifie d collectio n o f tropica l fish; and assignin g channel s t o radi o station s t o avoi d interference . I n th e graph belo w a n edg e betwee n tw o animal s indicate s tha t the y canno t share a habitat . Th e videotape , Geometry: New Tools for New Tech- nologies ha s a segmen t Connecting the Dots: Vertex Coloring whic h discusses th e minimu m numbe r o f habitats require d fo r thi s situation .

172 J O S E P H G . ROSENSTEI N

• Student s us e tre e diagram s t o represen t an d analyz e possibl e out - comes i n countin g problems , suc h a s tossin g tw o dice .

• Student s determin e whethe r o r no t a give n grou p o f dominoe s ca n be arrange d i n a lin e (o r i n a rectangle ) s o tha t th e numbe r o f dot s on th e end s o f adjacen t dominoe s match . Fo r example , th e dominoe s (03), (05) , (12) , (14) , (15) , (23) , (34 ) ca n b e arrange d a s (12) , (23) , (30), (05) , (51) , (14) , (43) ; and i f an eighth domin o (13 ) i s added, the y can b e forme d int o a rectangle . What if instead the eighth domino was (24) — could they then be arranged in a rectangle or in a line?

• Student s determin e th e minimu m numbe r o f block s tha t a polic e car ha s t o repea t i f i t mus t tr y t o patro l eac h stree t exactl y onc e on a give n map . Drawing Pictures With One Line contain s simila r real-world problem s an d a numbe r o f relate d gam e activities .

• Student s find th e bes t rout e fo r collectin g recyclabl e pape r fro m al l classrooms i n th e school , an d discus s differen t way s o f decidin g wha t is th e "best. " (Se e Drawing Pictures With One Line.)

• Student s mak e model s o f variou s polyhedr a wit h straw s an d string , and explor e th e relationshi p betwee n th e numbe r o f edges , faces , an d vertices.

8. Experimen t w i t h iterativ e an d recursiv e processes , w i t h t h e aid o f calculator s an d computers .

• Student s develo p a metho d fo r solvin g th e Towe r o f Hano i problem : There ar e thre e pegs , o n th e firs t o f which fiv e disk s ar e stacked , eac h smaller tha n th e one s underneath i t (se e diagram below) ; the proble m is t o mov e th e entir e stac k t o th e thir d peg , movin g disks , on e a t a time, fro m an y pe g t o eithe r o f th e othe r tw o pegs , wit h n o dis k eve r placed upo n a smalle r one . How many moves are required to do this? What if there were 6 disks ? How long would it take to do this with 64 disks? (A n ancien t legen d predict s tha t whe n thi s tas k i s completed , the worl d wil l end ; shoul d w e worry? )

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 173

Students vie w recursivel y Towe r o f Hanoi puzzle s wit h variou s num - bers of disks so that the y ca n express the number o f moves neede d to solve the puzzle wit h on e more dis k i n terms o f the numbe r o f moves needed fo r the puzzle wit h th e current numbe r o f disks.

• Student s attemp t t o list th e different way s the y coul d trave l 1 0 feet in a straight lin e if they wer e a robot whic h move d onl y in one or two foot segments , an d then thinkin g recursivel y determin e th e numbe r of differen t way s thi s robo t coul d trave l n feet .

• Student s develo p arithmeti c an d geometric progression s o n a calcu - lator.

• Student s fin d squar e root s usin g th e followin g iterativ e procedur e on a calculator. Mak e a n estimate o f the square roo t o f a numbe r B, divide th e estimate int o B , and average th e result wit h th e estimat e to ge t a new estimate . The n repea t thi s procedur e unti l a n adequat e estimate i s obtained. For example, if the first estimate of the square root of 10 is 3, then the second would be the average of 3 and 10/3, or 19/6 — 3.166. What is the next estimate of the square root of 10? How many repetitions are required to get the estimate to agree with the square root of 10 provided by the calculator?

• Student s develo p the sequence of areas and perimeters of iterations of the construction s o f the Sierpinsk i triangl e (to p figures ) an d the Koch snowflake (botto m figures) , an d discus s th e outcom e i f th e proces s were continue d indefinitely . (Thes e ar e discusse d i n mor e detai l i n the section s fo r earlie r grad e levels . Se e Unit 1 of Fractals for the Classroom for related activities. )

A A A / • \ /wWr\ /flw ^

Stag e 0 Stag e 1 Stag e 2 Stag e 3

Stag e 0 Stag e 1 Stag e 2

174 J O S E P H G . R O S E N S T E I N

• Student s recogniz e th e computatio n o f th e numbe r o f permutation s as a recursive proces s — that is , that th e numbe r o f ways of arrangin g 10 student s i s 1 0 times th e numbe r o f way s o f arrangin g 9 students .

9. Explor e m e t h o d s fo r storing , processing , an d communicatin g information.

• Student s conjectur e whic h o f th e followin g (an d other ) method s i s the mos t efficien t wa y o f handin g bac k correcte d homewor k paper s which ar e alread y sorte d alphabetically : (1 ) th e teache r walk s aroun d the roo m handin g t o eac h studen t individually ; (2 ) student s pas s th e papers around , eac h takin g thei r own ; (3 ) student s lin e themselves u p in alphabetica l order . Student s tes t thei r conjecture s an d discus s th e results.

• Student s investigat e an d repor t abou t variou s code s tha t ar e com - monly used , suc h a s zi p codes , UPC s (universa l produc t codes ) o n grocery items , an d ISB N number s o n books . ( A good sourc e for infor - mation abou t thes e an d othe r code s is Codes Galore by J. Malkevitch , G. Froelich , an d D . Proelich. )

• Student s writ e a Log o procedur e fo r makin g a rectangl e tha t use s variables, s o tha t the y ca n us e thei r rectangl e procedur e t o creat e a graphic scen e whic h contain s objects , suc h a s buildings , o f varyin g sizes.

• Student s ar e challenge d t o gues s a secre t wor d chose n b y th e teache r from th e dictionary , usin g a t mos t 2 0 yes-no questions. Is this always, or only sometimes possible?

• Student s us e Ven n diagram s t o solv e problem s lik e th e followin g one fro m th e Ne w Jerse y Departmen t o f Education' s Mathematics Instruction Guide (p . 7-13) . Suppose the school decided to add the springtime sport of lacrosse to its soccer and basketball offerings for its 120 students. A follow-up survey showed that: 35 played lacrosse, 70 played soccer, 40 played basketball, 20 played both soccer and bas- ketball, 15 played both soccer and lacrosse, 15 played both basketball and lacrosse, and 10 played all three sports. Using this data, complete a Venn diagram and answer the following questions: How many stu- dents played none of the three sports ? What percent of the students played lacrosse as their only sport? How many students played both basketball and lacrosse, but not soccer?

• Student s kee p a scrapboo k o f differen t way s i n whic h informatio n i s stored o r processed . Fo r exampl e a lis t o f event s i s usuall y store d b y date, s o th e scrapboo k migh t contai n a pictur e o f a pocke t calendar ; a queu e o f peopl e a t a ban k i s usuall y processe d i n first-in-first-ou t (FIFO) order , s o th e scrapboo k coul d contai n a pictur e o f suc h a queue. (How is this different from the waiting lines in a supermarket, or at a restaurant?)

A COMPREHENSIV E VIE W O F DISCRET E MATHEMATIC S 17 5

• Student s determin e whethe r i t i s possibl e t o hav e a yea r i n whic h there i s n o Frida y th e 13th , an d th e maximu m numbe r o f Frida y th e 13th's tha t ca n occu r i n on e calenda r year .

• Student s predic t an d the n explor e th e frequenc y o f letter s i n th e alphabet throug h examinatio n o f sampl e texts , compute r searches , and publishe d materials .

• Student s decod e message s wher e letter s ar e systematicall y replace d by othe r letter s withou t knowin g th e syste m b y whic h letter s ar e re - placed; newspaper s an d game s magazine s ar e goo d source s fo r "cryp - tograms" an d student s ca n creat e thei r own . The y als o explor e th e history o f code-makin g an d code-breaking . Th e videotap e Discrete Mathematics: Cracking the Code provides a goo d introductio n t o th e uses o f cryptograph y an d th e mathematic s behin d it .

10. D e v i s e , describe , an d tes t algorithm s fo r solvin g optimiza - t i o n an d searc h problems .

• Student s find th e shortes t rout e fro m on e cit y t o anothe r o n a Ne w Jersey map , an d discus s whethe r tha t i s the bes t route . (Se e Problem Solving Using Graphs.)

• Student s writ e an d solv e problem s involvin g distances , times , an d costs associate d wit h goin g fro m town s o n a ma p t o othe r towns , s o that differen t route s ar e "best " accordin g t o differen t criteria .

• Student s use binary representations of numbers to find winning strat - egy for Nim . (Se e Mathematical Investigations fo r other mathematica l games.)

• Student s pla n a n optima l rout e fo r a clas s trip . (Se e th e Short- circuiting Trenton lesso n i n th e Introductio n t o thi s Framework.)

• Student s devis e wor k schedule s fo r employee s o f a fast-foo d restau - rant whic h mee t specifie d condition s ye t minimiz e th e cost .

• Student s compar e strategie s fo r alphabetizin g a lis t o f words , an d test t o se e whic h strategie s ar e mor e efficient .

• Student s find a networ k o f road s whic h connect s a numbe r o f site s and involve s th e smalles t cost . In the example below, what roads should be built so as to minimize the total cost, where the number on each road reflects the cost of building that road (in hundreds of thousands of dollars) ?

176 J O S E P H G . R O S E N S T E I N

• Student s develo p a precis e descriptio n o f th e standar d algorith m fo r adding tw o two-digi t integers .

• Student s devis e strategie s fo r dividin g u p th e wor k o f addin g a lon g list o f number s amon g th e member s o f th e team .

References.

• Chavey , D . Drawing Pictures with One Line. Consortiu m fo r Mathematic s and It s Application s (COMAP) , Modul e # 2 1 , 1987 .

• Cozzens , M. , an d R . Porter . Problem Solving Using Graphs. Consortiu m for Mathematic s an d It s Application s (COMAP) , Modul e # 6 , 1987 .

• Crisler , N. , P . Fisher , an d G. Proelich , Discrete Mathematics Through Ap- plications. W . H . Freema n an d Company , 1994 .

• Jacobs , H . R . Mathematics: A Human Endeavor. W . H . Freema n an d Company, 1982 .

• Kenney , M . J. , Ed . Discrete Mathematics Across the Curriculum K-12, 1991 Yearbook o f the National Counci l of Teachers of Mathematics (NCTM) . Reston, VA : 1991.

• Malkevitch , J. , G. Proelich , an d D . Proelich , Codes Galore, Consortium fo r Mathematics an d It s Application s (COMAP) , Modul e # 1 8 , 1991 .

• Ne w Jerse y Departmen t o f Education . Mathematics Instruction Guide. D . Varygiannis, Coord . Januar y 1996 .

• Peitgen , Heinz-Otto , e t al . Fractals for the Classroom: Strategic Activities Volume One & Two. Reston , VA : NCT M an d Ne w York : Springer-Verlag , 1992.

• Rosenstein , J . G. , D . Franzblau , an d F . Roberts , Eds . Discrete Mathemat- ics in the Schools. Proceeding s o f a 199 2 DIMAC S Conferenc e o n "Discret e Mathematics i n th e Schools. " DIMAC S Serie s o n Discret e Mathematic s an d Theoretical Compute r Science . Providence , RI : American Mathematica l So - ciety (AMS) , 1997 .

• Seymour , D . Visual Patterns in Pascal's Triangle. Pal o Alto , CA : Dal e Seymour Publications , 1986 .

• Souviney , R. , e t al . Mathematical Investigations. Boo k One , Dal e Seymou r Publications, 1990 .

Video.

• Discrete Mathematics: Cracking the Code, Consortiu m fo r Mathematic s and It s Applications .

• Geometry: New Tools for New Technologies, videotape b y th e Consortiu m for Mathematic s an d It s Application s (COMAP) . Lexington , MA , 1992 .

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 17 7

10. Grade s 9-1 2 Overvie w

The five major theme s o f discrete mathematics , a s discussed i n th e K-1 2 Overview, ar e systemati c listing , counting , an d reasoning ; discret e mathematical modelin g usin g graph s (networks ) an d trees ; iter - ative (tha t is , repetitive ) pattern s an d processes ; organizin g an d processing information ; an d followin g an d devisin g list s o f instruc - tions, calle d "algorithms, " an d usin g t h e m t o fin d t h e bes t solutio n t o real-worl d problems .

The following discussion of activities at the 9-12 grade levels in discrete mathematics presupposes that corresponding activities have taken place at the K-8 grade levels. Hence high school teachers should review the discus- sions of discrete mathematics at earlier grade levels and might use activities similar to those described there before introducing the activities for these grade levels.

At th e hig h schoo l level , student s ar e becomin g familia r wit h algebrai c and functiona l notation , an d thei r understandin g o f al l o f th e theme s o f discrete mathematic s an d thei r abilit y t o generaliz e earlie r activitie s shoul d be enhance d b y thei r algebrai c skill s an d understandings . Thus , for example , the y shoul d us e formula s t o expres s th e result s o f problem s involving permutation s an d combinations , relat e Pascal' s triangl e t o th e coefficients o f th e binomia l expansio n o f (x + y)n, explor e model s o f growt h using variou s algebrai c models , explor e iteration s o f functions , an d discus s methods fo r dividin g a n estat e amon g severa l heirs .

At th e hig h schoo l level , student s ar e particularl y intereste d i n appli - cations; the y as k What is all of this good for? I n al l five area s o f discret e mathematics, student s shoul d focu s o n ho w discret e m a t h e m a t i c s i s used t o solv e practica l problems . Thus , fo r example , the y shoul d b e able t o appl y thei r understandin g o f countin g techniques , t o analyz e lot - teries; o f grap h coloring , t o schedul e traffi c light s a t a loca l intersection ; of path s i n graphs , t o devis e patro l route s fo r polic e cars ; o f iterativ e pro - cesses, t o analyz e an d predic t fish population s i n a pon d o r concentratio n o f medicine i n the bloodstream ; o f codes, t o understan d ho w bar-code scanner s detect error s an d ho w CD' s correc t errors ; an d o f optimization , t o under - stand th e 20 0 yea r ol d debate s abou t apportionmen t an d t o find efficien t ways o f schedulin g th e component s o f a comple x project .

Two importan t resource s o n discret e mathematic s fo r teacher s a t al l grade level s ar e th e 199 1 NCT M Yearbook , Discrete Mathematics Across the Curriculum K-12 an d th e DIMAC S Volume , Discrete Mathematics in the Schools edite d b y J . Rosenstein , D . Franzblau , an d F . Roberts . Usefu l resources a t th e hig h schoo l leve l ar e Discrete Mathematics Through Appli- cations b y N . Crisler , P . Fisher, an d G. Froelich ; For All Practical Purposes: Introduction to Contemporary Mathematics, b y th e Consortiu m fo r Mathe - matics an d it s Applications ; an d Excursions in Modern Mathematics b y P . Tannenbaum an d R . Arnold .

178 J O S E P H G . R O S E N S T E I N

11. Grade s 9-1 2 Indicator s an d A c t i v i t i e s

The cumulativ e progres s indicators for grad e 1 2 appear belo w in boldfac e type. Eac h indicato r i s followe d b y activitie s whic h illustrat e ho w i t ca n b e addressed i n th e classroo m i n grade s 9 , 10 , 11 , and 12 .

Building upo n knowledg e an d skill s gaine d i n th e precedin g grades , ex - periences wil l b e suc h tha t al l student s i n grade s 9-12 :

11. U n d e r s t a n d t h e basi c principle s o f iteration , recursion , an d mathematical induction .

• Student s relat e th e possibl e outcome s o f tossin g five coin s wit h th e binomial expansio n o f (x + yf an d th e fifth ro w o f Pascal' s triangle , and generaliz e t o value s o f n othe r tha n 5 .

• Student s develo p formula s fo r countin g path s o n grid s o r othe r sim - ple stree t maps .

• Student s find th e numbe r o f cut s neede d i n orde r t o divid e a gian t pizza s o tha t eac h studen t i n th e schoo l get s a t leas t on e piece .

• Student s develo p a precis e description , usin g iteration , o f th e stan - dard algorith m fo r addin g tw o integers .

12. U s e basi c principle s t o solv e combinatoria l an d algorithmi c problems.

• Student s determin e th e numbe r o f ways of spelling "mathematics " i n the arra y belo w b y following a path fro m to p t o botto m i n which eac h letter i s directl y below , an d jus t t o th e righ t o r lef t o f th e previou s letter.

M A A

T T T H H H H

E E E E E M M M M M M

A A A A A T T T T

I I I C C

• Student s determin e th e numbe r o f way s a committe e o f thre e mem - bers coul d b e selecte d fro m th e class , an d th e numbe r o f way s thre e people wit h specifie d role s coul d b e selected . The y generaliz e thi s activity t o finding a formul a fo r th e numbe r o f way s a n n perso n committee ca n b e selecte d fro m a clas s o f m people , an d th e numbe r of way s n peopl e wit h specifie d role s ca n b e selecte d fro m a clas s o f m people .

A COMPREHENSIV E VIE W O F DISCRET E MATHEMATIC S 17 9

• Student s find th e numbe r o f way s o f linin g u p thirt y student s i n a class, an d compar e tha t t o othe r larg e numbers ; fo r example , the y might compar e i t t o th e numbe r o f raindrop s (volum e = . 1 cc ) i t would tak e t o fill a spher e th e siz e o f th e eart h (radiu s = 650 7 KM) .

• Student s determin e th e numbe r o f way s o f dividin g 5 2 card s amon g four players , a s i n th e gam e o f bridge , an d compar e th e numbe r o f ways o f obtaining a flush (fiv e card s o f the sam e suit ) an d a ful l hous e (three card s o f on e denominatio n an d tw o card s o f another ) i n th e game o f poker .

• Student s pla y Ni m (an d simila r games ) an d discus s winnin g strate - gies usin g binar y representation s o f numbers .

13. U s e discret e m o d e l s t o represen t an d solv e problems .

• Student s stud y th e fou r colo r theore m an d it s history . ( The Math- ematicians J Coloring Book provide s a goo d backgroun d fo r colorin g problems.)

• Student s usin g grap h colorin g t o determin e th e minimu m numbe r of guard s (o r cameras ) neede d fo r museum s o f variou s shape s (an d similarly fo r placemen t o f law n sprinkler s o r motion-senso r burgla r alarms).

• Student s us e directe d graph s t o represen t tournament s (wher e a n arrow draw n fro m A t o B represent s " A defeat s B" ) an d foo d web s (where a n arro w draw n fro m A t o B represent s " A eat s B") , an d t o construct one-wa y orientations of streets in a given town which involv e the least inconvenienc e to drivers. ( A directed grap h i s simply a grap h where eac h edg e is thought o f as an arro w pointin g fro m on e endpoin t to th e other. )

• Student s us e tre e diagram s t o analyz e th e pla y o f game s suc h a s tic - tac-toe o r Nim , an d t o represen t th e solution s t o weighin g problems . Example: Give n 1 2 coin s on e o f whic h i s "bad, " find th e ba d one , and determin e whethe r i t i s heavie r o r lighte r tha n th e others , usin g three weighings .

• Student s us e grap h colorin g t o schedul e th e school' s final examina - tions s o that n o student ha s a conflict, i f at al l possible, o r t o schedul e traffic light s a t a n intersection .

• Student s devis e graphs for which there is a path tha t cover s each edge of the grap h exactl y once , and othe r graph s whic h have no such paths , based o n a n understandin g o f necessar y an d sufficien t condition s fo r the existence o f such paths, calle d "Eule r paths, " i n a graph. Drawing Pictures With One Line provide s backgroun d an d application s fo r Euler pat h problems .

180 J O S E P H G . ROSENSTEI N

• Student s mak e model s o f polyhedr a wit h straw s an d string , an d explore th e relationshi p betwee n th e number s o f edges , faces , an d vertices, an d generaliz e th e conclusio n t o plana r graphs .

• Student s us e graph s t o solv e problem s lik e th e a fire-station prob - lem" : Given a city where the streets are laid out in a grid composed of many square blocks, how many fire stations are needed to provide adequate coverage of the city if each fire station services its square block and the four square blocks adjacent to that one? Th e Mary - land Scienc e Cente r i n Baltimor e ha s a hands-o n exhibi t involvin g a fire-station proble m fo r 3 5 square block s arrange d i n a six-by-si x gri d with on e corne r designate d a park .

14. Analyz e iterativ e processe s w i t h t h e ai d o f calculator s an d computers.

• Student s analyz e the Fibonacc i sequence 1 , 1 , 2 , 3 , 5 , 8 , 1 3 , 2 1 , . . . a s a recurrence relatio n A n+2 — A n + An+i wit h connection s t o th e golde n ratio. Fascinating Fibonaccis illustrate s a variet y o f connection s be - tween Fibonacc i number s an d th e golde n ratio .

• Student s solv e problems involvin g compoun d interes t usin g iteratio n on a calculato r o r o n a spreadsheet .

• Student s explore examples of linear growth, usin g the recursive mode l based o n th e formul a A n+i — A n + d, wher e d i s th e commo n differ - ence, an d conver t i t t o th e explici t linea r formula , A n + i = A\ + n - d.

• Student s explor e example s o f population growth , usin g th e recursiv e model base d o n th e formul a A n+\ = A n x r , wher e r i s th e commo n multiple o r growt h rate , conver t i t t o th e explici t exponentia l formul a An+i = A\ x r

n , an d appl y i t t o bot h economic s (suc h a s interes t problems) an d biolog y (suc h a s concentratio n o f medicin e i n bloo d supply).

• Student s explor e logisti c growt h model s o f populatio n growth , usin g the recursiv e mode l base d o n th e formul a A n+1 = A n x ( 1 — A n) x r , where r i s th e growt h rat e an d A n i s th e fractio n o f th e carryin g capacity o f th e environment , an d appl y thi s t o th e populatio n o f fish in a pond . Usin g a spreadsheet , student s experimen t wit h variou s values o f the initia l valu e A\ an d o f the growt h rate , an d describ e th e relationship betwee n th e value s chose n an d th e lon g ter m behavio r o f the population .

• Student s explor e th e patter n resultin g fro m repeatedl y multiplyin g

.. r , b y itself . [ 1 0 J J

• Student s us e a calculato r o r a compute r t o stud y simpl e Marko v chains, suc h a s weathe r predictio n an d populatio n growt h models . (See Chapte r 7. 3 o f Discrete Mathematics Through Applications.)

A COMPREHENSIVE VIE W O F DISCRET E MATHEMATIC S 18 1

• Student s explor e graphica l iteratio n b y choosin g a functio n ke y o n a calculator an d pressin g it repeatedly , afte r choosin g an initia l number , to ge t sequence s o f number s lik e 2 , 4 , 8 , 1 6 , 3 2 , . . . o r 2 , \ / 2, v \ / 2 ,

Y \ A / 2, The y us e th e graph s o f th e function s t o explai n th e behavior o f th e sequence s obtained . The y exten d thes e exploration s by iteratin g function s the y progra m int o the calculator , suc h a s linea r functions, wher e slop e i s th e predicto r o f behavior , an d quadrati c functions f(x) = ax(l — x) , wher e 0 < x < 1 and 1 < a < 4 , whic h exhibit chaoti c behavior .

• Student s explor e iteratio n behavio r usin g th e functio n define d b y the tw o case s

f(x) = x + \ fo r x betwee n 0 an d \ f(x) = 2 — 2x fo r x betwee n \ an d 1

They us e th e initia l value s 1/2 , 2/3 , 5/9 , an d 7/10 , an d then , wit h a calculator o r computer , th e initia l value s .501 , .667, an d .70 1 (whic h differ b y a smal l amoun t fro m th e first grou p o f "nice " initia l values) . They compare the behavior o f the sequences generated b y these value s to th e sequence s generate d b y th e previou s initia l values .

• Student s pla y th e Chaos Game. Eac h pai r o f student s i s provide d with a n identica l transparenc y o n whic h hav e bee n draw n th e thre e vertices L , T , an d R o f a n equilatera l triangle . Eac h tea m start s b y selecting an y poin t o n th e triangle . The y rol l a di e an d creat e a ne w point halfwa y t o L i f the y rol l 1 or 2 , halfwa y t o R i f the y rol l 3 o r 4, an d halfwa y t o T i f the y rol l 5 o r 6 . The y repea t 2 0 times , eac h time usin g th e ne w poin t a s th e startin g poin t fo r th e nex t iteration . The teache r overlay s al l o f th e transparencie s an d ou t o f thi s chao s comes .. . th e familia r Sierpinsk i triangle . (Th e Sierpinsk i triangl e is discusse d i n detai l i n th e section s fo r earlie r grad e levels . Als o see Uni t 2 i n Fractals for the Classroom. The Chaos Game softwar e allows student s t o tr y variation s an d explor e th e gam e further. )

15. A p p l y discret e m e t h o d s t o storing , processing , an d commu - nicating information .

• Student s discus s variou s algorithm s use d fo r sortin g larg e number s of item s alphabeticall y o r numerically , an d explai n wh y som e sortin g algorithms are substantially faste r tha n others . T o introduce the topi c of sorting , giv e eac h grou p o f student s 10 0 inde x card s eac h wit h on e word o n it , an d le t the m devis e strategie s fo r efficientl y puttin g th e cards int o alphabetica l order .

• Student s discus s ho w scanner s o f ba r code s (zi p codes , UPCs , an d ISBNs) ar e able to detect error s in reading the codes, and evaluate an d compare ho w error-detectio n i s accomplishe d i n differen t codes . (Se e the COMA P Modul e Codes Galore or Chapte r 9 of For All Practical Purposes.)

182 J O S E P H G . ROSENSTEI N

• Student s investigat e method s o f erro r correctio n use d t o transmi t digitized picture s fro m spac e (Voyage r o r Mariner probes , o r the Hub - ble space telescope) ove r nois y or unreliable channels , o r t o ensur e th e fidelity o f a scratched C D recording . (Se e Chapter 1 0 of For Al l Prac - tical Purposes. )

• Student s rea d abou t codin g an d code-breakin g machine s an d thei r role i n Worl d Wa r II .

• Student s researc h topic s tha t ar e currentl y discusse d i n th e press , such a s public-ke y encryption , enablin g message s t o b e transmitte d securely, an d data-compression , use d t o sav e spac e o n a compute r disk.

16. A p p l y discret e m e t h o d s t o problem s o f voting , apportion - ment, an d allocations , an d us e fundamenta l strategie s o f op - timization t o solv e problems .

• Student s find th e bes t rout e whe n a numbe r o f alternat e route s ar e possible. Fo r example : In which order should you pick up the six friends you are driving to the school dance? In which order should you make the eight deliveries for the drug store where you work? In which order should you visit the seven "must-see" sites on your vacation trip? I n eac h case , yo u wan t t o find th e "bes t route, " th e on e whic h involves th e leas t tota l distance , o r leas t tota l time , o r leas t tota l expense. Student s creat e thei r ow n problems , usin g actua l location s and distances , an d find th e bes t route . Fo r a large r project , student s can tr y t o improv e th e rout e take n b y thei r schoo l bus .

• Student s stud y th e rol e o f apportionmen t i n America n history , fo - cusing o n th e 179 0 censu s (actin g ou t th e position s o f th e thirtee n original state s an d discussin g Georg e Washington' s first us e o f th e presidential veto) , an d th e dispute d electio n o f 1876 , an d discus s th e relative merit s o f differen t system s o f apportionmen t tha t hav e bee n proposed an d used . (Thi s activit y provide s a n opportunit y fo r math - ematics an d histor y teacher s t o wor k together. ) The y als o devis e a student governmen t wher e th e seat s ar e fairl y apportione d amon g al l constituencies. (Se e th e COMA P modul e The Apportionment Prob- lem o r Chapte r 1 4 of For All Practical Purposes.)

• Student s analyz e mathematica l method s fo r dividin g a n estat e fairl y among various heirs. (Se e Chapter 2 of Discrete Mathematics Through Applications, Chapte r 3 o f Excursions in Modern Mathematics, o r Chapter 1 3 of For All Practical Purposes.)

• Student s discus s various methods, such as preference schedule s or ap- proval voting , tha t ca n b e use d fo r determinin g th e winne r o f a n elec - tion involvin g thre e o r mor e candidate s (fo r example , th e pro m kin g or queen). Wit h preferenc e schedules , each voter rank s th e candidate s and th e individua l ranking s ar e combined , usin g variou s techniques ,

A C O M P R E H E N S I V E V I E W O F D I S C R E T E MATHEMATIC S 18 3

to obtain a group ranking ; preferenc e schedule s ar e used , fo r example , in rankin g sport s team s o r determinin g entertainmen t awards . I n ap - proval voting , eac h vote r ca n vot e onc e fo r eac h candidat e whic h sh e finds acceptable ; th e candidat e wh o receive s the mos t vote s then win s the election . (Se e th e COMA P modul e The Mathematical Theory of Elections o r Chapte r 1 1 of For All Practical Purposes.)

• Student s find an efficient wa y of doing a complex project (lik e prepar- ing a n airplan e fo r it s nex t trip ) give n whic h task s preced e whic h an d how muc h tim e eac h tas k wil l take . (Se e Chapte r 8 o f Excursions of Modern Mathematics o r Chapte r 3 o f For All Practical Purposes.)

• Student s find a n efficien t wa y o f assignin g song s o f variou s length s to th e tw o side s o f a n audi o tap e s o tha t th e tota l time s o n th e tw o sides ar e a s clos e togethe r a s possible . Similarly , the y determin e th e minimal numbe r o f sheet s o f plywoo d neede d t o buil d a cabine t wit h pieces o f specifie d dimensions .

• Student s appl y algorithm s fo r matchin g i n graph s t o schedul e whe n contestants pla y eac h othe r i n th e differen t round s o f a tournament .

• Student s devis e a strateg y fo r finding a "secre t number " fro m 1 to 100 0 usin g question s o f th e for m Is your number bigger than 837? and determin e th e leas t numbe r o f questions neede d t o find th e secre t number.

References.

• Bennett , S. , D . DeTemple , M . Dirks , B . Newell , J . Robertson , an d B . Tyus. The Apportionment Problem: The Search for the Perfect Democracy. Consortium fo r Mathematic s an d It s Application s (COMAP) , Modul e # 1 8 , 1986.

• Chavey , D . Drawing Pictures with One Line. Consortiu m fo r Mathematic s and It s Application s (COMAP) , Modul e # 2 1 , 1987 .

• Consortiu m fo r Mathematic s an d It s Applications . For All Practical Pur- poses: Introduction to Contemporary Mathematics. W . H . Freema n an d Company, Thir d Edition , 1993 .

• Crisler , N. , P . Fisher , an d G . Froelich , Discrete Mathematics Through Ap- plications. W . H . Freema n an d Company , 1994 .

• Francis , R . The Mathematician's Coloring Book. Consortiu m fo r Mathe - matics an d It s Application s (COMAP) , Modul e # 1 3 , 1989 .

• Garland , T . H . Fascinating Fibonaccis. Pal o Alto , CA : Dal e Seymou r Publications, 1987 .

• Kenney , M . J. , Ed . Discrete Mathematics Across the Curriculum K-12, 1991 Yearbook o f the National Counci l of Teachers of Mathematics (NCTM) . Reston, VA , 1991.

184 J O S E P H G . ROSENSTEI N

• Malkevitch , J . The Mathematical Theory of Elections. Consortiu m fo r Mathematics an d Its Applications (COMAP) . Modul e # 1 , 1985 .

• Malkevitch , J. , G . Froelich , an d D . Froelich . Codes Galore. Consortiu m for Mathematic s an d Its Applications (COMAP) . Modul e # 1 8 , 1991 .

• Peitgen , Heinz-Otto , e t al . Fractals for the Classroom: Strategic Activities Volume One & Two. Reston , VA : NCTM an d New York: Springer-Verlag , 1992.

• Rosenstein , J . G. , D. Franzblau, an d F. Roberts, Eds . Discrete Mathemat- ics in the Schools. Proceeding s o f a 199 2 DIMACS Conferenc e o n "Discret e Mathematics i n the Schools." DIMAC S Serie s on Discrete Mathematic s and Theoretical Compute r Science . Providence , RI : American Mathematica l So- ciety (AMS) , 1997.

• Seymour , D . Patterns in PascaVs Triangle. Poster . Pal o Alto , CA : Dal e Seymour Publications .

• Tannenbaum , P . an d R . Arnold . Excursions in Modern Mathematics. Prentice-Hall, 1992.

Software.

• The Chaos Game. Minnesot a Educational Compute r Consortiu m (MECC) .

D E P A R T M E N T O F MATHEMATICS , R U T G E R S UNIVERSIT Y

E-mail address : j oerQdimacs. r u t g e r s. ed u

Section 4 I n t e g r a t i n g Discret e M a t h e m a t i c s

into Existin g M a t h e m a t i c s Curricula , Grades K - 8

Discrete Mathematic s i n K- 2 Classroom s VALERIE A . D E B E L L I S

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Rhythm an d Pattern : Discret e Mathematic s wit h a n Artisti c Connection fo r Elementar y Schoo l Teacher s

R O B E R T E . JAMISO N

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Discrete Mathematic s Activitie s fo r Middl e Schoo l EVAN M A L E T S K Y

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Discrete M a t h e m a t i c s i n K - 2 Classroom s

Valerie A . DeBelli s

Introduction

This articl e describe s tw o K- 2 classroom s tha t I hav e observe d and/o r taught durin g th e 1996-9 7 schoo l year . Critic s hav e claime d tha t math - ematics taugh t i n primar y grade s (K-2 ) i s nothin g mor e tha n memorizin g facts, contain s littl e conten t beyon d computation , an d tha t topic s in discret e mathematics canno t b e thoughtfull y discusse d b y childre n a t thes e levels . 1 strongl y disagree . Fo r th e pas t te n years , I hav e bee n involve d wit h pro - fessional developmen t project s fo r K-1 2 teacher s o f mathematics , includin g the Leadershi p Progra m i n Discret e Mathematic s (se e Rosenstei n an d De - Bellis [7]) . Thi s experience , couple d wit h m y backgroun d i n mathematic s education, ha s provided man y opportunitie s t o collaborat e wit h K-1 2 teach - ers wh o ar e implementin g discret e mathematic s i n thei r classrooms . Base d on thes e experiences , I hav e com e t o believ e tha t no t onl y i s i t importan t to incorporat e discret e mathematic s int o existin g curriculum , bu t tha t K - 2 classroom s ar e a natura l plac e t o begi n developin g th e rudiment s o f th e subject.

T h e curren t K - 2 curriculu m

Traditional K- 2 mathematic s curricul a includ e topic s suc h a s counting , writing numerals , whol e numbe r operation s (addition , subtraction , multi - plication), fractions , estimation , plac e value , measurement , geometry , an d problem solving . Withi n th e pas t te n years , som e curriculu m developer s have als o included topic s i n probability an d statistic s fo r K- 2 childre n whic h typically focu s o n makin g prediction s abou t experiment s an d o n recordin g and interpretin g data . Th e followin g genera l summar y o f grad e leve l ex - pectations i n mathematic s i s base d o n m y revie w o f severa l curren t K- 2 mathematics curriculu m guide s fro m Ne w Jerse y publi c schools .

By th e en d o f kindergarten , childre n shoul d b e abl e t o coun t an d writ e numbers u p t o twenty , a s wel l a s ad d an d subtrac t thes e numbers . The y

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 .

187

https://doi.org/10.1090/dimacs/036/15

188 VALERI E A . DEBELLI S

should b e abl e t o measur e i n a d ho c unit s — fo r example , a des k ma y be thre e pencil s lon g — an d understan d spatia l relationship s suc h a s over , under, top , bottom , middle , left , right , inside , an d outside . The y shoul d b e able to identify plana r figures such as a circle, triangle, rectangle , an d square ; and sor t o r classif y object s b y attribut e — color , shape , o r size . Childre n i n kindergarten shoul d als o begi n flipping coin s an d recordin g outcomes .

By th e en d o f first grade , childre n shoul d b e abl e t o coun t an d writ e numbers u p t o on e hundre d an d ad d an d subtrac t two-digi t numbers . The y should begi n t o hav e som e part-whol e understandin g o f fraction s an d b e familiar wit h fractiona l amount s suc h a s 1/2 , 1/3 , an d 1/4 . The y shoul d be abl e t o identif y spatia l figures suc h a s a ball , cube , cone , can , an d box , and b e abl e t o acquir e informatio n fro m pictures , text , an d charts . The y should b e abl e t o identif y an d discus s notion s o f symmetr y an d perimete r in a square , rectangle , triangle , an d circle . The y shoul d b e abl e t o solv e two-step wor d problems .

By th e en d o f secon d grade , thes e sam e childre n shoul d b e abl e t o coun t and writ e number s u p t o 999 ; ad d an d subtrac t three-digi t numbers ; kno w multiplication fact s wit h 0 , 1 , 2 , 3 , 4 , an d 5 a s factors ; an d writ e fraction s symbolically an d wor k wit h mixe d numbers . The y shoul d als o kno w th e place valu e syste m fo r ones , tens , an d hundreds ; b e abl e t o mak e an d us e charts, tables , an d drawing s t o solv e problems ; identif y three-dimensiona l geometric shape s — cube , cylinder , sphere , cone , an d rectangula r prism ; and discus s are a an d volume . I t i s also during th e primar y schoo l year s tha t children lear n abou t systems : coins , clocks , calendars , maps , metri c system , standard measuremen t system , ba r graphs , an d pi e graphs .

"Young childre n ente r schoo l with informa l strategie s fo r solvin g mathe - matical problems , communicatio n skills , idea s abou t ho w number an d shap e connect t o eac h othe r an d t o thei r world , an d reasonin g skills . I n grade s K - 2, students shoul d buil d upo n thes e informa l strategies " (se e the New Jersey Mathematics Curriculum Framework [6] , pag e 83) . Cognitively , accordin g to Piaget , thi s populatio n acquire s knowledg e throug h though t an d actio n (see Inhelde r [3]) . A s a result , mathematica l concept s ar e taugh t throug h the physica l manipulatio n o f objects , throug h rol e playing , throug h stor y telling, an d throug h themati c teachin g approaches .

Existing curricula fo r th e primar y grade s already includ e natural connec - tions t o discret e mathematic s topics . Fo r example , durin g th e first markin g period, man y K- 2 grade s spen d tim e classifyin g an d sorting , includin g pat - tern detectio n (identif y th e pattern ) an d patter n projectio n (wha t come s next i n th e sequence) . I n fact , severa l K- 2 textbook s whic h clai m t o includ e discrete mathematic s topic s simpl y includ e sortin g activitie s an d nothin g more. Secon d grader s spen d tim e learnin g th e fundamental s o f geometry . The curriculu m usuall y include s topic s o n shape , size , wha t define s a n ob - ject, an d wha t make s tw o object s differen t fro m on e another . Bu t I hav e also observe d secon d grad e childre n explai n wha t make s a triangle , circle , and squar e th e same . Thes e childre n ar e capabl e o f doin g fa r mor e comple x

DISCRETE MATHEMATIC S I N K- 2 CLASSROOM S 189

mathematics tha n w e hav e traditionall y expected . Th e followin g account s serve t o demonstrat e wha t ca n b e don e i n K- 2 classrooms .

A visi t t o Grad e 2

The da y wa s "mat h day " (a n entir e da y devote d t o learnin g mathemat - ics) whe n I visite d Sharo n Heil' s secon d grad e classroo m a t th e Kossman n School in Long Valley, New Jersey. Th e schoo l has roughly five hundred chil - dren i n grade s K throug h 2 . Ms . Heil teache s i n a self-containe d classroo m of twenty-fou r students . Sh e describe d thi s clas s a s a trul y heterogeneou s group, comprising students fro m bot h far m familie s an d middle-managemen t families. Academically , th e student s hav e a wide range of abilities; some stu - dents receiv e academi c suppor t i n th e resourc e room , other s receiv e basi c skills assistanc e i n mathematics , language , and/o r reading , an d other s ar e high-achieving, articulat e proble m solvers . I n general , sh e feel s al l he r stu - dents ar e enthusiasti c learner s an d ver y curiou s abou t th e worl d aroun d them. I n th e classroo m description s tha t follow , th e name s o f th e childre n are fictitious s o tha t the y remai n anonymous .

Until participatin g i n th e 199 5 Leadership Progra m fo r Discret e Mathe - matics (se e Rosenstein an d DeBelli s [7]) , Ms. Heil had no t take n an y mathe - matics course s since graduating fro m colleg e over 20 years ago. T o her credit , she i s amon g man y elementar y schoo l teacher s wh o recogniz e th e nee d t o upgrade thei r ow n mathematica l learning . I t wa s no t eas y fo r her , bu t I witnessed th e benefit s — a teache r wh o provide s thoughtful , meaningfu l mathematical experience s t o he r students .

Sharon Hei l sa t o n a chai r nea r a carpete d ope n spac e i n he r classroom . The student s systematicall y pushe d thei r desk s t o th e sid e o f th e roo m an d lined u p nea r th e chalkboard , silentl y waitin g fo r instructions . Ms . Heil asked th e childre n t o randoml y si t o n th e floor i n fron t o f he r withou t an y parts o f thei r bod y touchin g on e another . The y wer e excite d becaus e the y saw he r holdin g a kickbal l an d though t th e ide a o f playin g wit h a bal l insid e the buildin g wa s neat ! Sh e told the m tha t thi s i s an activit y wher e everyon e is silent . "I' m goin g t o giv e th e bal l t o Annie . Yo u mus t pas s th e bal l fro m student t o studen t (withou t throwin g i t o r movin g fro m you r seat ) s o tha t everyone touche s i t a t leas t onc e an d get s i t bac k t o Annie. "

Ms. Hei l wa s imaginin g th e childre n a s vertice s i n a graph , wher e tw o children wer e joine d b y a n edg e i f the y wer e clos e enoug h t o han d th e bal l from on e t o th e othe r withou t changin g positions . Sh e wa s askin g the m to find a circui t whic h include d al l th e children ; soo n sh e woul d as k the m to find a Hamilto n circuit . Needles s t o say , sh e ha d initiate d thi s activit y without introducin g an y o f these terms . Th e childre n bega n t o pas s th e bal l to eac h othe r withou t talking . Whe n i t go t bac k t o Annie , th e teache r said , "raise you r han d i f you touche d th e bal l once. " Sixtee n childre n raise d thei r hands. "Rais e your han d i f you touched th e bal l twice." Eigh t childre n raise d their hand s an d th e teache r aske d the m t o stand . Thes e eigh t childre n ar e pictured a s Frank , Charlie , Zachary , Lisa , Deanne , Michael , Anthony , an d

190 VALERIE A . DEBELLI S

Annie i n Figur e 1 . Th e bal l wa s give n t o Jane t wh o wa s th e las t perso n t o touch th e bal l onc e (an d wa s sittin g o n th e floor ) befor e givin g i t t o Fran k who wa s th e firs t perso n t o touc h th e bal l twic e (an d wa s standing) . Th e teacher asked , "i s ther e a shorte r ... " an d wa s interrupte d b y Danie l wh o suggested tha t ther e wa s anothe r wa y to pas s th e ball . H e said, "Jane t give s it t o Jackie . Jacki e give s i t t o Lis a an d Lis a give s i t t o Annie. " Th e teache r asked thes e student s t o pas s th e bal l i n thi s fashio n t o sho w tha t suc h a path wa s possible . Afte r doin g so , Ms . Heil aske d Jackie , Lisa , an d Anni e to stan d an d al l other s t o sit . Figur e 1 indicate s th e tw o path s propose d by th e children ; th e origina l pat h consistin g o f eigh t childre n wh o touche d the bal l twic e (Fran k t o Charli e t o Zachar y t o Lis a t o Deann e t o Michae l t o Anthony t o Annie ) an d a shorte r pat h (Jacki e t o Lis a t o Annie ) introduce d by Daniel .

Frank Charli e Zachar y Lisa Deann e Michae l Anthon y Anni e

F I G U R E 1 .

A discussio n ensue d abou t ho w t o mak e a shorte r route . Lis a suggeste d that ther e i s no route whic h leave s fewer tha n thre e peopl e standin g because , "how ca n yo u coun t on e mor e perso n out ? You' d hav e t o thro w th e ball. " Daniel insiste d o n a ne w proposa l — Jane t t o Mat t t o Kenn y t o Maryan n to Anni e — the n independentl y realize d tha t thi s pat h wa s longe r tha n hi s original three-perso n path . Th e grou p conclude d tha t thre e wa s th e fewes t number o f childre n wh o mus t touc h th e bal l twice , unti l Danie l persiste d that th e bal l ca n b e passe d wit h onl y tw o student s touchin g i t twice . H e aggressively argued , "Jane t t o ... " bu t Cind y interrupted , "yo u ca n chang e the wa y you'r e passin g th e bal l t o onl y hav e Anni e touc h i t twice. " Danie l blurted, "yo u ca n jus t g o i n a circle. " Thes e suggestion s happene d simul - taneously an d th e lesso n tha t follow s wa s crafte d b y a gifte d teache r wh o encourages childre n t o explai n wha t the y ar e thinking .

Ms. Heil interrupted t o recognize appropriately th e thoughtful comment s that too k plac e and asked , "Okay , let's consider individuall y wha t Cind y an d Daniel eac h hav e said. " Th e teache r stoo d an d aske d Cind y t o exchang e positions wit h her . Al l childre n wer e no w sittin g o n th e floor , waitin g fo r their nex t instruction . Man y childre n wer e laughing becaus e the y though t i t was funny tha t th e teache r wa s sitting o n th e floor, wit h he r leg s crossed lik e all th e children , an d Cind y wa s no w i n th e teacher' s position . Cind y began , "Okay, Anni e give s t o Lisa , Lis a give s t o Robert , Rober t give s t o Sharon , Sharon give s t o Mark , etc. " Cind y orchestrate d th e movemen t o f th e bal l i n such a wa y tha t th e onl y perso n wh o touche d th e bal l twic e wa s Annie . Sh e worked outwar d fro m Annie , makin g sur e tha t everyon e touche d th e ball , but reserve d a pat h o f peopl e alon g th e fron t wal l whic h sh e late r use d a s the pat h tha t returne d th e bal l t o Annie . He r behavio r wa s ver y simila r t o

D I S C R E T E MATHEMATIC S I N K - 2 CLASSROOM S 191

that o f a mathematicia n a s sh e o r h e work s t o find a Hamilto n circui t i n a graph; tha t i s t o say , eac h decisio n abou t t o who m th e bal l shoul d nex t b e given i s made keepin g i n min d tha t everyon e neede d t o touc h i t onc e an d " a last path " wa s neede d i n orde r t o ge t bac k t o th e beginning . I though t t o myself, "A m I reall y i n a secon d grad e classroom? "

The teache r stoo d afte r th e tas k wa s complete d an d aske d wha t jus t happened? Th e childre n explained , "I f yo u d o i t th e firs t way , th e shortes t way w e coul d ge t i s thre e peopl e wh o touche d th e bal l twice , bu t i f yo u d o it Cindy' s way , yo u onl y ge t on e perso n wh o touche s th e bal l twice , Annie , so Cindy' s wa y i s shorter. "

"Now, wha t abou t Daniel' s comment . Daniel , wha t di d you say before? " He replied, "Yo u can just g o in a circle." Ms . Heil suggested, "Oka y everyone, let's ge t int o a circle. " Fro m a theoretical perspective , th e grap h represente d by th e childre n ha s bee n changed , bu t fro m a n educationa l perspective , Ms. Heil wa s presente d wit h a valuabl e opportunit y t o tak e th e lesso n int o uncharted territory , o f whic h sh e quickl y too k advantage .

All th e childre n sa t i n a larg e circl e o n th e floor. Th e bal l wa s give n to Annie . "No w ca n yo u pas s i t s o tha t everyon e touche s i t onc e an d i t gets bac k t o Annie? " Th e childre n passe d th e bal l an d whe n i t wa s re - turned t o Anni e th e teache r asked , "whic h wa y wa s easier? " The y shouted , "circle!" Why ? "Becaus e yo u kno w wher e you'r e going. " On e chil d actuall y explained, "becaus e you don't hav e to think abou t wher e to pass it next , yo u just ge t th e bal l fro m on e sid e an d pas s i t righ t t o th e next. " Thi s chil d wa s formulating a fundamenta l ide a i n compute r scienc e — tha t b y arrangin g many individua l units , eac h wit h a simpl e task , a large-scale , comple x tas k can b e performed . I n computer-scienc e terms , th e childre n wer e simulatin g cellular automata .

Ms. Heil continued, "I s there an y other wa y you could arrang e yourselve s so that ... " Anothe r chil d shouted , " a square". Th e childre n arrange d them - selves int o a square . The y agai n passe d th e ball . "I f I pas s th e bal l alon g the square , i s i t simila r o r differen t i f w e pas s i t o n a circle? " Severa l hand s were raise d immediatel y an d th e childre n responded , "similar. " On e chil d explained, "becaus e we'r e stil l passin g th e bal l t o someon e nex t t o you. " Another chil d shouted , " I thin k w e shoul d d o a triangl e becaus e w e coul d pass th e bal l ther e too. " Ms . Heil said , "Goo d idea! " Th e clas s arrange d itself int o a triangl e an d passe d th e bal l fo r a thir d time .

"So i s th e pat h i n th e triangl e simila r o r differen t t o th e pat h i n th e square?" Th e clas s responded , "Similar, " "Wha t abou t th e pat h i n th e tri - angle an d th e pat h i n th e circle? " "Similar. " "Wha t abou t th e shap e o f th e circle and th e shap e o f the square? " "Different" , the y shouted . "Wha t abou t the shap e o f th e circl e an d th e shap e o f th e triangle? " "Al l thei r shape s ar e different." "Ver y good! " th e teache r sai d a s sh e looke d a t m e i n surprise . "So a pat h i n a circle , square , o r triangl e i s similar eve n thoug h thei r shape s are different. " Thi s demonstrate d tha t secon d grader s ar e capabl e o f under - standing th e rudiment s o f topologica l equivalence .

192 VALERIE A . DEBELLI S

"What i f w e star t wit h Annie , bu t don' t en d there ? Coul d w e arrang e ourselves i n such a wa y that th e bal l start s wit h Anni e an d everyon e touche s it exactl y onc e bu t i t doesn' t hav e t o en d wit h Annie? " Th e student s wer e still arrange d i n thei r triangl e shape . The y looke d a t eac h othe r a s i f thi s was to o eas y a question . On e chil d said , "w e don' t hav e t o move . Jus t pas s it t o Anni e an d en d with Missy. " Miss y was the chil d who sat immediatel y t o the lef t o f Annie a s the bal l wa s passe d t o th e right . Anothe r chil d instantl y shouted, "w e coul d stan d i n a line. " Ms . Hei l began , "Okay , let' s ... " an d was interrupte d b y Danie l wh o said , "No , eve n i f yo u stan d i n a lin e yo u get i t bac k t o th e firs t person. " Th e teache r an d I wer e bot h confused . Di d Daniel se e a wa y fo r peopl e t o stan d i n a lin e an d stil l mak e a circuit ? Ms . Heil inquired , "Wha t d o you mean? " Danie l said , "Yo u just hav e t o giv e th e ball t o th e firs t person , th e firs t perso n give s i t th e thir d person , th e thir d to th e fifth , al l th e wa y t o th e end , an d the n tha t perso n jus t ha s t o pas s i t back t o th e one s who didn' t touc h i t yet. " I was truly amaze d a t thi s secon d grader's insight .

Ms. Heil said, "Danie l thinks tha t yo u can ge t th e bal l back t o th e begin - ning i f yo u stan d i n a lin e an d everyon e onl y touche s i t onc e excep t fo r th e first person . Wh o agree s with Daniel? " A few hands wer e raised, tentatively . "Okay Daniel , sho w u s wha t yo u mean. " Al l twenty-fou r student s stoo d i n a straigh t lin e excep t Daniel . H e gav e th e bal l t o Annie , wh o wa s standin g at on e end , an d said , "Anni e give s i t t o Prank , Pran k give s i t t o Michael , Michael give s i t ... " unti l th e bal l wa s passe d bac k t o Anni e wit h everyon e touching i t exactl y once . Figur e 2 depict s a simplifie d versio n o f th e pat h that Daniel , a second-grad e student , constructe d i n hi s mind ; Daniel' s pat h involved al l twenty-thre e children .

F I G U R E 2 .

"What jus t happene d here? " th e teache r asked . On e chil d explained , "even thoug h we'r e standin g i n a lin e yo u ca n ge t th e bal l bac k t o Anni e and onl y touc h i t once. " Th e lesso n conclude d b y introducin g th e word s "path" an d "circuit" . Whe n th e teache r introduce d th e wor d circuit , Pet e shouted, "i s tha t lik e a circui t breaker? " Childre n mak e connection s nat - urally i f they'r e allowe d t o investigat e thei r world . Th e word s circui t an d path wer e o n th e next' s wee k spellin g test .

I sa t bac k i n m y chai r i n amazement . Secon d grader s ar e quit e capabl e of intuitivel y constructin g path s an d circuit s i n quit e comple x ways . The y are abl e t o recogniz e tha t a ball' s pat h i s th e sam e i n a circle , i n a square , or i n a triangle . O f course , the y wer e unabl e t o discus s grap h isomorphism , but the y foun d way s tha t a circl e coul d b e th e sam e a s a squar e an d a s a triangle. Further , the y wer e abl e t o maintain , a t th e sam e time , tha t thes e objects hav e differen t shape s i n th e Euclidea n sense . The y wer e abl e t o

D I S C R E T E MATHEMATIC S I N K - 2 CLASSROOM S 193

identify an d generat e shorte r path s — b y finding a ne w wa y t o pas s th e ball tha t woul d involv e fewe r children . A secon d grad e classroo m ca n full y engage i n a dialogu e whic h i s filled wit h ric h mathematica l discourse .

This exampl e show s ho w widel y accessibl e topic s i n discret e mathemat - ics ca n be . A youn g person' s proble m ma y b e worde d a s follows : Give n N children, randoml y seate d o n th e floor, ho w d o w e pas s a bal l s o tha t eac h child touche s th e bal l onl y onc e an d s o tha t i t get s bac k t o th e first per - son wh o touche d th e ball ? A simila r challeng e fo r a mor e matur e proble m solver ma y b e worde d a bi t differently : Give n N point s i n th e plan e — eac h connected b y a n edg e wit h a fe w o f it s neighbors , find a Hamilto n circuit . Essentially, bot h population s (childre n an d adults ) ar e abl e t o discus s an d solve thes e problem s successfully . Havin g childre n thin k abou t suc h prob - lems durin g thei r primar y schoo l year s wil l provid e a foundatio n fo r late r mathematical development .

It migh t b e sai d tha t thi s secon d grad e classroo m wa s ful l o f gifte d children, o r a t leas t Danie l (th e chil d wh o generated man y interestin g path s during thi s lesson ) wa s quit e talented . Actually , non e o f the student s i n thi s class hav e bee n classifie d a s "gifted" , includin g Daniel . (I n th e Kossman n School, t o b e classifie d a s "gifted " th e chil d mus t scor e a t leas t a 13 5 o n the Wechsle r Intelligenc e Scal e fo r Childre n (W.I.S.C.). ) Mayb e educator s need t o evaluat e ho w w e determin e i f a chil d i s mathematicall y talented . I saw a fe w childre n i n thi s clas s wh o demonstrate d powerfu l mathematica l thinking an d wh o I woul d classif y a s "gifted. "

A visi t t o Kindergarte n

I wa s invite d int o Michel e Midura' s classroo m a t th e Irvin g Primar y School i n Highlan d Park , Ne w Jerse y t o teac h a discret e mathematic s les - son. Th e schoo l ha s roughl y fou r hundre d childre n i n grade s K throug h 2 . She teache s a self-containe d full-da y kindergarte n clas s wit h eightee n stu - dents, sixtee n o f who m wer e i n attendanc e o n th e da y o f m y lesson . Sh e describes th e clas s a s developmen t ally, culturally , an d economicall y diverse . Academically, th e student s hav e a wid e rang e o f abilities ; som e student s have learnin g disabilitie s whil e other s ar e readin g an d writin g o n a first grade level . Prio r t o m y arrival , sh e informe d m e tha t th e them e fo r th e month wa s sport s an d nutritio n an d suggeste d tha t whateve r mat h I did , I should someho w ti e i t t o on e o f thos e themes .

In keepin g wit h th e "sports " theme , I decide d t o introduc e th e notio n o f a tournamen t b y havin g each pai r o f children i n a small group rol l a giant di e to determin e a winner. I was uncertai n ho w muc h o f thi s topi c kindergarte n children woul d b e abl e t o understand , o r eve n whethe r the y woul d b e abl e to determin e i f ever y playe r i n thei r grou p compete d agains t ever y othe r player exactl y once . T o se e i f the y wer e capabl e o f bot h enumeratin g al l possibilities an d knowin g whe n the y foun d al l possibilities , I decide d t o begin wit h a combinatoric s activity .

194 VALERI E A . DEBELLI S

I entere d he r classroo m wit h a larg e duffl e ba g filled wit h sport s equip - ment, sneakers , an d severa l two-foo t lon g arrow s mad e fro m poster-boar d paper. Afte r thei r norma l routin e o f hangin g u p coats , turnin g i n home - work, selectin g ho t o r col d lunches , an d tellin g thei r mornin g new s (wha t Ms. Midur a call s "show-and-tell") , I a m abl e t o introduc e a mat h prob - lem. Th e childre n wer e sittin g o n th e floor arrange d i n a bi g square . I said , "Close you r eyes ! Kee p the m closed! " an d reache d int o m y bag . I pulle d out a se t o f plasti c bowlin g pin s wit h tw o plasti c bowlin g balls , stil l i n thei r original wrapper . I asked , "Wha t spor t woul d yo u b e playin g i f yo u neede d these?" The y simultaneousl y yelled , "Bowling! " Sixtee n littl e peopl e yellin g an answe r i n uniso n caugh t m e b y surprise . Thei r leve l o f excitemen t i s in - fectious. Thi s i s nothing lik e teaching undergraduates ! I place d th e bowlin g pins o n th e floor i n th e middl e o f th e squar e an d said , "Clos e you r eyes! " Several childre n bega n wigglin g wit h anticipation . "Kee p the m closed!" , I said. Afte r I pulle d ou t a tenni s racke t fro m th e ba g I asked , "Wha t spor t would yo u b e playin g i f yo u neede d this? " "Tennis!" , the y yelled . I place d the racke t o n th e floor nex t t o th e bowlin g pins . W e playe d th e "clos e you r eyes" routin e tw o mor e time s a s I pulle d ou t a pai r o f pin k sneaker s an d a pair o f Rebo k sneaker s an d place d the m o n th e floor.

"How man y way s ca n yo u choos e a pai r o f sneaker s an d a spor t t o play?", I asked . Ther e wa s dea d silence . I thought , "Uh , o h .. . thi s i s probably to o hard." I regrouped an d aske d a different question , "Ca n anyon e choose a pai r o f sneaker s an d a spor t t o play? " Al l sixtee n childre n raise d their hands . Anit a chos e th e pin k sneaker s an d bowlin g pins . I aske d i f anyone coul d find anothe r way . Jimm y chos e th e Rebo k sneaker s an d th e tennis racket . Bot h childre n wer e standin g i n fron t o f th e class , wearin g th e sneakers the y selecte d an d holdin g thei r chose n piec e o f sport s equipment . I pointe d t o eac h ite m an d repeated , "Okay , Anit a wear s pin k sneaker s an d bowls. Jimm y wear s what? " Th e childre n togethe r responded , "Rebok!" , "and plays?" , "Tennis!" , the y yelled . "Okay , ca n anyon e find a differen t way t o wea r sneaker s an d pla y a sport? " Sea n raise d hi s hand , walke d i n front o f th e fou r item s (no w o n th e floor) an d stare d a t them . Afte r a fe w seconds, I asked i f he would lik e a helpe r an d I noticed Ms . Midur a standin g behind al l th e childre n noddin g he r hea d yes . Sea n nodde d hi s hea d u p an d down and picke d the boy who was sitting next t o him. Togethe r the y selecte d the Rebo k sneaker s (becaus e pin k sneaker s wer e fo r girls ) an d th e bowlin g pins. I repeate d thei r choices , "Oka y no w w e hav e a differen t way . Rebo k sneakers an d bowlin g pins . Ca n anybod y find anothe r way? " Cind y selecte d pink sneaker s an d th e tenni s racket . I asked , "Wha t di d Anit a pick? " Th e students describe d he r selection . "Wha t di d Jimm y pick?" , "Wha t di d Sea n pick?", "Wha t di d Cind y pick? " Eac h tim e th e childre n describe d th e choic e of sneaker s an d sport s equipment .

Now I returned t o my original question , "Ho w many way s can you choos e a pai r o f sneaker s an d a spor t t o play? " "Four!" , the y yelled , continuin g t o respond i n unison. "Ho w did yo u kno w ther e wer e four?", I asked. On e chil d

D I S C R E T E MATHEMATIC S I N K - 2 CLASSROOM S 195

explained, "becaus e Anita' s wa y i s one , Jimm y i s two , Sea n i s three , Cind y is four , an d there' s n o othe r wa y t o mak e a matc h that' s different. " "Okay , what ar e th e differen t ways? " Togethe r th e childre n describe d eac h wa y o f matching a pai r o f sneaker s wit h a piec e o f equipment . Eac h tim e a ne w pair wa s mentioned , I place d a larg e fluorescent colore d arro w o n th e floor to sho w th e match . Afte r al l th e pair s wer e found , w e counte d th e numbe r of arro w head s togethe r an d discovere d fou r differen t ways . I said , "Clos e your eyes! " Thi s wa s no w a gam e fo r them . They'r e wigglin g an d gettin g excited becaus e the y kno w somethin g els e i s comin g ou t o f th e bag .

I place d a whiffl e bal l an d ba t o n th e floor nex t t o th e tenni s racket , removed th e arrows , an d asked , "No w ho w man y way s ca n w e matc h a pai r of sneaker s wit h a piec e o f sport s equipment? " Th e childre n enumerate d al l possibilities i n a simila r wa y describe d above . W e agai n place d th e arrow s on th e floor t o sho w al l si x possibilitie s an d coun t th e arro w heads . "Clos e your eyes! " Th e childre n wer e no w peekin g (an d tellin g m e tha t they'r e peeking) an d laughin g a s I pulle d a pai r o f men' s dres s shoe s fro m m y bag . I aske d them , "D o yo u kno w wha t thes e are? " N o on e responded . I said , "Geek sneakers! " an d the y al l starte d laughing . "No w ho w man y way s ca n we match a pai r o f sneaker s wit h a piec e o f sports equipment? " A variet y o f children suggeste d simultaneously , "ten" , "six" , "nine" , "twelve" . I aske d the youn g gir l wh o responde d "nine " t o explai n ho w sh e go t he r answer . After a bi t o f encouragement , sh e sai d tha t sh e counte d ever y objec t o n the floor (si x individua l sneaker s an d thre e piece s o f equipment) . The n on e child yelled , "No , it' s thre e plu s thre e plu s three. " I a m stunne d b y bot h responses.

I explaine d tha t mathematician s coun t al l sorts o f things. Sometime s w e count individua l things ; fo r example , i f we count al l the sneaker s an d al l th e equipment, w e find tha t ther e ar e nin e thing s altogether . Bu t sometime s w e count group s o f things , an d tha t i t i s goo d t o lear n ho w t o d o both . Her e we are countin g ho w man y way s ther e ar e o f formin g a grou p whic h ha s on e pair o f sneaker s an d on e kin d o f equipment . I aske d Carl o t o us e th e arrow s to sho w m e wha t h e mean t b y "thre e plu s thre e plu s three" . H e place d th e arrows o n the floor t o sho w each possibilit y an d the n proceede d t o coun t th e arrow heads ; altogethe r ther e ar e nine. H e showed tha t eac h pair o f sneaker s can b e matche d wit h thre e differen t piece s of sports equipmen t an d wa s abl e to describ e thi s matchin g a s "thre e plu s thre e plu s three" . I reviewe d thi s generalization wit h th e children , an d s o ende d th e combinatoric s activity .

Now convince d tha t kindergarte n childre n wil l b e abl e t o determin e i f all player s compete d agains t on e anothe r i n a tournament , I asked , "Fin d an X mad e fro m maskin g tap e o n th e floor an d si t o n to p o f it. " I n advance , eighteen smal l X' s wer e positione d i n thre e circle s (si x t o a circle ) s o tha t the fluorescent arrow s coul d b e place d betwee n an y tw o player s t o identif y the winne r an d lose r o f tha t competition . Al l sixtee n children , Ms . Midura , and he r classroo m assistan t playe d i n a tournament . Eac h grou p receive d one pai r o f gian t dic e an d fiftee n arrow s o f tw o length s — nin e lon g arrow s

196 VALERIE A . DEBELLI S

(to b e place d betwee n player s wh o ar e seate d acros s th e circl e fro m on e another) an d si x shor t arrow s (t o b e place d betwee n player s wh o ar e seate d next t o on e another). Playe r A rolled on e di e an d Playe r B rolled th e secon d die; togethe r the y determine d th e winne r — th e on e wh o rolle d th e large r number — and place d a n arro w o n the floor pointing fro m th e winner towar d the loser . Afte r al l fifteen arrow s wer e arrange d accordin g t o th e outcom e o f each competition , th e childre n wer e asked t o find a winning sequenc e — tha t is, a wa y o f listin g th e si x childre n i n th e grou p s o tha t eac h on e defeate d the nex t on e i n th e sequence ; th e winnin g sequenc e woul d b e a Hamilto n path i n th e directe d graph .

In thi s lesson , i t shoul d b e n o surpris e tha t th e childre n wer e unabl e to find a winnin g sequence . I di d severa l thing s wrong . First , th e siz e of th e group s (six ) wa s to o larg e fo r kindergarte n childre n t o eve n begi n to loo k fo r a winnin g sequence . Havin g fifteen arrow s pointin g i n man y directions containe d to o muc h informatio n fo r the m t o decipher . Second , the childre n wer e not comfortabl e wit h th e arrow s pointin g t o th e loser ; the y wanted the m t o poin t t o th e winne r an d afte r severa l protests , th e arrow s pointed t o eac h winner . Althoug h thi s doe s no t imped e on e fro m finding a Hamilton path , i t doe s introduc e anothe r cognitiv e ste p i n finding a winnin g sequence. Third , the y reall y like d gian t dic e an d I di d no t allo w enoug h time fo r th e childre n t o "play " befor e I aske d the m t o hol d a tournament . Seasoned teacher s o f primar y grade s alway s allo w tim e fo r pla y befor e the y ask childre n t o complete a task. Finally , I did no t revie w the skil l of finding a Hamilton pat h — that i s to say, constructing a path alon g a series of directe d edges s o a s t o touc h ever y verte x exactl y once . I woul d certainl y d o thi s activity differentl y th e nex t time , an d expec t tha t th e childre n woul d find Hamilton paths ; severa l kindergarte n teacher s i n th e Leadershi p Progra m have reported tha t the y were able to do this with the children in their classes .

Although th e lesso n di d no t achiev e wha t I initiall y intended , I learne d that childre n i n kindergarte n ar e abl e t o understan d issue s tha t aris e i n enumerating possibilitie s (sometime s a n exercis e i n creativ e thinking ) an d determining tha t al l possibilitie s hav e bee n exhausted . I als o learne d tha t children i n kindergarte n ar e quit e capabl e o f decidin g wh o i s a winne r an d how t o positio n a directe d edg e t o reflec t that . I als o believ e tha t childre n at thi s ag e ca n identif y a winning sequenc e i n a tournament wit h fou r o r five competitors, base d o n m y observation s o f why the y wer e unabl e t o complet e the tas k i n th e settin g describe d here . I n th e lon g run , suc h activitie s hel p children develo p strategies tha t ar e valuabl e fo r late r us e in proble m solving , as wel l a s fo r probability . No t t o includ e suc h activitie s throughou t th e primary grade s i s a seriou s omission .

In thi s classroo m I too k a ris k b y attemptin g a n activit y whic h I ha d not trie d befor e wit h childre n a t thi s grad e level . However , I hav e describe d this "failure " her e s o tha t teacher s wil l understan d tha t i t i s importan t fo r them t o tak e simila r risk s i n thei r classrooms . Teacher s wh o attemp t t o bring discret e mathematic s int o thei r classroom s wil l find tha t sometime s

D I S C R E T E MATHEMATIC S I N K - 2 CLASSROOM S 19 7

their lesson s ar e successful , an d sometime s the y ar e not . Nevertheless , the y should continu e t o tr y ou t ne w activities , sinc e ultimately thei r student s wil l benefit, a s the y fin d way s o f improvin g o n thei r faile d attempts , an d eve n their presen t student s wil l benefit , a s the proble m solvin g prepare s the m fo r future challenges .

A colorin g e x a m p l e

A popula r discret e mathematic s topi c amon g teacher s wh o atten d th e Leadership Progra m i s grap h coloring . W e introduc e th e topi c b y havin g groups o f teacher s wor k togethe r t o colo r a five-foo t ma p o f th e Unite d States. Eac h grou p i s provided wit h tw o hundre d circle s cu t fro m construc - tion paper . Ther e ar e te n differen t color s wit h twent y circle s o f eac h color . The initia l questio n pose d i s to simpl y colo r th e map . I n a shor t time , som e groups hav e nicel y colore d map s usin g al l te n colors , othe r group s ma y b e trying t o us e fewe r colors , an d ye t other s ma y defin e specifi c color s t o repre - sent characteristic s o f tha t stat e (i.e. , al l state s tha t borde r th e ocean , colo r orange o r green) . W e no w introduc e th e mapmake r problem . Imagin e tha t you ar e a mapmake r an d th e cos t t o produc e a ma p increase s base d o n th e number o f differen t color s yo u use . Further , sinc e ever y mapmake r want s individual region s to b e clearly viewed o n the map , n o two regions that shar e a borde r ma y b e colore d wit h th e sam e color . Wha t ar e th e fewes t numbe r of color s neede d t o colo r th e ma p o f th e Unite d States ? Why ? Ca n yo u identify area s o f th e ma p whic h caus e problems ? (I f yo u haven' t though t about thi s problem , tak e a fe w minute s t o thin k abou t i t befor e readin g th e rest o f thi s article! )

When I walke d int o Sharo n Heil' s secon d grad e classroom , I sa w ever y piece o f availabl e wall spac e fille d wit h students ' work , fro m floor t o ceiling . In th e fron t o f th e classroo m wa s a bulleti n boar d dedicate d t o ma p color - ing. Ther e wer e severa l colore d map s o f th e Unite d State s (partitione d int o states) an d severa l map s o f Ohi o (partitione d int o counties , supplie d b y a Leadership Progra m participan t fro m tha t state ) whic h wer e colore d b y th e students s o tha t n o tw o region s whic h shar e a borde r ha d th e sam e color . I though t t o myself , "I'v e see n tha t befor e .. . bu t I wonde r wha t secon d graders go t fro m th e experience. "

Ms. Hei l asked , "Doe s anyon e remembe r wha t w e di d whe n w e colore d the maps? " Instantly , severa l hand s wer e raised . On e chil d explaine d tha t you "colo r tw o state s wit h differen t color s i f they'r e nex t t o eac h other. " "Anything else?" , Ms . Heil asked . Anothe r chil d explained , "W e ha d t o decide i f corner s counte d o r not. " Severa l student s proudl y pointe d t o thei r map. Som e group s decide d tha t corner s "counted" , tha t is , the y shoul d b e considered a s shared borders , an d som e group s decide d tha t the y shoul d no t be considere d a s a commo n boundary . A t th e clos e o f th e discussion , Ms . Heil smile d an d whispered , "w e di d tha t ove r on e mont h ago. "

This discussio n wa s interestin g becaus e whe n w e giv e th e sam e exercis e to K- 8 teacher s i n ou r summe r institutes , the y to o begi n th e activit y wit h a

198 VALERIE A . DEBELLI S

similar struggle , namely , t o decid e i f a poin t shoul d b e considere d a share d boundary. A n exampl e o f thi s occur s i n th e Unite d State s ma p a t th e poin t where Arizona , Ne w Mexico , Colorado , an d Uta h meet . Mos t group s o f teachers initiall y colo r thes e fou r state s wit h fou r differen t colors . Whe n they ar e aske d t o colo r thei r map s usin g th e fewes t colors , the issu e arise s a s to wha t define s a border . Bot h population s (teacher s an d secon d graders ) had t o decid e and defin e fo r themselve s whether a point shoul d b e considere d a share d border . Al l teacher s wer e abl e t o mak e th e distinctio n tha t region s that mee t a t a poin t d o no t necessaril y hav e t o b e considere d a share d border; som e secon d grader s understoo d this , bu t other s di d not . Whe n you mak e a decisio n tha t tw o region s tha t mee t a t a poin t ma y hav e th e same color , th e numbe r o f color s yo u wil l nee d t o colo r a ma p ma y indee d be fewer . Henc e som e o f th e students ' map s wer e colore d wit h thre e colors , but other s use d mor e colors . Ms . Hei l late r explaine d tha t sh e di d no t focu s on "fewes t colors " an d tha t th e activit y wa s intende d t o introduc e colorin g and discus s wha t make s somethin g a border .

Map coloring is another exampl e tha t show s how widely accessible topic s in discret e mathematic s ca n be ; bot h adult s an d youn g childre n ca n engag e in mathematica l proble m solvin g an d experienc e simila r difficulties . I n th e end the y ma y resolv e the m quit e differently , bu t the y eac h hav e t o defin e aspects o f th e proble m tha t ar e no t clear . Workin g throug h th e "muck " o f problem situatio n i s on e o f th e mor e difficul t aspect s o f proble m solvin g t o teach; on e must b e willing to intimatel y engag e the proble m rathe r tha n pas - sively perceiv e i t (se e Levin e [4]) . Introductor y discret e mathematic s topic s seem t o invit e peopl e fro m th e non-mathematica l communit y t o thin k abou t their problem s because difficult problem s ar e easily understood an d typicall y require littl e prerequisit e knowledge ; i t ma y b e tha t discret e mathematic s can serv e t o attrac t under-represente d person s int o th e mathematic s field. It certainl y attract s adult s an d youn g childre n alike !

Traditionally, colorin g ha s bee n use d i n primar y grad e classroom s t o help develo p dexterity , creativity , an d artisti c talent . Now , wit h a n intro - duction t o discret e mathematics , teacher s o f K- 2 classe s ca n incorporat e mathematics int o thei r colorin g boo k activitie s b y askin g th e childre n t o color th e pag e s o tha t n o tw o region s tha t shar e a borde r hav e th e sam e color. Onc e i t i s clear tha t everyon e i s able t o colo r th e pag e i n this way , th e teacher ca n introduc e th e questio n o f usin g fewe r colors . Befor e the y begin , the childre n ca n b e encourage d t o tal k abou t ho w they migh t develo p a pla n to us e fewe r colors . The y ca n discus s wh y on e colo r o r tw o color s ma y no t be enoug h t o colo r th e picture , o r wh y N color s ma y b e to o many . Thes e early conversation s ca n hel p childre n begi n t o develo p mathematica l idea s about minimizatio n an d giv e practic e i n reasonin g abou t lowe r an d uppe r bounds. I n addition , suc h lesson s ca n als o help develo p powerfu l mathemat - ical proble m solvin g skills . Fo r a goo d descriptio n o f th e valu e o f colorin g in K- 4 classrooms , se e Case y an d Fellow s [1] .

D I S C R E T E MATHEMATIC S I N K - 2 CLASSROOM S 19 9

Coloring book s ar e als o a goo d plac e t o introduc e childre n t o construct - ing dua l graphs , a s par t o f a n introductio n t o th e topi c o f graphs . A typica l page i n a colorin g boo k ma y loo k lik e th e first bea r i n Figur e 3 . Primar y

F I G U R E 3 .

grade childre n ca n b e aske d t o plac e a do t insid e eac h regio n an d the n con - nect tw o dot s i f th e region s shar e a commo n border ; th e traditiona l kinder - garten curriculu m alread y include s learnin g notion s lik e "inside/outside " and "nex t to " a s geometr y topics . Thes e activitie s ca n provid e a visua l wa y to determin e i f childre n understan d suc h notions . I f th e teache r choose s th e picture wisely , she can the n hav e further discussion s with he r student s abou t the structur e o f th e graphs . Fo r example , i n Figur e 3 , a teache r coul d as k the childre n t o coun t th e numbe r o f vertice s an d edges ; t o describ e wha t parts o f th e grap h loo k th e sam e o r wha t part s o f th e grap h loo k different ; to determin e whethe r th e grap h i s connecte d (together ) o r disconnecte d (i n parts); o r whethe r the y ca n fol d a piec e of pape r i n suc h a wa y s o that ever y vertex wil l li e o n to p o f anothe r verte x an d ever y edg e o f th e grap h wil l li e on to p o f anothe r edge . Thes e experiences , i f implemente d i n thoughtfu l ways, ca n hel p develo p earl y notion s o f structur e an d symmetry .

T h e futur e o f t h e K - 2 curriculu m

The K- 2 curriculu m fo r th e twenty-firs t centur y need s t o includ e tech - nology topic s an d th e mathematic s tha t underlie s computing . Som e of thes e topics ma y b e paths an d circuit s i n graphs (consistin g of vertices an d edges) , vertex coloring , Eule r an d Hamilto n path s an d circuits , shortes t routes , counting, listin g an d sorting , an d recognizin g an d usin g pattern s i n num - ber an d geometry . However , I a m no t suggestin g school s shoul d simpl y ad d more mathematica l topic s t o a n alread y packe d curriculum . Rather , youn g children ca n lear n t o ad d number s i n th e contex t o f travelin g alon g path s in weighte d graph s (wher e eac h edg e i s assigne d a "weight " whic h may , fo r example, b e th e distanc e betwee n th e site s represente d b y th e vertice s a t the end s o f th e edge) , o r coun t "th e numbe r o f ways " — a n activit y tha t can b e don e instea d o f just countin g th e natura l numbers .

Primary grad e student s ca n establis h efficien t way s for dealin g with thei r environment, an d determin e wha t make s somethin g bette r (o r shorter , o r quicker) tha n somethin g else . I t i s durin g thi s tim e tha t the y ca n lear n

20 0 VALERIE A . DEBELLI S

how t o follo w directions , follo w classroo m rules , follo w a recipe , follo w a map, an d follo w a n algorithm . The y ca n lear n ho w t o coun t th e numbe r o f different way s t o mak e chang e fo r on e dollar , o r ho w t o systematicall y lis t the differen t way s t o arrang e thre e shirt s wit h thre e pair s o f pants . The y can mak e flipchart storybook s t o demonstrat e th e tota l numbe r o f outfit s one ca n wea r and , throug h suc h activities , com e t o kno w tha t mathematic s is a wa y o f thinking , no t a wa y o f memorizing . Al l o f thes e topic s ar e discussed i n detai l fo r th e K- 2 grad e levels , a s fo r othe r grad e levels , i n th e discrete mathematic s chapte r o f th e New Jersey Mathematics Curriculum Framework (se e Rosenstein , Caldwell , an d Crow n [6]) .

At th e primary grad e levels, children ca n als o be assisted an d encourage d to com e t o understan d wha t i t mean s t o b e a powerfu l proble m solver . A powerful proble m solve r i s on e wh o know s mor e tha n jus t a bunc h o f goo d strategies fo r solvin g a problem; i t i s a person wh o (amon g other things ) use s intuition, generate s conjectures , i s creative , an d perseveres . Youn g childre n can learn how to make a good prediction, ho w to remain comfortabl e eve n if a problem i s lef t unsolve d fo r severa l days , an d tha t sometime s goo d proble m solvers ge t wron g answers . The y ca n als o lear n tha t workin g o n a har d mathematical proble m i s sometimes frustrating , bu t tha t negativ e emotion s can b e regulate d b y th e proble m solve r t o a usefu l purpos e (se e DeBelli s [2]). Th e abilit y t o b e successfu l a t proble m solvin g i s n o longe r a highe r order thinkin g skil l tha t onl y mathematicall y talente d childre n ar e expecte d to demonstrate ; rather , al l citizen s o f th e twenty-firs t centur y wil l nee d thi s skill t o functio n i n a high-tec h world . Today' s kindergarte n childre n wil l graduate i n th e yea r 2009 .

Conclusions

I wa s quit e surprise d a t th e sophisticatio n wit h whic h primar y grad e students ca n behav e a s scientists . A s I walke d aroun d K- 2 classrooms , ob - serving othe r activitie s a s wel l a s thos e describe d i n thi s article , I watche d young childre n mak e conjectures , argu e wit h tea m member s fo r particula r outcomes, demonstrat e th e abilit y t o collec t an d recor d dat a accurately , verify tha t a n experimen t wa s ru n correctl y b y makin g sur e th e su m o f each componen t equalle d th e tota l numbe r o f experiment s conducted , an d demonstrate th e abilit y t o mak e th e distinctio n betwee n a predictio n an d a bes t prediction . The y als o intuitivel y discusse d fundamenta l notion s o f isomorphisms, algorithms , an d topologica l equivalence . The y wer e prou d o f themselves whe n eac h mathematica l tas k wa s completed , jus t a s th e teach - ers wer e wh o worke d o n th e sam e (o r similar ) problem s i n th e Leadershi p Program.

Certain idea s i n mathematic s — such a s "isomorphism" , "enumeration " (systematic listin g o f possibilities), o r th e abilit y t o generat e globa l comple x behaviors wit h simpl e loca l rule s — ar e ver y importan t an d shoul d b e de - veloped i n youn g children . I t shoul d no t b e tha t thes e discussion s happe n in K- 2 classroom s b y chance . Mathematicians , mathematic s educators , an d

DISCRETE MATHEMATIC S I N K- 2 CLASSROOM S 20 1

teachers nee d to collaborate to define wha t "bi g mathematica l ideas " ough t to b e learned a t eac h grad e level . Technolog y wil l continu e t o evolve an d new mathematica l discoverie s wil l unfold . Unles s schoo l system s allo w for the constan t infusio n o f new mathematica l topic s an d informatio n int o thei r curriculum, the y wil l foreve r b e teaching archai c topic s a t inappropriat e grade levels .

Finally, K- 2 discrete mathematic s topics , whe n introduce d b y goo d teaching methods , ca n serv e no t onl y t o buil d th e foundation s fo r importan t mathematical ideas , bu t als o ca n serv e a s a vehicle to help cove r traditiona l curriculum topics . K- 2 teachers nee d continue d suppor t fro m universit y and colleg e facult y member s wh o ar e bot h knowledgeabl e abou t th e conten t and understan d th e mathematical developmen t o f young children . A t th e same time, teacher s nee d t o remai n activ e i n the learnin g o f mathematics, at whatever leve l is appropriate fo r them. I t is only whe n teacher s themselve s are activ e proble m solver s who , fo r example, thin k abou t problem s the y cannot ye t solve, tha t the y ca n mode l th e desire d mathematica l behavior s for th e childre n i n their classes . Suc h activitie s an d collaboration s ca n onl y benefit th e children .

References

[1] Casey , Nancy , an d Michae l R . Fellows, "Implementin g th e Standards : Let' s Focu s on the Firs t Four" , thi s volume .

[2] DeBellis , Valeri e A. , Interactions between affect and cognition during mathematical problem solving: A two year case study of four elementary school children. Doctora l dissertation, Rutger s University , 1996 . An n Arbor, Michigan : Universit y Microfil m 96-30716.

[3] Inhelder , Barbel , "Som e aspect s o f Piaget's geneti c approac h t o cognition", i n Han s G. Furth , Piaget & Knowledge: Theoretical Foundations (2n d edition) , Universit y of Chicago Press , 1981 , p. 22 .

[4] Levine , Marvin , Effective Problem Solving, Prentic e Hall , 1994 . [5] Rosenstein , Josep h G. , " A Comprehensiv e Vie w of Discrete Mathematics : Chapte r 14

of th e New Jerse y Mathematic s Curriculu m Framework" , thi s volume . [6] Rosenstein , Josep h G. , Jane t H . Caldwell , an d Warre n D . Crown , New Jersey Mathe-

matics Curriculum Framework, Ne w Jerse y Mathematic s Coalition , 1996 . [7] Rosenstein , Josep h G. , an d Valeri e A . DeBellis , "Th e Leadershi p Progra m i n Discrete

Mathematics", thi s volume .

C E N T E R FO R MATHEMATICS , S C I E N C E , AN D C O M P U T E R EDUCATIO N ( C M S C E ) AND

C E N T E R FO R D I S C R E T E MATHEMATIC S AN D T H E O R E T I C AL C O M P U T E R S C I E N C E (DI -

MACS), R U T G E R S U N I V E R S I T Y

E-mail address: d e b e l l i s O d i m a c s . r u t g e r s . e d u

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

R h y t h m a n d P a t t e r n : Discret e M a t h e m a t i c s wi t h an Artisti c Connectio n fo r

E l e m e n t a r y Schoo l Teacher s

Robert E . Jamiso n

[It is easy to] appreciate sunsets, and the ocean waves, and the march of the stars across the heavens. As we look into these things we get an aesthetic pleasure from them directly on observation. There is also a rhythm and a pattern between the phenomena of nature which is not apparent to the eye, but only to the eye of analysis; and it is these rhythms and patterns which we call Physical Laws.

—Richard Feynman , The Character of Physical Law [11 ]

1. Introductio n

Over th e pas t tw o years, I hav e ha d th e privileg e o f offering a cours e en - titled Connecting Mathematics with Art, Music, and Nature t o tw o cadres of elementary school teachers participating i n the Ocone e County Lea d Teache r program.1 Eac h cadr e o f abou t twent y teacher s dedicate d ever y Monda y night fo r tw o year s t o th e project . Althoug h simila r i n spiri t t o th e elemen - tary mathematic s specialis t progra m suggeste d b y th e NCT M (Nationa l Council o f Teacher s o f Mathematics) , th e Ocone e projec t focuse s o n gen- eralists, rathe r tha n thos e wh o alread y hav e a specia l affinit y o r gif t fo r mathematics, wh o ca n the n becom e leader s i n thei r school s fo r introducin g new an d mor e successfu l approache s t o mathematic s instruction . I n thi s way th e principl e tha t "mathematic s i s fo r everyone , ca n b e learne d b y ev - eryone, an d enjoye d b y everyone " i s emphasized . Quit e naturally , man y o f the topic s wer e i n discret e mathematics . Som e ar e fortunatel y becomin g a common par t o f th e curriculum : buildin g polyhedr a wit h Polydron shape s [18, 20 , 21] , makin g tessellation s [31 , 3 3 , 34 , 38] , an d classifyin g stri p

1991 Mathematics Subject Classification. Primar y 00A35 . x The Lea d Teache r progra m wa s develope d a t th e Universit y o f Chicago , an d cus -

tomized fo r th e Schoo l Distric t o f Ocone e County , Sout h Carolina , b y An n Stafford , a district offic e staf f professional , an d Sybi l Sevic , a classroo m teacher .

20 3

https://doi.org/10.1090/dimacs/036/16

20 4 ROBERT E . JAMISO N

patterns b y symmetr y type s [6 , 9] . Th e topic s t o b e discusse d her e ar e les s standard: drawin g exercise s fo r regula r polygons , movemen t exercise s t o de - velop symmetry concepts , an d connection s betwee n modula r arithmeti c an d music.

Currently, I am usin g man y o f the sam e idea s an d activitie s i n a n under - graduate geometr y cours e fo r pre-servic e elementar y schoo l teachers . Th e course ha s fou r mai n goals :

1. t o broade n th e participants ' vie w o f mathematics ; 2. t o graduall y stretc h thei r leve l o f comfor t wit h mathematica l idea s

and abstractions ; 3. t o introduc e th e participant s t o a developmen t ally appropriat e mode l

of education ; an d 4. t o giv e new meaning t o mathematics b y connecting i t t o subject s tha t

have emotiona l conten t lik e ar t an d music .

This articl e describe s som e o f th e activitie s tha t I us e an d th e mathe - matics underlyin g them . I t i s addresse d primaril y t o mathematician s an d mathematics educator s workin g with elementar y schoo l teachers or students ; elementary schoo l teacher s ma y als o fin d activitie s her e t o tr y i n thei r class - rooms.

As Peter Hilto n [17 ] has said, "Geometr y i s a natural sourc e of question s and algebr a i s a sourc e o f tool s t o answe r them . Whe n w e teac h algebr a before geometry , w e as k student s t o answe r question s tha t n o on e woul d ever ask . An d late r i n geometry , w e giv e the m problem s tha t the y hav e no hop e o f eve r solving. " Thi s speak s t o th e ide a tha t ther e i s a preferre d order i n th e introductio n o f mathematica l concepts . I stan d firml y wit h Rudolf Steine r [30] , Jean Piage t [8] , and th e va n Hiele s [10 ] in believing thi s order i s development ally determine d an d tha t i n roug h outlin e i t follow s th e historical developmen t o f th e subject .

The impuls e fo r man y o f th e activitie s I describ e her e come s fro m th e Montessori an d Waldor f [2 , 3 , 4 , 5 , 2 5 , 26 , 30 , 37 ] educationa l move - ments, whic h I believ e hav e muc h o f valu e t o offe r al l schools . I n th e lea d teacher course s a s wel l a s i n othe r courses , I hav e no t onl y use d idea s fro m Montessori an d Waldor f educatio n bu t hav e als o integrated workshop s give n by experience d Montessor i an d Waldor f teachers . Th e benefi t o f thes e idea s is that the y stres s th e valu e of proper foundation s fo r concep t formatio n an d learning.

In order t o clarif y m y approach, fo r th e sak e of discussion, le t m e sugges t five stage s i n th e learnin g process :

1. encounte r 2. observatio n 3. reflectio n 4. understandin g 5. creativit y

DISCRETE MATHEMATIC S WIT H A N ARTISTI C CONNECTIO N 20 5

These term s hav e rathe r broa d meaning s i n genera l use , s o le t m e describ e more specificall y wha t I wan t the m t o mea n here . Suppos e tha t o n m y wa y to wor k I pas s a particula r ol d bric k building . I se e i t everyday , perhaps , just ou t o f th e corne r o f m y eye , withou t payin g muc h attention . I t i s jus t there. The n I hav e encountere d it . On e da y somethin g cause s m e t o paus e and notic e th e building . "Oh, " I think , "tha t i s a n interestin g building , with rathe r attractiv e bric k work. " No w I hav e observe d it . I f I continu e to notic e th e building , lookin g fo r differen t pattern s i n th e brickwork , an d wondering ho w the y wer e made , the n I hav e begu n t o reflect . I n a garde n walk behin d m y house , I successfull y incorporat e a bric k patter n lik e on e o f those i n th e building . Thi s demonstrate s understanding . I f I no w choos e t o invent a patter n o f m y own , the n I hav e becom e creative .

In thi s analysis , I see rule formatio n a t th e thir d stag e an d skil l develop - ment a t th e fourth . Thes e ar e th e primar y focu s o f ou r curren t educationa l system, reinforce d b y constan t testing . Bu t the y depen d ver y muc h o n earlier experience s an d observation s t o mak e the m meaningful . Thes e foun - dational experience s ma y li e severa l year s bac k o r ma y requir e repetitio n over a lon g period. 2

Unfortunately ou r curren t educationa l syste m offer s fe w incentive s for , say, first grad e teacher s t o provid e th e foundationa l experience s s o essentia l to a fourt h grad e teacher' s success . Nonetheless , I strongl y encourag e th e teachers i n m y classe s t o consciousl y provid e experience s fo r thei r students ' later developmen t eve n i f i t wil l no t b e teste d i n thei r classes . Thi s i s on e of th e mai n idea s underlyin g th e activitie s presente d here .

A secon d mai n ide a i s th e valu e o f kinestheti c an d sensoria l learning . This i s a particularl y stron g featur e o f th e Montessor i material s an d th e morning "concentratio n exercises " i n th e Waldor f schools . Th e ide a i s tha t by activel y involvin g ou r bodie s an d senses , th e meaningfulnes s o f learnin g is enhanced . Fo r tha t reason , th e activitie s whic h follo w involv e movemen t and colo r a s essentia l features .

The thir d mai n ide a behin d th e activitie s i s th e artisti c element . Th e goal i s no t jus t t o loo k fo r application s o f mathematic s i n art . No r i s i t to us e ar t a s a suga r coatin g fo r th e bitte r pil l o f mathematics . Rathe r I hop e t o captur e th e artisti c spiri t i n mathematic s a s somethin g beauti - ful an d creative . Lookin g fo r symmetr y pattern s i n medieva l architectur e should stimulat e th e students ' estheti c sens e an d sens e o f histor y a s muc h as piqu e thei r mathematica l curiosity . Thu s th e goa l i s no t a dominanc e o f one subjec t ove r th e others , bu t a balance d blen d i n whic h eac h i s see n t o offer somethin g o f value .

2 This i s on e reaso n wh y bot h Montessor i an d Waldor f hav e th e sam e teache r sta y with a clas s fo r mor e tha n on e year . Montessor i teache s i n three-yea r cycle s wit h grade s 1, 2 , an d 3 togethe r followe d b y a cycl e wit h grade s 4 , 5 , an d 6 together . Th e Waldor f teacher stay s wit h th e sam e clas s fo r eigh t year s an d s o ca n enjo y i n th e eight h grad e th e blossoms fro m seed s plante d i n th e first .

20 6 R O B E R T E . JAMISO N

2. Drawin g P o l y g o n s an d Thei r Diagonal s

Drawing Regula r Polygons . Th e firs t tas k her e is deceptively simple : to dra w th e regula r n-gon s (fo r n < 10) . Wha t make s i t interestin g i s tha t the drawin g i s t o b e don e freehand , withou t liftin g th e crayo n fro m th e paper an d withou t turnin g th e paper. 3 Th e goa l i s t o develo p a n intuitiv e feel fo r th e regula r polygon s i n th e hand s an d finger s s o tha t th e drawin g comes wit h eas e an d freedom . Student s mus t recal l a menta l imag e o f th e regular n-go n an d us e thei r understandin g o f th e figur e t o reproduc e i t o n paper. Thi s require s student s t o activel y reflec t o n their previou s experienc e with th e regula r n-go n an d ca n lea d t o som e satisfying insights . Also , i n ou r everyday experience , ther e ar e enoug h encounter s wit h triangles , squares , hexagons, an d octagon s tha t mos t peopl e ca n dra w the m rathe r well . Th e 5-, 7- , 9- , an d 10 - gon s ar e les s familia r an d th e missin g experienc e wit h these figure s need s t o b e provided .

For mos t people , drawin g th e pentago n an d heptago n i s already difficult , and i t i s bes t t o approac h th e tas k i n stages . Yo u migh t begi n b y havin g students trac e aroun d templates 4 wit h (colored ) pencil . Thi s provide s th e student wit h model s fo r late r us e a s wel l a s a valuabl e kinestheti c encounte r with th e shapes . Th e tracin g shoul d b e don e i n on e motio n an d i f possibl e without liftin g th e pencil .

"Helping figures " shoul d als o b e use d i n th e beginnin g (se e Figur e 1) . For example , i n drawin g a n octagon , on e ca n star t wit h a squar e an d cu t of f the corner s (Figur e l a ) . Dually , on e coul d "pus h out " th e side s o f a squar e by addin g fou r ne w corner s abov e th e center s o f side s o f th e square . Th e decagon ca n b e derive d fro m th e pentago n i n a simila r way . T o get th e pen - tagon, i t ma y be helpful t o start wit h the 5-pointe d sta r (th e pentagram) an d connect it s points . Th e 5-pointe d sta r resemble s a perso n wit h leg s sprea d and arm s out-stretched , a s i n th e celebrate d d a Vinc i drawing—makin g th e anthropomorphic connectio n explici t ca n strengthen th e connectio n betwee n the abstrac t worl d o f mathematic s an d th e direc t experienc e o f th e child . The 9-go n ca n b e buil t u p b y addin g trapezoid s t o th e side s of an equilatera l triangle (Figur e l b ) , bu t gettin g th e proportion s jus t righ t i s rathe r tricky . There i s n o wa y t o reduc e th e 7-go n t o a smalle r polygon—sinc e 7 is prime .

After a se t o f tracing s i s completed , an d th e student s hav e create d th e shapes b y modifyin g simple r polygons , th e freehan d drawin g ca n begin . A t first, allo w th e student s t o lif t th e crayo n an d tur n th e paper . A s under - standing, skill , an d confidenc e grow , as k the m t o reduc e th e us e o f thes e aids. Th e accurac y o f th e fina l drawing s ca n b e visuall y checke d b y turnin g the pag e t o se e whethe r th e figur e look s th e sam e n o matte r whic h sid e i s chosen a s base .

31 say "crayon " her e because , a s th e student s discover , crayo n i s mor e forgivin g tha n pencil, an d produce s attractive , colorfu l drawings . I highl y recommen d th e beeswa x bloc k crayons fro m Stockmar—lightl y rubbin g th e pape r produce s a lovely , sof t colore d back - ground. O f course , th e us e o f ruler s i n thi s exercis e i s taboo !

4E.g., th e Montessor i "geometr y cabinet " contain s a tra y o f regula r n-gon s ( n < 10) .

DISCRETE MATHEMATIC S WIT H A N ARTISTI C CONNECTIO N 20 7

FIGURE 1 . Usin g "helpin g figures" t o dra w regula r n-gons : (a) derivin g a regula r octago n b y cuttin g th e corner s o f a square; (b ) derivin g a regula r 9-go n b y addin g trapezoid s t o the side s o f a n equilatera l triangle .

Several importan t mathematica l idea s hav e crep t int o th e exercis e b y this stage. 5

• Th e drawin g require s tha t segment s hav e equa l lengt h an d tha t an - gles b e equal . Tha t is , a polygo n i s regula r i f an d onl y i f i t i s bot h equilateral an d equiangular . Fo r triangles , thes e tw o propertie s ar e equivalent, bu t fo r quadrilateral s the y defin e tw o different classes , th e rhombi an d th e rectangles . Thi s lead s nicel y int o a discussio n o f th e different classe s o f quadrilaterals , th e natur e o f mathematica l defi - nitions, an d tha t all-importan t question : "I s a squar e a rectangle? " (See [7 , pp. 133-140 ] fo r a goo d discussio n o f th e Aristotelia n theor y of definition s versu s th e moder n theory. )

• Th e helpin g figures use d alon g th e wa y illustrat e relationship s amon g the regular n-gon s based on the factor s an d divisibilit y properties o f n. The geometr y the n become s a visibl e expression o f certain arithmeti c relationships.

• Checkin g th e accurac y b y turnin g th e drawin g invoke s a "transfor - mational" definitio n o f regularit y (i n mathematica l terms , a figure i s regular i f an d onl y i f it s grou p o f symmetrie s act s transitively) . I t also prepare s th e wa y fo r a discussio n o f rotation s an d reflection s i n general.

• Certai n pedagogica l issue s ar e als o addressed . Holdin g th e pape r fixed require s th e studen t t o sens e equa l length s an d equa l angle s i n a variet y o f orientations—no t jus t i n th e natura l horizontal-vertica l frame o f referenc e give n b y th e bilatera l symmetr y o f th e student' s

51 discus s thes e issue s wit h th e in-servic e teacher s bu t urg e the m no t t o discus s the m with thei r schoo l children . The y ma y serv e a s guide s fo r observin g a child' s progress , bu t the prope r qualitie s shoul d b e instille d b y example , no t b y edict .

208 ROBERT E . JAMISO N

body. No t liftin g th e crayo n require s concentratio n an d force s stead y adjustment o f perception o f length an d angle . Thi s require s effort an d forces a ver y consciou s encounte r wit h th e polygon . Th e drawing s are als o a n exercis e i n neatness , patience , precision , an d prid e i n workmanship, qualitie s whic h ar e valuabl e i n mathematic s a s wel l a s other endeavors .

Diagonals an d Sta r Polygons . Onc e th e student s ar e comfortabl e with th e freehan d drawings , the y ca n b e aske d t o dra w th e diagonal s o f th e regular n-gons , yieldin g th e variou s sta r polygons , a s i n Figur e 2 . Color , o f course, ca n b e use d t o brin g ou t specia l relationships . Student s shoul d b e urged t o us e colo r no t i n a rando m way , bu t rathe r a s a n ai d i n revealin g the inheren t order .

In th e Waldor f school s ther e i s a lovel y string exercis e that complement s this activity . Stan d a grou p o f n childre n i n a circle . Numbe r the m 0 t o n — 1 (o r 1 t o n wit h younge r children ) an d b e sur e the y remembe r thei r numbers. Checkin g t o se e tha t student s ar e standin g o n th e vertice s o f a regula r polygo n involve s recallin g som e interestin g geometry : everyon e should b e th e sam e distanc e fro m th e cente r an d everyon e shoul d b e th e same distanc e t o thei r tw o neighbors . I als o as k m y student s t o chec k tha t each paralle l clas s o f diagonal s i s indee d parallel . Now , tak e a larg e bal l o f yarn an d giv e it t o the firs t child , tha t is , the chil d wit h numbe r zero . Decid e on a numbe r k o f "steps " t o tak e i n tossin g th e yar n around . The n hav e th e children coun t of f i n tur n an d tos s th e yar n t o ever y fcth child . Eac h chil d holds ont o th e stran d o f yar n whe n received , s o tha t i n th e en d som e sta r polygon i s formed . (Th e yar n wil l b e droppe d occassionally , an d ther e wil l be lot s o f laughte r an d giggling. ) Holdin g ont o th e corners , th e childre n ca n slowly lowe r th e yar n t o th e floo r t o se e the patter n emerge . (Se e Figur e 2. ) It i s surprising , eve n fo r adults , wha t a differen t experienc e i t i s t o "draw " these diagonal s b y tossin g yar n rathe r tha n b y usin g penci l an d paper .

This exercis e ca n b e tailore d t o fi t a variet y o f purpose s an d age s o f children. Fo r example , wit h younge r children , I woul d hav e the m coun t of f the integer s i n thei r natura l order . Hav e al l th e childre n coun t togethe r ou t loud, a s rhythmically a s possible. Thi s helps focus thei r attentio n an d instill s a kinestheti c sens e o f number . Sinc e th e yar n i s tosse d wheneve r a multipl e of k i s calle d out , thi s reinforce s th e multiple s o f k an d i s preparatio n fo r learning th e multiplicatio n tables . Her e i s ho w th e patter n fo r n = 1 2 an d k = 5 starts :

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 etc . In thi s case , th e yar n i s tosse d whe n 5 , 10 , 15 , 20 , et c ar e calle d out . Th e yarn wil l b e tosse d fro m chil d 0 t o chil d 5 t o chil d 1 0 t o chil d 3 t o chil d 8 , and s o on .

Implicit i n this version i s an indirec t encounte r wit h divisio n an d remain - ders. Wit h olde r children , yo u ma y wis h t o mak e thi s mor e explicit . Befor e tossing th e yarn , hav e eac h chil d writ e hi s numbe r o n a nam e tag . Agai n

D I S C R E T E MATHEMATIC S W I T H A N ARTISTI C C O N N E C T I O N 20 9

FIGURE 2 . Sta r polygon s create d b y yarn-tossing fo r n = 12 ; (a) k = 3 ; (b ) jf e = 5 .

have th e childre n coun t of f th e integer s i n thei r natura l order . Thi s time , however, hav e the m cal l ou t thei r number s individually , on e afte r th e other , rather tha n i n chorus . Thu s fo r n = 12 , chil d numbe r 3 wil l cal l ou t 3 th e first tim e around , 1 5 the secon d tim e around , 2 7 the thir d tim e around , an d so on . Thes e ar e precisel y th e positiv e integer s tha t leav e a remainde r o f 3 on divisio n b y 12 . Thi s versio n o f th e exercis e i s a preparatio n fo r modula r arithmetic, discusse d belo w i n Sectio n 5 .

When yo u hav e chose n a number n o f childre n t o wor k with , i t i s best t o systematically g o throug h al l value s o f k fro m 1 to n — 1 . Childre n shoul d note tha t k an d n — k alway s give the sam e shape , bu t trace d ou t i n opposit e orders. Th e valu e n = 1 2 seem s t o b e a goo d on e t o star t wit h fo r severa l reasons. Fo r younge r childre n i t i s relate d t o th e familia r cloc k face . Fo r older childre n wh o ar e learnin g geometri c constructions , i t i s possibl e wit h only moderat e difficult y t o construc t a regula r dodecago n wit h straightedg e and compass . Henc e the y ca n construc t accurat e diagram s o f th e pattern s they firs t forme d wit h th e string . Moreover , fo r n = 12 , severa l familia r geometric shape s appear : a hexago n fo r k = 2 an d 10 , a squar e fo r k — 3 and 9 , an d a triangl e fo r k = 4 an d 8 .

The exercis e ca n b e furthe r modifie d t o investigat e man y othe r mathe - matically significan t questions .

1. Whic h polygon s ar e th e same ? 2. Doe s th e yar n alway s com e bac k t o wher e i t started ? 3. Fo r whic h k wil l everyon e ge t th e yarn ? 4. I f onl y som e childre n ge t th e yarn , ca n yo u predic t ho w many ? 5. I f onl y som e childre n ge t th e yarn , ca n yo u predic t whic h ones ? 6. Suppos e w e star t tossin g th e yar n fro m a chil d othe r tha n 0 . Wh o

will ge t th e yar n then ? 7. Fo r whic h n wil l everyon e always ge t th e yarn ? 8. Fo r whic h k d o yo u us e u p th e mos t yarn ?

210 R O B E R T E . JAMISO N

All of these ar e foundation s fo r mor e advance d concepts . Questio n 1 is a question abou t isomorphism o f figure s tha t ar e th e sam e bu t loo k different ; Questions 2 , 3 , 4 , an d 7 ar e relate d t o question s o f divisibilit y an d primes . Questions 5 an d 6 ar e th e basi s o f th e ide a o f subgroups and cosets.

I hav e don e thi s yar n exercis e wit h elementar y schoo l children , parent s of elementary schoo l children, elementar y schoo l teachers, an d mos t recentl y with Compute r Scienc e majors i n a Discrete Mathematics class . I t i s fun an d stimulating fo r al l these groups , bu t obviousl y I emphasize differen t stage s of the learnin g proces s wit h th e differen t groups . Wit h th e childre n an d thei r parents, I leav e i t mostl y a t th e "encounter " stage . W e d o th e exercis e an d admire th e patterns . I as k a fe w leadin g question s abou t wha t the y expec t to happe n an d generall y ge t som e goo d answer s back . Wit h th e teacher s I hav e worke d mor e a t th e "observation " leve l — explicitl y callin g thei r attention t o th e pattern s tha t ar e developing . I n m y Discret e Mathematic s class, I wa s aimin g a t reflectio n an d understanding . W e openl y discusse d modular additio n an d multiplication , congruenc e modul o k , an d equivalenc e classes, al l topic s i n th e course . M y poin t i s that fo r al l o f thes e groups , th e same mathematica l idea s wer e bein g presente d — onl y a t differen t level s of explicitness a s appropriate t o th e group' s degre e of mathematical awareness .

3 . Polyhedr a an d Schlege l diagram s

Three dimension s ar e reall y mor e concret e tha n tw o dimension s sinc e the physica l spac e w e liv e i n i s thre e dimensional . Youn g children , hav - ing onl y recentl y mastere d th e difficul t task s o f holdin g themselve s uprigh t and walking , hav e a n intuitiv e fee l fo r thre e dimension s tha t surprise s man y adults. Thu s encounter s wit h polyhedr a ca n begi n ver y earl y an d naturall y come before th e stud y o f polygons. I n the pre-schoo l an d earl y grade s teach - ers shoul d encourag e childre n t o explor e o n thei r ow n wit h Polydrons (se e [20, 21] ) o r othe r buildin g materials . Teacher s shoul d kee p colorfu l model s of polyhedra o n th e shelve s of their classroom s just a s decoration, mayb e fo r occassional discussion . I n th e earl y stage s i t i s th e encounter s wit h polyhe - dra rathe r tha n thei r forma l stud y whic h matters. Youn g children ar e just a s fond o f learnin g impressiv e word s lik e "icosahedron " a s the y ar e o f learnin g "brontosaurus" an d the y shoul d b e casuall y introduce d t o th e name s an d models o f th e regula r solids . (Bu t pleas e don' t qui z the m o n it! )

A goo d tas k fo r childre n o f al l age s i s t o as k the m t o coun t th e numbe r of face s (o r edge s o r vertices ) o f som e polyhedra l model . I t instill s a n ap - preciation fo r systemati c countin g an d ca n b e use d t o teac h quit e a bi t o f combinatorics. I n orde r t o encourag e an d guid e th e children' s explorations , the teacher s nee d a fairl y soun d knowledg e o f polyhedr a themselves .

There ar e numerou s excellen t book s o n buildin g polyhedr a [18 , 20 , 2 1 , 22, 24 , 28] , so I need no t g o into that here . However , I would lik e to discus s drawing th e Schlegel diagrams o f polyhedr a (se e Figur e 3) . Thi s exercis e stretches th e visua l imagination , provide s a n occasio n fo r th e artisti c us e

D I S C R E T E MATHEMATIC S W I T H A N ARTISTI C C O N N E C T I O N 21 1

FIGURE 3 . Schlege l diagram s fo r th e tetrahedro n (a) , an d the cub e (b) .

of color , an d lay s a experientia l backgroun d fo r tw o importan t subjects : topology an d it s step-child—grap h theory .

The Schlegel diagram o f a polyhedro n i s what yo u woul d se e i f you hav e very goo d periphera l visio n an d yo u pu t you r ey e righ t i n th e middl e o f one fac e o f a wir e fram e mode l o f th e polyhedron . I n othe r words , i t i s a perspective projection o f the vertice s an d connectin g edge s ont o th e plan e o f the face. I n practice, i t i s hard t o view the Schlegel diagram this way, becaus e the outsid e face s appea r s o skinny . Ther e i s anothe r wa y t o thin k o f th e Schlegel diagram whic h is often mor e useful. Imagin e the polyhedron i s made out o f rubber. No w pum p ai r int o th e polyhedro n unti l i t bulge s an d round s out lik e a sphere. No w imagine th e face s disappearin g s o that onl y th e edge s and vertice s ar e left . Pu t you r hand s throug h on e fac e an d pul l outwar d o n the edge-verte x skeleto n unti l i t lie s flat i n a plane . Tha t i s th e Schlege l diagram. Thi s approac h suggest s topology—"rubbe r shee t geometry. " I t also bring s ou t tw o importan t principle s o f discret e mathematics . First , i n the grap h o f a polyhedron , i t i s only th e incidenc e relation s betwee n vertice s and edge s tha t matters , no t th e exac t length s o f th e edge s an d th e angle s betwen them . Second , a polyhedro n ca n b e though t o f a s a tessellatio n o f the sphere .

There ar e actuall y tw o kinds of Schlegel diagrams: thos e with al l vertice s "finite" an d thos e wit h on e verte x "a t infinity. " (Se e Figur e 4. ) Th e verte x at infinit y ca n be visualized usin g a spherical model. Imagin e the polyhedro n again a s vertice s an d edge s stretche d ou t o n a sphere . Pul l al l th e vertice s except on e int o th e singl e hemispher e yo u ar e lookin g at . Th e othe r verte x remains o n th e othe r side—a t infinity , s o to speak—wit h th e edge s goin g t o it wrappin g aroun d th e sphere . No w cu t thes e edge s an d tak e wha t i s i n the hemispher e yo u ca n se e an d flatten i t ou t int o th e plane . Th e cu t edge s dangle of f int o th e outsid e regio n o f th e diagram , runnin g of f t o mee t a t

212 R O B E R T E . JAMISO N

W a b

FIGURE 4 . Schlege l diagram s fo r th e octahedron , (a ) Verte x at infinity ; (b ) al l vertice s finite.

infinity a t th e missin g vertex . I hav e recentl y illustrate d thi s fo r pre-servic e elementary teacher s b y havin g the m wor k i n group s makin g drawing s o n transparent plasti c spheres 6 wit h non-permanen t overhea d markers . I t wa s time consuming , bu t ver y instructive .

The drawing s o f Schlege l diagram s presen t numerou s challenges , an d the tim e spen t drawin g th e regula r polygon s first i s ver y helpful . Becaus e perspective projectio n distort s angle s an d lengths , mos t o f th e polygon s i n a Schlege l diagra m wil l no t b e regular . However , ther e ar e usuall y severa l key regula r polygon s i n eac h diagram . Drawin g thes e first i n th e correc t position make s th e res t o f the diagra m easie r t o draw . Fo r example , t o dra w the octahedro n i n Figur e 4b , star t wit h tw o equilatera l triangles , oppositel y oriented, wit h on e insid e th e other . No w "stitch " bac k an d fort h betwee n these tw o triangle s creatin g a "seam " o f edge s tha t alternate s betwee n th e vertices o f th e inne r an d oute r triangle . Thi s "seam " i s wha t i s know n i n graph theor y a s a Hamiltonia n cycle . Thi s i s a n excellen t plac e t o us e colo r to mak e th e "seam " stan d out .

Two commo n stumblin g block s ar e 1 ) drawing face s tha t ar e no t convex , and 2 ) leavin g som e vertice s wit h onl y tw o edges . Fo r example , th e regula r dodecahedron ha s twelv e pentagona l faces . Student s drawin g th e Schlege l diagram fo r th e first tim e ofte n en d u p wit h som e o f thes e face s nonconvex . Although suc h drawing s ma y b e correc t a s graphs, the y ar e no t tru e Schlege l diagrams sinc e perspectiv e projectio n doe s preserv e convexity . I encourag e the student s t o thin k abou t ho w the y woul d hav e t o mov e th e vertice s an d edges i n thei r drawing s t o mak e al l th e face s convex . Thi s provide s a n excellent opportunit y t o emphasiz e visualizatio n an d th e "rubbe r sheet " nature o f the drawin g a s well as the importanc e o f an aestheticall y appealin g product.

6These Lendrt Sphere s wer e originall y produce d i n Hungary , bu t Ke y Curriculu m Press ha s take n ove r thei r productio n i n th e USA . Editors' note: Yo u ma y b e abl e t o find inexpensive Christma s ornament s o r toy s fo r thi s purpose .

DISCRETE MATHEMATIC S WIT H A N ARTISTI C CONNECTIO N 21 3

The secon d stumblin g bloc k i s mor e seriou s becaus e i t i s geometricall y impossible fo r a thre e dimensiona l figure t o hav e a verte x attache d t o onl y two edges . I n fact , countin g th e numbe r o f edge s a t eac h verte x i s a goo d way t o detec t error s i n a drawing . I n a regula r polyhedron , eac h verte x will hav e th e sam e degre e - tha t is , numbe r o f edge s a t tha t vertex . I n a pyramid, ther e wil l be vertice s o f different degrees . Explorin g th e degree s of vertices a s wel l a s th e numbe r o f sides o r face s i n a variet y o f polyhedr a ca n lead t o th e discover y o f many beautifu l relationships , suc h a s the celebrate d Euler formula : V — E + F = 2 , wher e V i s th e numbe r o f vertices , E th e number o f edges, and F th e numbe r o f faces. I n keepin g with th e philosoph y of thi s pape r tha t eve n introductor y materia l shoul d provid e a n encounte r with dee p an d significan t mathematics , I wan t t o poin t ou t tha t I alway s write th e Eule r formul a as V — E + F = 2 becaus e i t i s thi s for m whic h generalizes t o highe r dimension s an d bes t display s th e rol e o f 2 (th e Eule r characteristic o f th e plane ) a s a topologica l invariant .

4. Movemen t an d S y m m e t r y

The followin g activit y wa s inspire d b y a highl y geometrica l for m o f dance-movement know n a s eurythmy, taugh t onl y i n th e Waldor f schools . These exercise s ar e intende d t o giv e a kinestheti c sens e o f th e symmetrie s of th e regula r polygons .

Divide th e clas s int o group s o f six , eac h consistin g o f a leade r an d five children wh o wil l for m th e vertice s o f a regula r pentagon . A s i n th e yarn - tossing exercise , i t i s valuabl e t o revie w th e propertie s o f regula r polygons : equal distanc e t o neighbor s an d parallel s i n order . Th e first exercis e i s quit e simple: eac h perso n i n th e pentago n i s t o wal k (counter-clockwise ) alon g an edg e o f th e pentago n unti l h e o r sh e reache s th e positio n previousl y occupied b y hi s o r he r neighbor . Al l five childre n ar e t o wal k a t once , t o a steady rhythm , sa y of three beats , clappe d b y the leade r (o r teacher). Befor e beginning, as k th e childre n t o poin t t o wher e the y ar e going—thi s help s t o avoid mishap s an d clea r u p misunderstandings . Th e en d effec t i s a rotatio n 72 degree s counter-clockwise . Repeatin g thi s exercis e yield s a rotatio n 14 4 degrees counter-clockwise .

Now come s a n exercis e tha t tie s i n wit h th e drawin g o f diagonals . Thi s time th e childre n ar e t o wal k a diagonal t o th e positio n o f the second perso n from the m counter-clockwis e aroun d th e pentagon . Agai n the y ar e t o wal k in a straigh t lin e t o a stead y rhythm . Mor e beat s ar e required , becaus e the distanc e i s further . Th e en d resul t i s muc h les s clear ! As k th e childre n first i f the y thin k the y wil l collide . Som e wil l thin k the y wil l collid e a t th e center o f th e pentagon . Bu t i f they hav e draw n th e diagonal s o f a pentago n beforehand, som e wil l remembe r tha t th e diagonal s d o no t mee t a t th e center o f th e pentagon , bu t rathe r pas s aroun d it . Thus , i f th e pentago n i s large enough , the y wil l no t collide , bu t steadil y mov e pas t eac h other . I n fact, th e whol e pentago n contract s spirall y an d the n expand s again . Thi s i s

214 R O B E R T E . JAMISO N

quite a beautifu l form . Th e stead y rhyth m i s essentia l t o insur e a smoot h transformation o f th e pentagon .

Let u s mov e o n t o reflections . Hav e th e leade r choos e a perso n i n th e pentagon an d stan d o n th e edg e opposit e t o for m a mirro r line . No w th e pentagon i s t o b e reflected throug h th e mirro r line . Agai n as k th e childre n to poin t t o wher e the y ar e going . Thi s wil l invariabl y lea d t o considerabl e confusion, whic h i s importan t t o straighte n out . I n particular , th e vertex - person o n th e mirro r lin e ofte n want s t o know : "Wher e d o I go ? D o I exchange wit h th e leader? " Th e answer , o f course , i s "No" . A poin t o n th e mirror stay s fixed an d doe s no t move . Th e leade r i s needed becaus e i t take s two points t o determin e th e mirro r line . Doin g the reflectio n i s slightly mor e tricky tha n th e rotatio n becaus e (1 ) th e distance s t o wal k ar e no t th e same , and (2 ) exchangin g childre n will collide unles s the y wal k aroun d eac h other .

Having practice d bot h rotation s an d reflections , i t i s no w possibl e t o explore the m a little more . As k the childre n t o not e who m the y ar e standin g next to . No w d o a reflectio n an d loo k again . The y wil l hav e th e sam e neighbors, bu t lef t an d righ t wil l be reversed. Thi s illustrates tha t reflection s reverse orientatio n wherea s rotation s preserv e orientation .

Now as k th e childre n t o tak e not e o f th e spo t wher e the y ar e standing . Do tw o consecutiv e reflection s aroun d differen t mirro r line s an d as k th e children ho w th e pentago n ha s move d fro m it s origina l position . I t wil l have rotated , illustratin g th e fac t tha t th e produc t o f two reflection s (whos e mirrors intersect ) i s a rotation .

These activitie s ca n b e repeate d wit h group s o f mor e tha n si x children , of course . Th e smalle r groups , however , ar e mor e manageable , an d a n od d number work s bes t fo r walkin g th e diagonals. 7

I us e thes e exercise s a s a precurso r t o th e stud y o f symmetr y groups . This usually includes a full stud y of the symmetries of the equilateral triangl e and th e square , th e classificatio n o f th e symmetrie s o f friez e patterns , an d a brie f discussio n o f th e symmetrie s o f regula r polyhedr a an d wallpape r patterns. I n fact, th e two kinds of Schlegel diagrams of the regular polyhedr a help t o illustrat e tw o kind s o f rotationa l symmetries :

1. diagram s with al l vertices finite illustrat e rotationa l symmetr y aroun d a face-to-fac e axis ;

2. diagram s wit h on e verte x a t infinit y illustrat e rotationa l symmetr y around a ver t ex-to-vertex axis .

Symmetry abound s i n art i n the for m o f ornaments [9 , 3 1, 33]. I usuall y go throug h a serie s o f slide s o f medieva l architectura l monument s wit h m y class, askin g the m t o identif y th e symmetr y group s o f friez e pattern s [9] . Round singin g or clapping rhythmi c pattern s i s an excellen t wa y to illustrat e

7Needless t o say , thes e exercise s nee d t o b e don e outdoor s wher e ther e i s plent y o f room. I hav e don e thes e exercise s wit h in-servic e an d pre-servic e elementar y teachers , all women , wh o enjoye d the m immensely . I hav e als o trie d the m wit h a mi x o f mal e and femal e mat h an d mat h educatio n major s i n a Moder n Algebr a course ; thi s wa s les s successful, becaus e th e me n especiall y wer e muc h mor e resistan t an d self-conscious .

D I S C R E T E MATHEMATIC S W I T H A N A R T I S T I C C O N N E C T I O N 21 5

translational symmetr y aurally . Baroqu e musi c i s ful l o f sequence s [27 , pp . 230-242] whic h ca n als o b e use d t o illustrat e th e concep t o f translationa l and eve n reflectiona l symmetr y i n a non-geometri c spac e havin g tim e an d pitch a s it s "dimensions. "

My intentio n i s no t t o forc e a complet e understandin g o f thes e rathe r subtle ideas . Rathe r th e inten t i s t o slowl y an d carefull y prepar e th e intel - lectual groun d i n th e studen t fo r th e plantin g o f a n intellectua l see d whic h may tak e year s t o ripen . Whe n i t does , i t grow s wit h th e strengt h o f a self-discovered ide a rathe r tha n a s a n ide a impose d fro m outside .

5. Modula r A r i t h m e t i c

Many elementary-school mathematic s curricul a include a section on arith - metic i n othe r bases .

A much mor e meaningfu l alternativ e tha t i s closely related t o bas e arith - metic i s modula r arithmetic . I n fact , I hav e introduce d i t t o m y in-servic e teachers a s th e stud y o f wha t happen s t o th e las t digi t i n computation s in othe r bases . Bu t thi s i s onl y i n passing , becaus e ther e ar e man y muc h more importan t an d seriou s application s o f modula r arithmetic—especiall y in codin g theor y [12 , 1 3 , 14 , 16] . Moreover , ther e ar e man y immediat e examples i n everyda y lif e wit h whic h student s ar e familiar . Her e ar e som e sample problem s whic h convinc e beginner s tha t the y alread y kno w som e modular arithmetic :

1. Yo u leav e o n a 5 hour tri p a t 10:0 0 am . Wha t tim e wil l yo u arrive ? 2. Wha t da y o f th e wee k wil l i t b e 1 0 days fro m today ? 3. Octobe r 4 i s a Monday : wha t da y o f th e wee k i s Octobe r 23 ? 4. Wha t i s th e dat e exactl y 3 weeks afte r Jun e 20 ?

These example s mak e i t eas y t o gras p th e ide a o f "clock " arithmetic . However, the y al l involv e addition . I n introducin g multiplication , i t i s help- ful t o recal l tha t multiplicatio n o f integer s i s jus t repeate d addition . Thi s makes th e definitio n appea r les s arbitrary . I als o ti e thi s i n wit h th e diago - nals of a polygon . Fo r example , th e "multiple s o f 5 mod 12 " ar e obtaine d b y taking th e sequenc e o f "5-ste p diagonals " o f a dodecago n (show n i n Figur e 2b). A t thi s stage , student s spontaneousl y as k abou t subtractio n an d divi - sion. An d the y discove r tha t whil e subtractio n alway s make s perfec t sense , division i s fa r mor e problematic .

At this point, i t is helpful t o systematize the study by having the student s write ou t table s fo r additio n an d multiplicatio n modul o m fo r smal l m —say, addition fo r m < 6 and multiplicatio n fo r m < 13 . Th e student s ar e quick t o spot man y patterns . I n th e additio n tables , row s an d column s ar e obtaine d just b y cycli c permutato n an d henc e eac h elemen t appear s exactl y onc e i n each ro w an d column . Th e situatio n i s more comple x fo r multiplication , an d hence th e nee d t o writ e ou t a large r numbe r o f tables. I n orde r t o hel p brin g out th e patterns , I as k m y student s t o writ e eac h "0 " i n re d an d eac h " 1" in blue .

216 R O B E R T E . JAMISO N

There i s alway s a re d borde r o f zeroes . Eve n i f thi s i s discarded , som e elements ca n occu r mor e tha n onc e i n a ro w o r column . This , o f course , spells troubl e fo r division . Student s ar e le d t o gues s tha t th e modul i fo r which thi s ba d situatio n doe s no t occu r ar e precisel y th e primes . Henc e for prim e moduli , modula r arithmeti c i s ver y simila r t o regula r rationa l arithmetic wit h al l fou r operation s defined . Othe r pattern s tha t ca n b e elicited ar e a s follows .

1. Th e table s ar e symmetri c abou t th e mai n diagona l an d thi s mean s the operation s ar e commutative ;

2. Th e row s an d column s wit h "extra " zeroe s al l correspon d t o number s which occu r mor e tha n onc e i n som e row s an d columns ;

3. A row or column contain s a (blue ) on e if and onl y if it doe s not contai n an "extra " (red ) zero .

A lovel y illustratio n o f a practica l us e o f thes e idea s i s the Internationa l Standard Boo k Numbe r (ISBN ) whic h ever y boo k possesse s [23 , 16 , pp . 36-39]. Thi s consist s o f a ten-digi t cod e a i , a2, a 3 , . .. , aio o f whic h th e firs t nine digit s identif y th e language , publisher , an d catalogu e numbe r o f th e book. Th e las t digi t ai o i s a "chec k digit " whic h allow s single-digi t error s and eve n th e transpositio n (reversal ) o f tw o adjacen t digit s t o b e detected . It i s als o possibl e t o determin e a missin g digi t i f it s positio n i s known . Thi s all follow s fro m th e fac t tha t ai o i s chose n s o tha t th e equatio n

10ai + 9a 2 + 8a 3 + .. . 4- 2a9 + ai o = 0 (mo d 11 )

holds. Notic e tha t ai o ma y b e require d t o tak e th e valu e 10 ; if this happens , it i s represented b y an X in the ISB N number . I n introducin g modula r arith - metic, I illustrat e th e abov e equalit y wit h severa l ISB N numbers . Later , we actuall y solv e fo r missin g digits . Thi s involve s solvin g a linea r equatio n kx + b = 0 (mo d 11) , an d inevitabl y lead s t o a health y discussio n o f wha t i t means t o solv e suc h a n equation .

6. Modula r A r i t h m e t i c i n Musi c

I wil l giv e her e a sketch y accoun t o f connection s betwee n musi c an d modular arithmetic , al l possibl e t o describ e withi n th e real m o f elementar y mathematics. A detaile d discussio n ca n b e foun d i n [1 ] or [37 , pp. 88-103] . Unfortunately, thi s materia l take s a lon g tim e t o cover , an d onl y a t a ver y slow pace , becaus e mos t o f m y student s hav e a poo r musica l background , and w e mus t star t fro m scratch . However , eve n thos e wit h a n extensiv e musical backgroun d fin d tha t th e carefu l mathematica l treatmen t put s th e theory int o a ne w an d cleare r light . I n eithe r case , I conside r th e tim e wel l spent becaus e musi c i s suc h a n importan t backgroun d fo r mathematics .

The simples t wa y tha t modula r arithmethi c enter s musi c i s through th e cyclic namin g o f th e note s a s A,B,C,D,E,F,G . Whe n G i s passed , w e star t over agai n wit h A . Thu s statin g tha t C (not e 3 ) i s fou r note s abov e F (not e 6) correspond s t o th e equatio n 3 = 4 + 6 (mo d 7) . Th e whit e key s o n the pian o ge t th e name s A,B,C,D,E,F, G i n cycli c order . Th e blac k key s ar e

D I S C R E T E MATHEMATIC S W I T H A N ARTISTI C C O N N E C T I O N 21 7

C# D # F # G # A#

C D E F G A B C

F I G U R E 5 . (a ) A standar d octav e o n th e piano ; eac h whit e or blac k ke y represent s a hal f step . Recal l tha t th e C-majo r scale i s th e sequenc e o f tone s playe d o n th e whit e key s only . (b) Th e Circle of Fifths show n a s diagonal s o f a dodecagon .

named usin g sharps (o r flats). (Se e Figure 5a. ) A scale is a sequence of notes, beginning a t a fundamenta l not e an d ascendin g t o th e nex t occurrenc e o f the sam e not e name . Thu s a C majo r scal e begin s o n a C an d end s a t th e next highe r C . Becaus e Wester n scale s typicall y hav e 8 notes , th e interva l between the beginning and ending note of a scale is called an octave. Anothe r important interva l i s that betwee n th e fundamenta l an d th e fifth not e o f th e scale. Thi s i s the fifth interva l of the scale and i s usually simpl y calle d a fifth. For example , th e string s o f a violi n ar e tune d i n fifths: G , D , A , E . Notic e that "fifth " i s use d her e a s a n ordina l number , no t a s a fraction . Sinc e thi s is a frequen t sourc e o f confusion , i t provide s a n opportunit y t o clarif y th e distinctions betwee n differen t type s o f number s an d ou r word s fo r them .

All thi s ma y appea r quit e arbitrary , bu t i t i s not . Th e octav e i s th e first harmoni c (o r overtone ) an d th e fifth i s th e secon d harmoni c ove r th e fundamental. Vibrationa l energ y i s easil y passe d betwee n a ton e an d it s harmonics, an d thi s sound s pleasan t (o r harmonious! ) t o ou r ears . Pitc h i s determined b y rat e o f vibration . Th e octav e vibrate s twic e a s fas t a s th e fundamental wherea s th e fifth vibrate s 3/ 2 a s fast . Standar d pitc h fo r a violin A strin g i s 44 0 vibration s pe r second . Thu s th e violi n E strin g a fifth above i s 66 0 vibration s pe r second , an d th e 'cell o A strin g a n octav e belo w is 22 0 vibration s pe r second .

The characteristi c soun d o f a scal e i s determined b y th e frequenc y ratio s between th e note s o f th e scale . I f w e wis h t o buil d a majo r scal e startin g on D instea d o f C , w e ar e force d t o ad d sharp s i n orde r t o kee p th e sam e frequency ratio s a s i n C major . I n musi c theory , th e basi c frequenc y ratio s (intervals) ar e expresse d i n term s o f whol e step s an d hal f steps . O n th e piano, a hal f ste p i s th e interva l betwee n a not e an d th e not e immediatel y next t o it . Thu s B- C an d E- F ar e hal f step s becaus e ther e ar e n o black note s

218 ROBERT E . JAMISO N

in between , bu t G- A i s a whol e ste p sinc e i t consist s o f th e tw o hal f step s G-G(t an d Gtf-A . (Se e Figur e 5a) . Ther e ar e twelv e hal f step s i n a n octav e and seve n hal f step s i n a fifth . Thu s determinin g whic h not e i s a fift h abov e a give n fundamenta l i s reall y a proble m i n additio n modul o 12 .

There i s a systemati c wa y o f listin g th e majo r scale s wit h startin g note s rising a fift h eac h time . Thi s i s calle d th e Circle of Fifths. Sinc e a fift h i s seven hal f steps , thi s ca n b e viewed a s the 7-ste p diagonal s o f the dodecago n (Figure 5b) . Notic e tha t thi s i s th e patter n tha t woul d b e forme d b y th e yarn exercis e i n Sectio n 2 with n = 1 2 an d k = 7 .

Writing out th e first eigh t scale s with successivel y mor e sharps illustrate s cyclic rotation again , an d reveal s several beautifu l patterns , a s shown below :

c G D A E B Ftf ctt

D A E B F« Ctt Gtt Dft

E B Ftt ctt Gtt Dtt Att EJt

F C G D A E B Ft!

G D A E B Ffi ctt G)J

A E B Fit ctt Gtt Dtt Att

B n ctt Gtt Dtt Att Ett B«

C G D A E B Ftt Ctt

The ne w sharp s alway s appea r i n th e tabl e i n th e sam e (seventh ) position . Visually, th e patter n o f sharp s i n th e tabl e i s essentiall y a pai r o f triangles . There ar e als o several diagonal relationships whic h ca n be brought ou t nicel y with th e us e of color , sa y b y makin g al l th e Ffl' s th e sam e color . Notic e als o that th e las t fou r note s o f eac h scal e ar e th e sam e a s th e firs t fou r note s o f the nex t scale .

A mor e involve d connection , bot h mathematicall y an d musically , i s th e calculation o f frequency interval s fro m basi c harmonics. A s noted above , th e frequency ratio s fo r th e octav e an d th e fift h ar e 2: 1 an d 3:2 . I n fact , ever y musical interva l ca n b e expresse d a s a simpl e rati o o f tw o smal l integer s [15, 35] . Thi s discover y reall y date s bac k t o Pythagora s [7 , p . 72 ] i n th e 5th centur y BC . I t represent s th e firs t expressio n o f a physica l la w i n math - ematical terms . Mathematically , followin g on e interva l b y anothe r involve s the multiplicatio n o f fraction s givin g thei r frequenc y ratios . Thi s lead s t o certain rathe r surprisin g musica l consequence s o f the uniqu e factorizatio n o f integers int o primes . Fo r example , i f w e g o al l th e wa y aroun d th e Circl e of Fifths , w e wil l g o throug h twelv e fifth s an d en d u p seve n octave s higher . Since the frequenc y rati o fo r a fift h i s 3:2 an d fo r a n octav e i s 2:1 , we shoul d have

(3/2)1 2 = 2 7,

or gl2 _ _ 2(12+7 ) _ 21 9

The firs t equatio n ma y see m plausible ; indeed , (3/2) 1 2 i s 129.746... , whic h is convincingl y clos e t o 12 8 = 2 7. Th e secon d equation , however , say s tha t

DISCRETE MATHEMATIC S WIT H A N ARTISTI C CONNECTIO N 21 9

some larg e numbe r ha s tw o differen t prim e factorizations , a n impossibility . Musically, thi s mean s tha t goin g aroun d th e Circl e o f Fifth s wil l no t retur n us t o th e origina l C , bu t t o a ton e just a shad e higher . Thi s mathematicall y inescapable fac t say s tha t i t i s impossible t o tun e a pian o s o that al l octave s are tru e an d al l fifths ar e true. 8

7. T h e M a t h e m a t i c s Underlyin g t h e A c t i v i t i e s

The activitie s describe d i n thi s pape r ar e no t intende d merel y a s enrich - ment exercise s o r a s entertainmen t t o mak e mathematic s mor e palatable . They ar e designe d t o provid e kinestheti c experience s leadin g t o dee p an d significant mathematics . Eve n fo r th e majorit y o f student s wh o wil l no t go o n t o highe r mathematics , I thin k i t i s stil l advisabl e tha t thei r limite d mathematical experienc e b e base d o n soun d an d significan t mathematics . In thi s section , I woul d lik e t o indicat e som e o f th e deepe r mathematica l concepts underlyin g th e activities .

The basi c mathematical ide a underlying these activities i s that o f a group of transformations. Th e concep t o f grou p is , o f course , th e formalizatio n of th e genera l notio n o f symmetry . Th e cycli c group s captur e rotationa l symmetry an d o f cours e periodi c behavio r whic h i s commonl y know n a s rhythm. Par t o f m y goa l i s to mak e m y student s awar e o f th e man y divers e contexts i n whic h periodi c behaviou r an d rhyth m occur . Ther e i s rhythm i n the column s alon g th e nav e o f a Romanesqu e churc h a s wel l a s i n musi c an d in th e season s o f th e year . Ther e i s als o a rhyth m i n th e wa y th e number s are arrange d i n th e (Cayley ) table s fo r additio n an d multiplicatio n modul o n. W e als o se e rhyth m i n th e rotationa l symmetr y o f man y flowers.

Polyhedra an d Schlege l diagrams lea d naturally t o graphs an d importan t notions i n grap h theory : th e "fundamenta l theorem " tha t th e su m o f th e degrees i s twic e th e numbe r o f edges , planarity , an d Euler' s formula .

The introductio n o f modular arithmeti c naturall y lay s the foundatio n fo r a whol e hos t o f algebrai c an d numbe r theoreti c notions : finite fields, codin g theory, an d prim e factorizatio n amon g them . Scales revea l man y pattern s related t o arithmeti c modul o 7 an d 12 . An d th e stud y o f frequenc y ratio s leads t o prim e factorization , logarithms , an d th e arithmeti c o f th e rational s modulo 1 .

It i s important t o poin t ou t tha t I d o no t g o int o a prolonge d discussio n of th e underlyin g mathematic s wit h th e elementar y schoo l teacher s i n m y class. I onl y wis h t o giv e the m a hin t o f th e broade r significanc e tha t lie s down th e road . I expec t tha t the y wil l say eve n les s to thei r schoo l children . What i s important i s that a hos t o f meaningful experience s shoul d b e a par t

8 This fac t an d other s relate d t o i t involvin g th e tunin g o f third s le d t o a hos t o f compromise tuning s o r temperament s i n th e baroqu e period . Th e moder n solutio n i s th e "equal temperament " system , i n whic h al l interval s excep t th e octav e ar e just slightl y ou t of tune , bu t al l equall y so . Al l half-step s o n th e pian o ar e tune d i n th e frequenc y rati o of th e twelft h roo t o f 2 , a n irrationa l number , an d interval s ar e sometime s measure d i n a logarithmic scal e givin g 120 0 "cents " t o eac h octave .

220 ROBERT E . JAMISO N

of each student' s backgroun d s o that a s more advance d mathematica l topic s are introduce d i n middl e school , hig h school , o r college , th e studen t wil l have som e persona l experienc e t o connec t wit h them .

8. Conclusio n

There i s really a curiou s parado x her e an d a seriou s lopsidednes s i n ou r educational system . W e ar e no w expectin g o f al l sixt h grader s a deepe r understanding o f arithmeti c tha n wha t th e mos t learne d me n i n Europ e possessed in 1500 ! Onl y after 149 4 did Hindu-Arabic numeral s finally replac e Roman numeral s i n al l Medic i accoun t book s [36 , p . 81] . Simo n Stevin' s La Disme introducin g decima l notatio n appeare d i n onl y 158 5 [36 , p . 89] . And i n 163 7 Descartes stil l referre d suspiciousl y t o th e negativ e root s o f a n equation a s "fals e roots " [36 , p . 96] .

Our expectatio n ma y b e reasonable , bu t i t i s b y n o mean s trivial . An d unfortunately i t i s no t balance d b y a supportin g expectatio n i n geometry . Even mor e unfortunatel y i t i s no t balance d b y a supportin g expectatio n in music . Th e Greek s i n establishin g th e quadriviu m understoo d th e vita l connections o f thes e areas :

arithmetic — number s a t res t geometry — figures a t res t

music — number s i n motio n astronomy — figures i n motio n

It i s sad tha t th e routin e computationa l aspect s o f arithmetic hav e com e to dominat e ou r elementar y mathematic s curriculu m an d tha t musica l skil l has com e t o b e regarde d a s a specia l talent . Th e Suzuk i metho d o f violi n instruction ha s give n th e li e to th e limitin g ide a tha t i t take s specia l inbor n musical skil l t o pla y a n instrument . Suzuki' s philosoph y i s tha t talen t ca n be traine d an d th e succes s o f hi s instructiona l "mothe r tongue " method , based o n slo w an d carefu l steps , imitation , an d positiv e reinforcemen t give s evidence tha t h e i s right . I t i s importan t t o remembe r tha t th e goa l o f th e Suzuki metho d i s no t t o produc e musica l specialist s (i.e. , concer t violinists ) or musica l consumer s (i.e. , musi c appreciators ) bu t "beautifu l huma n be - ings." Th e goa l i s to ope n u p a n avenu e o f enjoyment an d expressio n fo r th e child, t o develo p a skil l tha t ca n enric h a whol e life .

I woul d lik e t o sugges t tha t th e sam e shoul d b e th e goa l o f mathemat - ical instruction . Th e rea l goa l o f th e educationa l syste m shoul d b e t o hel p students develo p th e intellectua l an d emotiona l skill s necessar y t o hav e th e freedom t o choos e thei r ow n future s wisely .

Acknowledgments

I woul d lik e t o than k Furma n Universit y fo r continuin g t o suppor t th e development o f thi s cours e i n An n Stafford' s Lea d Teache r progra m afte r Clemson Universit y eliminate d it s in-servic e mathematic s offering s du e t o budgetary constraints . Thank s ar e als o du e t o Marjori e Senecha l an d th e NSF Regiona l Geometr y Institut e o f 199 3 a t Smit h College , wher e th e ide a for thi s pape r first too k shape . I a m als o gratefu l fo r th e hospitalit y o f th e

D I S C R E T E MATHEMATIC S W I T H A N ARTISTI C C O N N E C T I O N 22 1

Mathematics Departmen t a t Cornel l Universit y durin g m y sabbatica l yea r there an d fo r stimulatin g interaction s wit h Davi d Henderson , Bo b Connel - ley, To m Rishel , an d Mari a Terrel l a t a n NS F Institut e hel d ther e i n 1994 . Special thank s ar e du e t o tw o of the editors , Debora h Franzbla u an d Josep h Rosenstein, fo r thei r patien t encouragemen t o f the writin g o f this repor t an d for thei r editoria l assistanc e i n puttin g thi s pape r int o final form .

References

Willi Apel , Harvard Dictionary of Music, Th e Belkna p Pres s o f Harvar d Universit y Press, Cambridg e MA , 1969 . Se e article s o n Acoustic s (p . 9) , Intervals , Calculatio n of (p . 419) , Pitc h (p . 679) , Temperament s (p . 835) . Hermann vo n Baravalle , Geometric Drawing and the Waldorf School Plan, Waldor f School Monographs , 1967 . Hermann vo n Baravalle , The Teaching of Arithmetic and the Waldorf School Plan, Waldorf Schoo l Monographs , 1967 . Henry Barnes , "Learnin g t h a t Grow s wit h th e Learner : A n Introductio n t o Waldor f Education", Educational Leadership, October , 1991 , 52-54. Henry Barne s e t al. , "Waldor f Education : a Symposium, " Teachers College Record, Vol. 81 , Nr. 3 , Sprin g 1980 , 322-370 . Richard G . Brow n , Transformational Geometry, originall y publishe d b y Silver , Bur - dett, k, Gin n Inc. , 1973 . Reprinte d b y permissio n b y Dal e Seymour , Pal o Alto , CA . Lucas Bunt , Philli p S . Jones , Jac k D . Bedient , The Historical Roots of Elementary Mathematics, Dover , Mineol a NY , 1988 . Richard W . Copeland , How Children Learn Mathematics: Teaching Implications of PiageVs Research, Macmillan , Ne w York , 1974 . Donald Crowe , "Symmetry , Rigi d Motions , an d Patterns, " HiMA P Modul e 4 , COMAP, Arlingto n MA , 1986 . Mary L . Crowley , "Th e va n Hiel e Mode l o f th e Developmen t o f Geometri c Thought" , in Learning And Teaching Geometry K-12, NCT M 198 7 Yearbook , NCTM , Resto n VA, 1987 , pp . 1-16 . Richard Feynman , The Character of Physical Law, M.I.T . Press , Cambridg e MA , 1965, p . 13 . Joseph A . Gallian , "Ho w Computer s ca n Rea d an d Correc t I D Numbers" , Math Horizons, Winte r 1993 , pp . 14-15 . Joseph A . Gallian , "Th e Mathematic s o f Identificatio n Numbers" , The College Math- ematics Journal, 2 2 (1991) , 194-202 . Joseph A . Gallian , "Assignin g Driver' s Licens e Numbers, " Mathematics Magazine, 64 (1991) , 13-22 . G. D . Halse y an d Edwi n Hewitt , "Mor e o n th e Superparticula r Ratio s i n Music" , Am. Math Monthly 7 9 (1972) , 109 6 -1100 . Raymond Hill , A First Course in Coding Theory, Oxfor d Applie d Mathematic s an d Computing Scienc e Series , Clarendo n Press , Oxford , 1986 , pp . 3 6 - 39 . Peter Hilton , Lectur e a t th e Howar d Eve s 80t h Birthda y Conference , Universit y o f Central Florida , May , 1991 , Orlando , Florida . Peter Hilto n an d Jea n Pederson , Build Your Own Polyhedra, Addison-Wesley , Ne w York, 1988 . Jay KapprafT , Connections: The Geometric Bridge Between Art and Science, McGraw-Hill, Ne w York , 1991 . Marilyn Komar c an d Gwe n Clay , Exploring with Polydron: Book 1 (Grades 3-9), Cuisenaire, Ne w Rochell e NY , 1991 .

222 ROBERT E . JAMISO N

[21] Marily n Komar c an d Gwe n Clay , Exploring with Polydron: Book 2 (Grades 3-9) Cuisenaire, Ne w Rochelle , NY , 1991 .

[22] Mar y Laycock , Dual Discovery Through Straw Polyhedra, Creativ e Publications , Pal o Alto C A 1970 .

[23] Josep h Malkevitch , Gar y Proelich , an d D . Proelich , Codes Galore, Consortiu m fo r Mathematics an d it s Application s (COMAP) . Modul e # 1 8 , 1991 .

[24] Davi d Mollet , "Ho w th e Waldor f Approac h Change d a Difficul t Class, " Educational Leadership, October , 1991 , 55-56.

[25] Han s R . Niederhause r an d Margare t Frohlich , Form Drawing, Mercur y Pres s of Rudol f Steiner College , Sacrament o CA , 1974 .

[26] Pete r an d Susa n Pearce , Polyhedra Primer, Dal e Seymour , Pal o Alt o CA , 1978 . [27] Walte r Piston , Harmony, W.W . Norto n & Co. , Ne w York , 1962 . [28] Anthon y Pugh , Polyhedra: A Visual Approach, Dal e Seymour , Pal o Alt o CA , 1990 . [29] Victori a Pohl , How to Enrich Geometry Using String Designs, NCTM , Resto n VA ,

1986. [30] Ren e M . Querido , Creativity in Education: The Waldorf Approach, H . S . Daki n Co. ,

San Francisco , 1984 . [31] Issa m El-Sai d an d Ays e Parman , Geometric Concepts in Islamic Art, Dal e Seymour ,

Palo Alt o CA , 1976 . [32] Dori s Schattschneider , Visions of Symmetry (Notebooks, Periodic Drawings, and Re-

lated Work of M.C Escher), W . H . Freeman , Ne w York , 1990 . [33] Dal e Seymour , Geometric Design - Step by Step, Dal e Seymour , Pal o Alt o CA , 1988 . [34] Dal e Seymou r an d Jil l Britton , Introduction to Tessellations, Dal e Seymour , Pal o

Alto CA , 1989 . [35] A . L . Leig h Silver , "Musimatic s o r th e Nun' s Fiddle, " Am. Math Monthly 78(1971) ,

351-357. [36] Dir k J . Struik , A Concise History of Mathematics, Dover , Mineol a NY , 1987 . [37] Beng t Ulin , Finding the Path: Themes and Methods for the Teaching of Mathematics

in a Waldorf School, Th e Associatio n o f Waldor f School s o f Nort h America , Wilton , N.H., 1991 .

[38] Joh n Willson , Mosaic and Tessellated Patterns: How to Create Them, Dover , New York, 1983 .

D E P A R T M E N T O F MATHEMATICA L SCIENCES , C L E M S O N U N I V E R S I T Y , C L E M S O N , S C

29634-1907 E-mail address: rejamQclemson.ed u

DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Discrete M a t h e m a t i c s Activitie s fo r Middl e Schoo l

Evan Maletsk y

There i s a bod y o f knowledg e tha t ha s com e t o b e know n a s discret e mathematics an d muc h o f i t i s accessibl e t o middle-schoo l students . Man y related topic s ca n alread y b e foun d i n th e existin g curriculu m an d other s can b e readil y integrate d int o it . Discret e mathematic s problem s ten d t o be simpl y state d an d easil y motivated . The y offe r a rich , ne w sourc e o f diversified problem-solvin g experience s tha t rang e acros s al l abilit y levels . Furthermore, the y serv e to portra y mathematic s fro m a broade r perspectiv e than man y typica l practic e exercises .

It i s equall y importan t t o not e tha t problem s i n discret e mathematic s can b e incorporate d int o man y o f th e hands-o n activitie s tha t alread y ar e part o f th e establishe d classroo m scene . Thi s articl e focuse s o n tha t con - nection throug h th e tw o centra l idea s o f counting an d change. Countin g i s viewed throug h numbe r patterns , computation , manipulation , an d visual - ization, an d thes e ar e connecte d t o chang e throug h th e mathematica l ide a of iteration. I t i s th e notio n o f iteratio n — arithmetic , algebraic , an d geo - metric — tha t bring s aliv e th e subjec t o f mathematics , an d i t i s throug h hands-on activitie s tha t i t i s made real . Emphasizin g thi s combinatio n whe n we teach offer s a dynami c vie w of the disciplin e s o greatly neede d b y today' s middle schoo l students .

This articl e begin s wit h a samplin g o f discret e mathematic s activitie s arising fro m a simpl e countin g proble m involvin g pape r folding , the n move s through other s tha t ca n b e analyze d b y graphs , an d end s wit h som e appli - cations o f iteratio n throug h geometri c transformations . Th e example s illus - trate th e importanc e o f bot h conten t an d pedagog y an d sho w ho w discret e mathematics ca n b e designe d an d wove n int o th e broa d fabri c o f middle - school mathematics .

Counting

Almost ever y middl e schoo l studen t an d teache r has , a t on e tim e o r another, use d th e foldin g o f pape r t o explor e a mathematica l relationship .

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 .

22 3

https://doi.org/10.1090/dimacs/036/17

22 4 EVAN MALETSK Y

This first illustratio n show s some differen t way s one simpl e pape r mode l ca n be tie d int o th e aren a o f discret e mathematics .

Cut ou t som e 2x8-inc h strip s o f paper , on e fo r eac h student . Hav e the m fold th e strip s i n hal f an d i n hal f agai n a s show n i n Figur e 1 . Le t the m visualize i n thei r min d wha t th e stri p woul d loo k lik e unfolded .

F I G U R E 1 . Foldin g a stri p o f pape r

Ask the student s t o mentall y coun t al l the rectangle s tha t the y visualize , including th e squares . Afte r writin g thei r individua l answers , le t the m com - pare an d discus s thei r answer s wit h othe r students . Onc e a n agreemen t i s reached i n thei r groups , the y ca n unfol d th e strip s an d chec k thei r answer s by actuall y countin g fro m th e model . Finally , a s a writin g activity , hav e your student s describ e th e algorithm s the y use d fo r thei r counting , bot h i n the abstrac t an d i n th e concret e case .

This activit y i s muc h mor e tha n jus t on e o f visualization . I t involve s analysis an d systemati c counting . On e approac h migh t b e t o lette r th e squares (a s i n Figur e 2a ) an d mak e a lis t o f the 1 0 different rectangle s usin g successive letters , four , three , two , an d on e a t a tim e (Figur e 2b) . Anothe r approach migh t b e t o sho w th e solutio n i n a grap h wit h 4 vertice s an d 1 0 edges (Figur e 2c) . Si x edge s connec t differen t vertices , denotin g differen t starting an d endin g squares . Fou r edge s connec t vertice s t o themselves , indicating th e sam e startin g an d endin g square .

A B C D

(a)

ABCD AB C A B A

BCD B C B

CD c

(b) (c )

F I G U R E 2 . (a ) A n unfolde d an d labele d piec e o f paper , (b ) Systematic listin g o f rectangles , (c ) Usin g th e edge s o f a graph t o represen t startin g an d endin g square s o f eac h rec - tangle.

The lis t reveal s that , fo r tw o successiv e folds , th e answe r i s 10 , th e su m of th e first fou r countin g numbers . Compar e th e numbe r 1 0 for tw o fold s t o

DISCRETE MATHEMATIC S ACTIVITIE S FO R MIDDL E SCHOO L 22 5

the numbe r 3 fo r on e fold :

Folded once : 1 + 2 = 3 . Folded twice : 1 + 2 + 3 + 4 = 10 .

Ask you r student s t o fol d th e stri p i n hal f a thir d tim e an d as k fo r a n educated gues s a s to ho w man y rectangle s wil l b e i n th e unfolde d stri p now . See ho w man y student s ca n find an d exten d th e pattern .

Folded thre e times : 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 .

Some middl e schoo l student s ma y wan t t o explor e thi s proble m furthe r and loo k fo r a genera l solution . Fo r n successiv e folds , th e numbe r o f rect - angles i s th e su m o f th e first 2 n countin g numbers .

Folded n times : 1 + 2 + 3 + 4 + . . . + 2 n = 2 n ~ 1 (2 n + 1) .

Given th e formula , thi s paper-foldin g activit y no w offer s student s a n additional importan t experienc e wit h exponent s an d othe r algebrai c sym - bolism. Fo r example , wit h fou r successiv e folds , ther e ar e 13 6 differen t rectangles, sinc e wit h n — 4 ,

2 n - i ( 2 n + i) = 2 3 ( 2 4 + 1 ) = 8(1 6 + 1 ) = 8(17 ) = 136 .

Are th e countin g number s tha t com e fro m thi s paper-foldin g activity , such a s 3 , 10 , 36 , an d 136 , specia l i n an y othe r way ? Yo u ma y recogniz e them fro m anothe r discret e mathematic s topi c alread y i n th e middl e schoo l mathematics curriculum . The y ar e member s o f th e se t o f triangula r num - bers.

Figure 3 a show s th e triangula r array s whic h accoun t fo r th e nam e "tri - angular numbers" . Triangula r array s suc h a s thes e ca n b e easil y buil t an d vividly displaye d o n a n overhea d projector . Figur e 3 b show s ho w th e trian - gular number s ar e calculate d b y summin g th e row s o f th e triangula r arrays .

Another featur e o f th e triangula r number s emerge s i f w e loo k a t a dif - ference table . I n a differenc e table , w e first recor d th e difference s betwee n successive triangular number s — these are called "firs t differences" . The n w e record th e differenc e betwee n successiv e first difference s — thes e ar e calle d "second differences. " Fo r th e triangula r numbers , secon d difference s ar e al l 1, a s i n Figur e 3c .

Compare thi s t o th e familia r squar e number s wher e th e secon d differ - ences ar e al l 2 , a s i n Figur e 4 . Her e w e se e anothe r topi c fro m discret e mathematics, finite differences , closel y connectin g t o th e existin g middl e school curriculum .

We ca n als o loo k a t othe r geometri c array s — squares , pentagons , hex - agons, etc. — and introduc e other sequence s of "figurat e numbers " — square numbers, pentagona l numbers , hexagona l numbers , etc . Thes e geometri c arrays lea d t o countin g activities , numbe r pattern s t o explore , discoverie s

22 6 EVAN M A L E T S K Y

o 1 3 6 1 0 1 5

21 28

1 = 1 1 + 2 = 3 1 + 2 + 3 = 6 1 + 2 + 3 + 4 = 1 0 1 + 2 + 3 + 4 + 5 = 1 5 1 + 2 + 3 +4 + 5 + 6 = 2 1 1 + 2 + 3 + 4 + 5 + 6 + 7 = 2 8

1 +2 + 3 + . . . + n = n(n+l)/2

(b)

(a)

Triangular Numbers 1 ^ _ _ ^ 3 ^ ^ _ ^ ^ 6 ^ ^ 1 0 ^ ^ - 1 5 v ^ ^ 2 1

Firs t difference s 2 ^ ^ _ ^ ^ 3 " ^ _ _ ^ ^ 4 ^ ^ 5 ^ ^ _ ^ ^ 6 ^ ^ _ _ ^ ^ Secon d difference s 1 1 1 1 l

(c)

F I G U R E 3 . (a ) Th e triangula r numbers , (b ) Calculatin g th e triangular numbers , (c ) Differenc e tabl e fo r triangula r num - bers.

Square Numbers 1 ^ ^ 4 ^ _ ^ 9 ^ ^ 1 6 ^ ^ 2 5 ^ . ^ 36

Firs t difference s 3 ^ _ ^ 5 v - ~ _ ^ 7 ^ ~ _ - ^ 9 " ^ _ ^ ^ ^ ^

Secon d difference s 2 2 2 2 2

FIGURE 4 . Differenc e tabl e fo r th e squar e numbers .

to mak e an d test , an d mor e question s wort h investigating . Fo r example , will pentagona l number s hav e successiv e secon d difference s tha t ar e al l 3 ? For hexagona l numbers , wil l th e successiv e secon d difference s al l b e 4 ? Th e answer i s ye s fo r al l figurate number s o f thi s type . I n fact , an y secon d degree, quadrati c expressio n suc h a s n(n + l ) / 2 mus t hav e constan t secon d differences, a n idea worth challengin g your students to explore as a calculato r activity.

Let u s g o bac k t o th e folde d stri p o f pape r fo r som e mor e countin g activities. Hav e you r student s labe l th e square s o n on e sid e wit h th e digit s 1, 2 , 3 , an d 4 . Tea r apar t th e fou r square s an d th e student s hav e a nic e model fo r som e countin g problems .

One goo d questio n i s th e following : ho w man y differen t four-digi t num - bers ca n b e forme d usin g th e squares ? Le t student s wor k i n team s arrangin g the digits , makin g lists , finding an d applyin g countin g procedures , an d writ - ing about thei r methods . Thi s can lead nicel y into the topics of permutation s

DISCRETE MATHEMATIC S ACTIVITIE S FO R MIDDL E SCHOO L 22 7

and factorials , sinc e man y student s wil l discove r tha t th e answe r 2 4 i s ex - pressed a s 4 x 3 x 2 x 1 . Yo u ca n als o as k ho w man y number s wit h 4 , 3 , 2, and 1 digits ca n b e formed . Th e answe r her e i s

(4 x 3 x 2 x 1 ) + ( 4 x 3 x 2 ) + ( 4 x 3 ) + 4 - 2 4 + 2 4 + 1 2 + 4 - 64 .

For student s a t a highe r level , as k the m t o tur n th e stri p ove r an d pu t the digit s 5 , 6 , 7 , an d 8 on th e back , wit h th e 8 behind th e 1 , before tearin g the square s apart . As k th e sam e tw o question s note d above . Her e th e algorithmic thinkin g i s ever y bi t a s importan t a s th e numerica l answer s o f

8 x 6 x 4 x 2 = 38 4 an d ( 8 x 6 x 4 x 2 ) + ( 8 x 6 x 4 ) + ( 8 x 6 ) + 8 = 632 .

For thos e wh o wan t a rea l challenge , labe l bot h side s o f th e stri p 1 through 8 a s note d above , bu t don' t tea r th e strip s apart . Foldin g onl y o n existing creases , ho w man y differen t number s wit h 1 , 2 , 3 , an d 4 digit s ca n be formed ? A t thi s level , some difficult analysi s i s called fo r b y the students . Let the m discus s an d explor e th e proble m mentall y befor e the y star t foldin g and formin g number s i n thei r hand s wit h thei r pape r strips .

On anothe r day , revie w thes e result s an d the n offe r a variation . Mar k the fou r square s o f a newl y folde d stri p o f pape r wit h th e digit s 1 , 2 , 3 , and a decima l poin t (a s i n Figur e 5) . Separat e th e square s an d thin k abou t possible arrangement s usin g on e o r mor e o f th e squares . Wha t ar e th e different decima l number s tha t ca n b e formed ?

1 2

. 2

2 3

3 1 2

1 . 3

Marking the squares Five possible arrangements

F I G U R E 5 . Creatin g decima l number s wit h digit s 1 , 2 , 3 , and "." .

Counting th e differen t possibilitie s ca n b e a n interestin g an d challeng - ing activit y fo r th e middle-schoo l student . Bu t couche d i n th e for m o f a class gam e o r competition , muc h mor e classroo m excitemen t an d enthusi - asm ca n b e generated . Middl e schoo l teacher s fro m th e Rutger s Universit y Leadership Progra m i n Discret e Mathematic s an d other s hav e relate d bac k to m e severa l gam e variation s the y hav e use d i n thei r classe s wit h grea t success. On e o f th e mor e interestin g format s use s n o pape r othe r tha n the strip s use d t o introduc e th e activity . Ever y da y fro m ther e on , be - gin th e clas s wit h a numbe r fro m th e set , sa y 2 . Se e ho w fa r yo u ca n ge t around th e class , askin g eac h studen t fo r th e nex t large r decima l tha t ca n be forme d fro m th e set , befor e a mistak e i s made . Whe n on e occurs , stop . The followin g day , tr y again . Challeng e th e student s t o ge t throug h t o th e

228 EVAN M A L E T S K Y

largest decima l withou t an y mistakes . Thi s ma y see m easy , bu t experienc e proves otherwise . Th e first fe w correc t choices , i n orde r startin g wit h 2 , ar e 2,2.1,2.13,2.3,2.31,3,3.1,3.12,...

From th e poin t o f vie w of mathematical content , thi s activit y deal s wit h the importan t skil l of ordering decimals . Bu t eve n mor e important , student s must creat e them , an d t o d o s o require s th e abilit y t o pla y freel y an d imag - inatively wit h number s an d shape s i n situation s involvin g discret e choices . This skil l need s t o b e develope d an d nurture d thoroughl y i n th e middl e grades b y embeddin g i t withi n th e existin g curriculu m an d aroun d famil - iar classroo m experiences . Thes e kind s o f simpl e exercises , whil e bot h fu n and challengin g fo r student s a t thi s age , la y th e foundatio n tha t wil l enabl e them, i n late r years , to approac h mor e profoun d an d intriguin g applications .

Graphs

Many problem s ca n bes t b e approache d throug h model s i n th e for m o f graphs. Grap h model s offe r a kin d o f organizationa l structur e tha t ca n b e utilized i n man y problem-solvin g experience s involvin g bot h manipulative s and counting . Le t u s loo k a t a n example .

Five cubes of different colors are arranged in a row. How many different arrangements are possible?

Many student s familia r wit h countin g kno w thi s i s a permutatio n prob - lem and kno w the answer to be 5 ! = 1 2 0 . But , whe n asked for a n explanatio n or meaning , the y hav e littl e t o sa y becaus e the y reall y se e nothing . Earl y counting experience s o f thi s typ e nee d t o b e don e wit h concret e material s and modele d i n diagra m for m fo r bette r understanding . I n th e followin g example, w e use fiv e blocks , on e o f each o f th e color s gree n (G) , orang e (O) , red (R) , yello w (Y) , an d blu e (B) .

You migh t begi n b y arrangin g th e cube s i n a ro w an d discussin g thei r order. Hav e student s sugges t an d sho w othe r orderings . Pu t th e cube s i n your han d an d as k ho w man y choice s ther e ar e fo r th e firs t position . Ho w many choice s remai n fo r th e second , an d the n th e third , an d th e fourth , and th e fifth positions ? Connec t thes e question s t o th e block s an d t o th e diagram i n whic h th e number s ar e entere d on e a t a tim e (se e Figur e 6) , an d to thei r product .

/ / / / /

G O R Y B /

F I G U R E 6 .

A systemati c listin g o f al l solution s i s ofte n accessibl e an d usefu l i n solving man y countin g problems . However , a listin g o f th e 12 0 choice s her e

D I S C R E T E MATHEMATIC S A C T I V I T I E S F O R MIDDL E SCHOO L 22 9

seems a bi t tedious . Thi s i s on e plac e wher e a grap h ca n b e useful . Th e vertices represen t th e cube s an d th e edge s sho w al l th e possibl e connection s (see Figur e 7a) . Ever y on e o f th e 12 0 possibl e arrangemen t o f th e cube s is a distinct , directe d pat h o f fou r edge s connectin g th e five vertices . Th e arrangement GORY B ca n b e represente d b y a pat h a s show n i n Figur e 7b .

O O

R ^/J \ \ Y R * ^ _ \ Y

(a) (b ) F I G U R E 7 . (a ) Grap h wit h vertice s representin g blocks , (b ) Directed pat h representin g th e linea r arrangemen t o f blocks , GORYB.

Many goo d countin g question s ca n b e asked . Ho w man y o f thes e path s start a t G ? Ho w man y star t a t G an d en d a t B ? Ho w man y hav e G nex t t o B? Ho w man y d o no t hav e G nex t t o B ?

Situations ca n b e analyze d an d answer s ca n b e foun d fro m th e graph . Have students trac e ou t path s fo r give n arrangement s an d arrang e th e cube s for give n paths . (Thes e requir e ver y differen t skills. ) Hav e student s coun t the numbe r o f edge s i n th e complet e grap h an d explai n wha t th e numbe r means. Connec t th e answe r t o th e proble m o f choosin g tw o cube s fro m th e set o f five. Se e i f the y recogniz e th e answe r a s a triangula r number .

Many discret e mathematic s problem s ar e alread y i n th e textbook s an d other availabl e literatur e a s example s addressin g teachin g method s o r class - room issues . Th e EQUAL S projec t a t th e Lawrenc e Hal l o f Scienc e a t th e University o f Californi a a t Berkeley , throug h it s publications , Get It To- gether, suggest s a n interestin g cooperativ e learnin g activit y simila r t o th e one just described . I t is an arrangemen t proble m involvin g six colored cubes .

Four student s independentl y receiv e critical information , tha t the y alon e possess, abou t th e arrangement . Al l students mus t participat e becaus e eac h student ha s informatio n t o contribut e an d need s t o d o s o a t th e righ t time . The tas k i s t o arrang e th e si x colore d cube s i n a ro w i n th e correc t order .

a: Gree n i s no t nex t t o yello w an d purpl e i s no t nex t t o green . b : Orang e i s no t nex t t o yello w an d gree n i s no t nex t t o blue . c: Yello w i s no t nex t t o red , blu e no t nex t t o purple , an d re d no t nex t

to orange . d: Purpl e i s no t nex t t o yellow , blu e no t nex t t o orange , an d gree n no t

next t o red .

23 0 EVAN MALETSK Y

The proble m offer s a n excellen t exampl e o f a cooperativ e learnin g sit - uation i n th e aren a o f discret e mathematics . On e approac h i s hands-on , with th e solutio n emergin g throug h th e arrangin g an d rearrangin g o f th e colored cubes . Anothe r approac h i s t o dra w a complet e grap h wit h 6 ver - tices representin g th e color s an d 1 5 edges representin g al l possibl e way s an y two colore d cube s migh t touc h eac h other , whe n arrange d i n a row . Clearly , in an y give n arrangemen t o f th e cubes , onl y som e o f thes e connection s wil l be made . On e b y one , th e student s remov e thos e edge s no t allowe d b y th e restrictions the y wer e given . I n all , 1 0 edge s wil l b e eliminate d fro m th e graph. Th e 5 edge s tha t remai n revea l th e onl y possibl e sequence , ordere d left-to-right o r right-to-left , show n i n Figur e 8 .

FIGURE 8 . Thi s pat h show s th e onl y tw o possibl e arrange - ments o f th e si x cube s whe n place d i n a row : YBRPO G o r GOPRBY.

Encourage student s t o mak e u p simila r set s o f condition s o n thei r own . Let the m chec k on e another' s suggestions . Hav e the m describ e algorithm s for creatin g problem s tha t wil l ensur e uniqu e solutions . Thes e ar e som e o f the importan t component s o f th e critica l thinkin g require d fo r doin g math - ematics.

How ca n discret e mat h problem s suc h a s these, involvin g th e orderin g of colored cubes , b e modifie d t o assig n length s t o th e edges ? Suppose , instea d of havin g five colore d cubes , team s o f student s selec t five whol e number s i n the rang e 0 t o 100 . Imagin e th e number s a s th e name s o f citie s whic h ar e connected b y airplan e flights.

Begin b y havin g th e team s arbitraril y plac e thei r five vertices , identifie d by thei r choic e o f numbers . Next , hav e the m assig n distance s t o th e edge s corresponding t o th e difference s betwee n th e number s o n th e connecte d ver - tices representin g cities . Her e ar e som e possibl e investigation s t o consider .

• Tr y t o find th e shortes t rout e connectin g al l five cities. Wher e woul d you star t an d wher e woul d yo u end ? Wha t abou t a roun d tri p tha t takes yo u throug h al l five cities ?

• Wher e woul d yo u star t an d en d fo r th e longes t route , withou t repeat - ing an y connections ? I s th e sam e sequenc e th e bes t fo r th e longes t round trip ?

DISCRETE MATHEMATIC S ACTIVITIE S FO R MIDDL E SCHOO L 23 1

Have th e team s tr y t o fin d algorithm s fo r solvin g thes e problems . As k whether thei r procedure s woul d change for a n even instead o f an odd numbe r of cities. I n middle school, student s nee d th e experienc e o f exploring, trying , testing, an d expressin g thei r idea s i n situation s lik e thes e a s muc h a s the y need t o lear n an d appl y know n algorithm s fro m discret e mathematics .

Figure 9 shows a complete graph, weighte d o n the edge s by the distance s for th e five citie s numbere d 6 , 32 , 19 , 84 , an d 61 .

F I G U R E 9 . Grap h showin g five citie s wit h distances .

There ar e 5 ! = 1 2 0 directe d path s tha t connec t th e five vertice s an d 4! = 2 4 directe d trip s throug h the m bac k t o th e startin g point .

Finding th e shortes t path s an d circuit s throug h th e five vertice s i n thi s situation doe s no t requir e a grea t effort , especiall y i f on e realize s tha t tour s among th e point s o n th e grap h correspon d t o route s alon g th e rea l numbe r line. Findin g th e longes t path s an d circuit s require s mor e thinkin g an d testing. Searchin g fo r appropriat e algorithm s fo r an y se t o f verte x value s poses som e interestin g challenges .

Iteration

When th e dynamic s o f chang e i s buil t int o a hands-o n activit y fo r th e mathematics classroo m throug h som e iterativ e process , th e experienc e be - comes al l th e mor e powerful . On e reaso n i s tha t numerical , geometric , an d algebraic relationship s an d connection s ofte n emerg e fro m a singl e experi - ence, a s i n th e followin g activity .

Start wit h a n equilatera l triangl e cu t fro m paper . Mar k a verte x P an d repeat th e followin g foldin g procedur e throug h severa l stages :

When the vertex P appears in a triangle, fold it to the midpoint of the opposite side and then unfold. (Se e Figur e 10) .

The outlin e o f th e folde d pape r a t eac h stag e i s a trapezoid , bu t thes e trapezoids chang e throug h successiv e stages . Ho w ar e the y changing ? Wha t do yo u see ?

From a measurement poin t o f view, th e trapezoid s ar e growin g i n height . Start wit h a triangl e whos e are a i s 1 square uni t an d watc h th e area s o f th e trapezoids change .

23 2 EVAN MALETSK Y

Z5^ Stage 0 Stage 1 Stage 2 Stage 3 Stage 4

F I G U R E 10 . A trapezoi d foldin g activit y

The firs t triangl e folde d ove r ha s a n are a o f 1/4 . The secon d folde d triangl e ha s a n are a o f (1/4) 2 o r 1/16 . The thir d ha s a n are a o f (1/4) 3 o r 1/64 , an d s o on .

Subtract thes e successiv e power s o f 1/ 4 fro m th e origina l are a o f 1 t o find wha t are a remain s fo r th e trapezoi d a t eac h stage :

3 . 1 5 , 6 3 4 1 6 6 4

Stage 1 Stag e 2 Stag e 3

> 2 5 5 256

Stage 4

What els e i s changin g a s th e proces s i s repeate d ove r an d over ? Th e unfolded stage s revea l othe r interestin g pattern s o f a discret e nature , a s i n Figure 11 .

Stage 0 Stage 4

In thi s form , w e ca n vie w triangle s an d trapezoid s i n quit e a differen t way, a s show n i n thi s table :

Stage Number o f triangle s Number o f trapezoid s

0 1 0

1 2 1

2 3 3

3 4 6

4 5

n n + 1

n(n + l ) / 2

Here again, w e find the triangular number s embedded i n a counting prob- lem centere d aroun d a geometri c activity . Lookin g a t th e fold s themselves , still anothe r visio n ma y appear . Le t you r student s describ e wha t the y see .

One imag e i s tha t o f a strangel y distorte d ladder . Whe n yo u clim b it , each successiv e ste p i s hal f a s hig h an d eac h successiv e run g hal f a s wide . When yo u loo k up , yo u foreve r se e reduced version s o f exactly wha t yo u sa w before. An d th e climb , step-by-step , i s endless !

You can quickl y see how some more powerful notions , such as perspectiv e in ar t an d limit s i n mathematics , ca n b e brough t int o play . Student s nee d

DISCRETE MATHEMATIC S ACTIVITIE S FO R MIDDL E SCHOO L 23 3

to see , think , an d tal k abou t concept s suc h a s thes e fro m a n intuitiv e poin t of vie w durin g th e middl e schoo l years . B y choosin g a goo d visua l mode l and askin g th e righ t questions , on e ca n brin g togethe r a hos t o f relate d mathematical idea s i n a singl e activity . An d i t i s no t surprisin g tha t man y of thes e tur n ou t t o b e discret e i n nature .

Suppose the folding proces s is changed a bit, a s described i n the followin g algorithm.

Every time you have a triangle, fold each vertex to the midpoint of the opposite side. Cut off the corners and keep only the middle triangular piece at each stage. (Se e Figur e 12. )

AAA Stage 1 Stag e 2 Stag e 3 Stag e 4

F I G U R E 12 .

A new set of figures i s generated an d ne w sets of number pattern s emerge . Stage Number o f triangle s Area Perimeter

0 1 1 1

1 1

1/4 1/2

2 -I

1/16 1/4

3 1

1/64 1/8

4 1

1/256 1/16

n 1

(1/4)" (1/2)"

By interchangin g wha t i s kep t an d discarde d i n th e foldin g an d cuttin g process, a n entirel y differen t sequenc e o f figure s i s created , a s show n i n Figure 13 . Thi s time , kee p the corne r piece s an d discar d th e middl e piec e a t each stag e wit h eac h triangle . No w th e proces s lead s t o a n entirel y differen t structure, a fractal calle d th e Sierpinsk i triangle .

Stage Number o f triangle s Area Perimeter

0 1 1 1

1 3

3/4 3/2

2 9

9/16 9/4

3 27

27/64 27/8

4 81

81/256 81/16

n 3 "

(3/4)" (3/2) n

As a n alternativ e approac h i n th e classroom , hav e you r student s dra w these tw o set s o f figure s o n triangula r do t paper , Choos e a larg e triangl e where the dots divide the sides into units that numbe r a power of 2. Thi s way the spacin g o f the dot s wil l facilitate drawin g severa l repeate d reduction s b y one-half. Fo r man y students , bot h type s o f activitie s woul d b e worthwhile . Indeed, seeing , drawing , an d visualizin g experience s al l nee d t o occu r mor e

234 EVAN M A L E T S K Y

A A A .4 Stag e 1 Stag e 2 Stag e 3 Stag e 4

F I G U R E 13 .

often i n th e mathematic s classroo m t o improv e ou r students ' abilitie s i n visual literacy .

The distinct , discret e stage s o f growt h clearl y sho w a n underlyin g prop - erty o f fractals , tha t o f self-similarity. Copie s o f th e figure appea r withi n itself a t al l scales . Thre e reduce d image s o f th e initia l stag e ca n b e see n in stag e 1 . Thre e reduce d image s o f stag e 1 ca n b e see n i n stag e 2 . Thre e reduced image s o f stag e 2 ca n b e see n i n stag e 3 , an d s o on .

The intricat e structur e o f th e emergin g fracta l ca n b e measure d b y it s fractal dimension. Fo r th e Sierpinsk i triangle , thi s complexit y measuremen t is approximatel y 1.58 . Se e Volum e 1 of [2 ] fo r a n introductio n t o th e topi c of fractals .

Is there a n underlyin g structur e her e tha t i s independent o f the shap e of the initia l figure? Tha t is , i f w e star t wit h a differen t figure an d repeatedl y put togethe r thre e copie s o f th e figure, scale d t o one-half , wha t d o w e get ? Have student s explor e thi s questio n startin g wit h othe r figures, suc h a righ t triangle, a scalen e triangle , o r eve n a square , rathe r tha n a n equilatera l triangle.

Start wit h a squar e cu t fro m paper . Cu t i t i n hal f verticall y an d hori - zontally. Us e th e rebuildin g proces s show n i n Figur e 1 4 with thre e reduce d copies a t eac h stag e place d i n th e shade d cells .

E L EL QL Stage 0 Stag e 1 Stag e 2 Stag e 3 Stag e 4

F I G U R E 14 . Iteratio n base d o n a squar e

It doe s not tak e many stage s to se e a familiar shap e emerging . Hav e your students think , talk , an d writ e abou t th e similaritie s an d th e difference s between th e changin g structure s bein g generate d fro m square s an d thos e that wer e generate d abov e fro m triangles . I n bot h cases , o f course , th e limi t structure i s th e Sierpinsk i triangle .

As a final activity , hav e student s pu t thei r ow n persona l twis t t o th e rebuilding ste p i n the iteratio n process , whic h ca n b e abbreviate d a s Reduce, Replicate, and Rebuild.

D I S C R E T E MATHEMATIC S A C T I V I T I E S F O R MIDDL E SCHOO L 23 5

Mentally labe l th e thre e cell s A, B , an d C , a s show n i n Figure 15 . Whe n the reduce d image s ar e droppe d bac k int o th e appropriat e cells , conside r possible rotations. I n the sequence of figures show n in Figure 15 , the reduce d copy i n cel l A i s alway s rotate d 270 ° clockwis e a t eac h stage . Thos e copie s placed i n cell s B an d C alway s remai n i n thei r origina l orientation , which , for convenience , ca n b e calle d a rotatio n o f 0° .

\ 2 7 0 "

B C IJ. E A i k Stag e 0 Stag e 1 Stag e 2

F I G U R E 15 .

Stag e 3 Stag e 4

Four choice s o f rotatio n ar e possibl e fo r eac h o f th e thre e cells . Tha t gives 4 x 4 x 4 = 6 4 different rebuildin g code s usin g rotations . Thi s ca n lea d to th e exploratio n o f a whol e famil y o f relate d fractal s wit h man y differen t structures. Hav e student s creat e thei r ow n persona l fractal s b y makin g individual choice s o f rotation s fo r cell s A , B , an d C . The y ca n cu t ou t an d tape togethe r thei r image s o r dra w th e firs t fe w stage s o n grap h paper . Th e first fou r stage s ca n b e readil y draw n usin g 2 x 2-inc h initia l square s o n 1/8-inch grap h paper .

When reflection s ar e considered , anothe r fou r transformation s o f th e square ca n b e explored . Se e Figur e 16 .

(a) (b) (c)

FIGURE 16 . (a ) Horizonta l reflectio n abou t th e vertica l axis, (b ) Vertica l reflectio n abou t th e horizonta l axis , (c ) Reflection abou t th e lower-left , upper-righ t diagonal , (d ) Re - flection abou t th e upper-left , lower-righ t diagonal .

In th e sequenc e o f iteration s show n i n Figur e 17 , th e reduce d cop y i n cell A i s reflecte d abou t th e upper-left , lower-righ t diagona l a t eac h stage . Those copie s i n cell s B an d C remai n i n thei r origina l orientation .

In all , four rotation s an d fou r reflection s ca n b e made in each of the thre e square cells . Wit h eigh t transformation s possibl e i n eac h cell , ther e mus t b e 8 x 8 x 8 = 51 2 differen t rebuildin g codes . Wil l al l 51 2 differen t buildin g codes produc e differen t fractals ? Th e answe r i s no . Becaus e o f symmetry , some image s wil l be duplicated . Ho w many distinc t fracta l image s wil l ther e be? Th e questio n i s lef t fo r th e reade r t o investigat e an d answer .

236 EVAN MALETSK Y

1 ^ B C i.

Stage 0 Stage 1 Stage 2

F I G U R E 17 .

Stage 3 Stage 4

Here we are, answering and asking yet another countin g problem emerg - ing fro m a n iterative , geometri c activity . Th e middl e schoo l curriculu m i s fertile groun d fo r increase d attentio n t o situation s involvin g discret e math - ematics. Th e problems ar e all around u s if we but loo k fo r them .

References

[1] Erickson , T\ , Get It Together: Math Problems for Groups — Grades 4-12, Lawrenc e Hall o f Science , Berkele y CA , 1989.

[2] Pietgen , H-O. , Jurgens , EL , Saupe , D. , Maletsky , E. , Perciante , T. , an d Yunker , L . Fractals for the Classroom: Strategic Activities, Volumes One and Two, Springer - Verlag, Ne w York, 1991.

[3] Sobel , M. , and Maletsky , E. , Teaching Mathematics: A Sourcebook of Aids, Activities, and Strategies, Ally n & Bacon , Needha m Height s MA , 1988.

M O N T C L A I R STAT E U N I V E R S I T Y , U P P E R M O N T C L A I R , N E W J E R S E Y 0704 3

E-mail address: m a l e t s k y e Q a l p h a . m o n t c l a i r . e d u

Section 5 I n t e g r a t i n g Discret e M a t h e m a t i c s

into Existin g M a t h e m a t i c s Curricula , Grades 9-1 2

Putting Chao s int o Calculu s Course s R O B E R T L . DEVANE Y

Page 23 9

Making a Differenc e wit h Differenc e Equation s J O H N A . D O S S E Y

Page 25 5

Discrete Mathematica l Modelin g i n th e Secondar y Curriculum : Rationale an d Example s fro m Th e Core-Plu s Mathematic s Projec t

E R I C W . H A R T

Page 26 5

A Discret e Mathematic s Experienc e wit h Genera l Mathematic s Student s B R E T H O Y E R

Page 28 1

Algorithms, Algebra , an d th e Compute r La b P H I L I P G . L E W I S

Page 28 9

Discrete Mathematic s I s Alread y i n th e Classroo m — Bu t It' s Hidin g J O A N R E I N T H A L E R

Page 29 5

Integrating Discret e Mathematic s int o th e Curriculum : A n Exampl e J A M E S T . S A N D E F U R

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

P u t t i n g Chao s int o Calculu s Course s

Robert L . Devane y

Our goa l i n thi s pape r i s to giv e a brie f descriptio n o f ho w som e elemen - tary idea s whic h typicall y belon g t o th e real m o f discret e mathematic s ma y be easil y an d beneficiall y incorporate d int o th e standar d calculu s course . These idea s com e fro m dynamica l system s theory . The y for m a unifie d thread tha t begin s wit h th e basi c topic s i n th e calculu s an d culminate s i n a modern treatmen t o f Newton' s method .

There ar e man y reason s fo r incorporatin g idea s fro m dynamica l system s theory i n th e calculu s curriculum . On e reaso n i s the fac t tha t i t i s becomin g increasingly importan t fo r mathematic s an d scienc e student s t o understan d and b e abl e t o us e suc h numerica l algorithm s a s Newton' s method . I n th e same vein , i t i s importan t tha t thes e student s understan d th e limitation s o f computer implementation s o f thes e algorithm s (e.g. , the y ma y fai l t o con - verge, round-of f erro r ma y affec t results , etc.) . Anothe r reaso n fo r includin g dynamical idea s i s th e eas e wit h whic h student s ma y b e expose d t o top - ics of contemporar y researc h interes t i n mathematic s (iteratio n o f quadrati c functions o f a rea l variabl e remain s a n activ e fiel d o f researc h interest! ) A third reaso n i s tha t dynamic s provide s a natura l aren a i n whic h t o coupl e theoretical result s fro m calculu s wit h compute r experimentation .

In thi s pape r w e wil l presen t a threa d o f idea s whic h run s throug h such topic s a s iteration , graphica l analysis , attractin g an d repellin g periodi c points, an d chaos . Thi s threa d terminate s wit h Newton' s method , wher e w e show tha t al l o f thes e dynamica l idea s ma y b e combine d t o presen t a coher - ent treatmen t o f thi s algorithm , it s result s a s wel l a s it s limitations . Thi s can als o serve a s a "jumping-of f point " fo r studen t compute r experiment s o r research project s involvin g Newton' s metho d i n th e plan e an d th e relate d concept o f Juli a set s o r th e Mandelbro t set .

1. Iteratio n

Iteration i s on e o f th e basi c operation s o f dynamica l systems . Give n a function F, th e basi c questio n i s wha t happen s whe n w e compos e F wit h

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 . Partially supporte d b y NS F Gran t ESI-9255724 .

239

https://doi.org/10.1090/dimacs/036/18

24 0 ROBERT L . DEVANE Y

itself man y time s i n succession . Tha t is , give n a numbe r xo calle d th e seed, the basi c questio n i s wha t happen s t o th e followin g sequence :

Xi = F(XQ)

x2 = F( Xl) = F(F{x 0))

xs = F(x 2) = F(F(F(x 0)))

xn — b \x n—\).

This sequenc e i s calle d th e orbit o f xo - Th e questio n i s then : Wha t i s th e fate o f orbits ?

For example , i f F{x) = x 2, th e orbi t o f 1/ 2 i s

1 XQ

2 1 4 1 16 1 256

1 Xn cy2 n '

Thus w e se e tha t th e orbi t o f th e see d x o = 1/ 2 unde r F(x) = x 2 tend s t o 0. O n th e othe r hand , th e fat e o f th e orbi t o f x o = 2 i s muc h different : I t tends t o infinity .

x0 Xl

= 2 = 4

= 16

= 256

~ 9 n

= 22 .

For simplicity , w e ofte n writ e F n t o mea n th e n-fol d compositio n o f F with itself . Tha t is , F 2 ( x 0 ) = F(F(x 0)) an d F

3 ( x 0 ) = F(F(F(x 0))). There ar e al l sort s o f possibilitie s fo r th e behavio r o f th e orbi t o f a give n

seed. Th e poin t x ma y b e a fixed point, i.e. , F(xo) = XQ, or a periodic point, i.e. , F n (xo) = xo - I n th e latte r case , n i s th e period o f x o an d th e orbit o f x o i s calle d a cycl e o f orde r n . Fo r example , 0 i s a fixe d poin t for F(x) = x 2, sinc e it s orbi t i s th e constan t sequenc e 0,0,0 , O n th e other hand , 0 i s periodi c wit h perio d 2 fo r th e functio n F(x) = x 2 — 1 . The orbi t o f 0 i n thi s cas e i s th e repeatin g sequenc e 0 , —1,0, —1,0, —1,

P U T T I N G CHAO S I N T O CALCULU S COURSE S 241

Cycles an d fixed point s ar e amon g th e mos t importan t type s o f orbit s i n any dynamica l system . I n applications , suc h orbit s correspon d t o cycli c o r periodic behavio r o r to equilibriu m points .

Other type s o f orbit s includ e thos e tha t ar e asymptoti c t o fixed o r pe- riodic point s an d orbit s whic h behav e randoml y o r chaotically . Al l of thes e orbits ar e often presen t i n even the simplest o f dynamical systems , includin g Newton's method .

With thi s i n mind , i t i s natura l t o introduc e th e concep t o f iteratio n whenever compositio n o f function s i s defined . On e may introduc e th e con - cept o f orbi t a t thi s tim e an d the n begi n t o hin t abou t som e o f th e con - temporary researc h topic s th e clas s i s abou t t o experience . Thi s i s nothin g but advertisin g a t thi s point , bu t i t serve s th e purpos e o f convincin g th e students tha t somethin g excitin g wil l occu r i n thi s course .

Iteration i s particularly eas y t o illustrat e usin g technology . Arme d wit h a compute r wit h a simpl e program , a spreadsheet , o r a calculator , student s may easil y experimen t wit h a variet y o f iterations . Th e "sequence " mod e and we b plot o f the Texas Instrument s TI-8 2 graphin g calculato r i s particu - larly usefu l fo r iteration . Othe r calculator s ca n be programme d t o produc e similar results .

2. Graphica l Analysi s

Most student s wh o ente r calculu s ar e quit e familia r wit h th e concep t of th e grap h o f a function . Nevertheless , mos t instructor s find i t usefu l to revie w a numbe r o f basi c graph s tha t student s mus t kno w earl y i n th e course. A t thi s juncture , i t i s natural t o introduc e th e concep t o f graphica l analysis, a procedur e fo r determinin g orbit s geometricall y usin g th e grap h of a function .

How doe s on e use the grap h o f a functio n t o displa y orbits ? Th e proce - dure i s quite eas y (se e Figure 1) . Star t wit h th e graph o f F an d superimpos e the diagona l lin e y = x. Th e orbit o f a given point xo will be displayed alon g this diagonal . T o begin, dra w a vertical lin e fro m th e diagona l t o the graph , starting a t (xo , XQ) an d endin g a t (xo , F(#o)). Th e second ste p i s to dra w a horizontal lin e from thi s poin t bac k to the diagonal, reachin g th e diagonal a t (F(xo)JF(xo)), whic h give s th e second poin t o n the orbit. Thu s th e proces s of computin g F i s given geometricall y b y going verticall y fro m th e diagona l to the graph, an d then goin g horizontally fro m th e graph t o the diagonal. T o compute furthe r iterate s o f F th e process i s the same : mov e verticall y fro m (F(XQ)1 F(XO)) o n th e diagona l t o th e graph , an d the n horizontall y bac k t o the diagona l t o reac h th e poin t (F 2(xo),F2(xo)) o n th e diagonal . Thu s w e see displaye d o n the diagona l th e first thre e point s o n th e orbi t o f #o unde r F. Continuin g i n thi s fashion , w e ofte n se e a "staircase " diagra m whic h displays th e orbit o f XQ alon g th e diagonal. Figur e 1 displays thi s procedur e for F(x) = y/x.

Note tha t graphica l analysi s immediatel y yield s th e fixed point s o f F: these ar e point s o f intersectio n o f th e grap h wit h th e diagonal , i.e. , point s

242 ROBERT L . DEVANE Y

x Sx J/x 1 / z z

F I G U R E 1 . Graphica l analysi s o f F(x) = y/x

xo whic h satisf y F(XQ) = x$. I n th e cas e o f y/x, th e fixed point s ar e 0 an d 1. Not e that , b y graphica l analysis , th e orbi t o f an y non-zer o see d tend s to th e fixed poin t a t 1 . Th e staircas e ascend s towar d 1 if th e see d satisfie s 0 < XQ < 1 ; i t descend s towar d 1 if XQ > 1 . I n thi s way , graphica l analysi s gives a powerfu l too l fo r visualizin g orbit s geometrically .

For simpl e function s suc h a s F(x) = x 2 , x 3 , yfx, an d 1/x , student s ca n check easil y th e fat e o f an y orbi t usin g graphica l analysis . Thi s provide s a good exercis e for th e studen t i n that i t reinforce s th e concep t o f the grap h o f a functio n (indee d i t necessitate s a n accurat e graph ) a s wel l a s introducin g a ne w concep t simultaneously . I n Figur e 2 w e displa y graphica l analysi s applied t o F(x) = cosx . Not e that , a s wit h th e squar e roo t function , al l orbits her e als o ten d t o a fixed point . Thi s fixed poin t i s impossibl e t o determine algebraically , however , sinc e w e mus t solv e th e transcendenta l equation cos x = x. Nevertheless , graphica l analysi s yield s it s existenc e together wit h th e fac t tha t al l orbit s ten d t o it .

F I G U R E 2 . Graphica l analysi s o f F(x) = cos(x )

Not al l iteration s ar e a s wel l behaved . I n Figur e 3 we displa y th e resul t of iteratio n o f th e functio n F(x) = 4x(l — x). Not e ho w complicate d th e

PUTTING CHAO S INT O CALCULU S COURSE S 24 3

FIGURE 3 . Graphica l analysi s o f F(x) = 4x( l - x)

graphical analysi s is : Thi s i s on e vie w o f wha t mathematician s no w cal l chaos.

Incidentally, graphica l analysi s i s a built i n feature o f the TI-8 2 graphin g calculator, an d i t ca n b e programme d o n mos t othe r calculators .

3. Iteratio n vi a C o m p u t e r

At thi s point , les s tha n tw o week s int o th e standar d calculu s course , the studen t ma y b e expose d t o a laborator y experimen t tha t foreshadow s much of the remainin g topic s i n dynamics. I f a computer la b is available, th e student ma y b e asked to report o n the behavior o f orbits o f Fc(x) = cx(l — x) for variou s value s o f th e paramete r c . A s lon g a s c > 1 , i t i s a fac t tha t al l orbits o f point s XQ with XQ < 0 o r XQ > 1 ten d t o — oo. Thi s fac t ma y b e seen usin g graphica l analysis . A s a n example , th e orbi t o f xo = 2 unde r F(x) = 2x( l - x) i s give n b y x 0 = 2 , xx = - 4 , x3 = - 4 0 , x4 = -3280 , and th e orbi t o f XQ — 1.1 i s

xz # 4

^ 5

.—

—

1.1

-0.22

-0.5368

- 1 . 6 4 9 9 . . .

- 8 . 7 4 4 2 . . .

- 1 7 0 . 4 1 0 9 . . .

Therefore, al l o f th e interestin g dynamica l behavio r occur s fo r 0 < xo < 1 . For variou s value s o f c i n th e rang e 1 < c < 4 , th e studen t ma y observ e a wide variet y o f dynamica l behaviors .

24 4 ROBERT L . DEVANE Y

The experimen t i s to start wit h a given point i n the uni t interva l (usuall y xo = 1/2 , th e critica l point ) an d recor d th e behavio r o f th e orbi t o f xo fo r various value s o f th e paramete r c . Th e progra m t o prin t thi s orbi t i s quit e simple. Se e Figure 4 . Thi s i s typical o f programs i n dynamica l systems : th e code i s ofte n quit e simpl e bu t th e outpu t i s ofte n quit e remarkable . Usin g the progra m ITERATE , th e studen t ma y b e aske d t o comput e th e orbi t o f 0.5 for variou s c values under iteratio n o f Fc(x) = cx(\ — x). Severa l orbits of xo = 0. 5 ar e displaye d i n Tabl e 1 . Not e tha t whe n c = 1.5 , th e orbi t o f 1/ 2 tends t o a fixed point . Indeed , usin g graphica l analysis , i t i s eas y t o chec k that th e orbi t o f an y poin t x o wit h 0 < x o < 1 behaves i n thi s manner . O n the othe r hand , mos t o f thes e orbit s ten d t o a perio d 2 cycl e whe n c = 3. 1 and t o a perio d 4 cycl e whe n c = 3.5 .

i n p u t x V/.V/.V/ . (seed ) i n p u t c Y/oY/oY/ o (parameter ) i n p u t n Y/oY/oY/ o (ma x numbe r o f i t e r a t i o n s ) i = 0 do while i <= n

print i, x x = c*x*(l-x) i = i+1

end

F I G U R E 4 . Th e progra m ITERAT E

One o f th e principa l benefit s tha t ma y b e derive d fro m thi s o r simila r experiments i s tha t student s se e first-hand th e notio n o f convergenc e an d non-convergence. Fo r value s o f c < 3.6 , i t appear s tha t al l orbit s eventuall y tend somewher e (perhap s t o a fixed poin t o r a cycle) . Fo r 3. 6 < c < 4 , many orbit s see m t o wande r aimlessl y abou t th e uni t interval , althoug h there ar e c-value s fo r whic h convergenc e ma y b e observed . Th e c-value s for whic h n o convergenc e o r cycli c behavio r i s observe d i s wha t w e wil l describe belo w a s chaoti c behavior . I n an y event , th e phenomeno n o f non - convergence arise s naturall y i n th e settin g o f iteration , unlik e traditiona l calculus settings , wher e limit s which fai l t o exis t ofte n see m lik e pathologica l cases t o students .

We remar k tha t iteratio n demand s a differen t notio n o f convergenc e a s convergence t o cycle s i s clearly allowable . However , experiment s suc h a s th e above mak e thi s concep t eas y t o digest .

4. A t t r a c t i n g an d R e p e l l i n g Cycle s

The experimen t i n th e previou s sectio n highlight s th e fac t tha t fixed an d periodic point s com e i n two quite distinc t varieties . Fo r al l c > 1 , Fc ha s tw o fixed point s i n th e uni t interval , a t 0 an d a t x c = (c — l ) / c . Whe n c < 3 we may easil y find x c becaus e th e orbi t o f an y othe r poin t i n 0 < x < 1 tend s

PUTTING CHAO S INT O CALCULU S COURSE S

Iterate c = 1.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

TABLE 1.

0.5 0.375 0.3515625 0.341949462 0.337530041 0.335405268 0.334362861 0.333846507 0.333589525 0.33346133 0.333397307 0.333365314 0.333349322 0.333341327 0.33333733 0.333335331 0.333334332 0.333333832 0.333333583 0.333333458 0.333333395 0.333333364

0.333333348 0.333333341 0.333333337 0.333333335 0.333333334

0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333

c = 3.2 0.5 0.8 0.512 0.7995392 0.512884056 0.799468803 0.513018994 0.799457618 0.513040431 0.79945583 0.513043857 0.799455544 0.513044405 0.799455499 0.513044492 0.799455491 0.513044506 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549 0.513044509 0.79945549

c = 3.5 0.5 0.875 0.3828125 0.826934814 0.500897694 0.874997179 0.382819903 0.826940887 0.500883795 0.874997266 0.382819676 0.826940701 0.500884222 0.874997263 0.382819683 0.826940706 0.500884209 0.874997263 0.382819683 0.826940706 0.50088421 0.874997263 0.382819683 0.826940706 0.50088421 0.874997263 0.382819683 0.826940706 0.50088421 0.874997263 0.382819683 0.826940706 0.50088421 0.874997263 0.382819683 0.826940706 0.50088421 0.874997263 0.382819683 0.826940706

The orbit of .5 for various c-values. This orbit is attracted to a fixed point when c = 1.5, to a cycle of period 2 when c; = 3.2, and to a cycle of period 4 when c = 3.5.

246 ROBERT L . DEVANE Y

to it . Whe n c > 3 , thi s cease s t o b e th e case . I n th e forme r cases , x c i s called a n attractin g fixed point ; i n th e latter , x c i s repelling . Th e compute r experiment abov e show s tha t attractin g fixed point s ar e visibl e wherea s repelling fixed point s ar e invisibl e t o th e computer . T o b e precise , a fixed point x o fo r F i s calle d attracting i f ther e i s a n interva l (a , b) containing x o and havin g th e propert y tha t i f x G (a, b) then F(x) G (a, b) and moreover , Fn(x) — > xo a s n — > oo. Th e fixed poin t i s repelling i f th e orbi t o f an y x G (a, b) (excep t x = xo ) eventuall y leave s (a , 6). Attractin g an d repellin g periodic cycle s ar e define d analogously .

This i s wher e calculu s enter s th e picture . I t i s eas y t o se e tha t a fixed point x o fo r F i s attractin g i f |F ;(xo)| < 1 . Similarly , x o i s repellin g i f | F ' ( x 0 ) | > 1 . Se e Figur e 5 .

(a) (b )

F I G U R E 5 . (a ) Th e fixed poin t p i s attracting : \F'(p)\ < 1 . (b)The fixed poin t p i s repelling : |F'(p) | > 1 .

As an example, we know that F(x) = x 2 ha s two fixed points, a t 0 and a t 1. W e hav e -F'(O ) = 0 , s o 0 i s a n attractin g fixed point . Also , F ' ( l ) = 2 , s o 1 i s repelling. Thi s i s illustrated nicel y b y graphica l analysis . Se e Figur e 6 .

The determinatio n o f whether a cycle is attracting o r repelling provides a nice application of the chain rule. Suppos e xo lies on a cycle of period n fo r F. Then, arguin g a s above , x o i s an attractin g periodi c poin t i f |(F n ) / (xo)| < 1 . But, b y th e chai n rule ,

(Fn)'(x0) = F'(F n-1(x0)) ••••• F'(F(x 0)) • F'(x 0)

For example , th e point s 0 and — 1 li e on a cycl e of period 2 for G(x) = x 2 — 1. This cycl e i s attractin g sinc e

(G2)'(0) = G'(G(0) ) • G"(0 )

= G ' ( - 1 ) - G ' ( 0 )

= - 2 - 0

= 0

PUTTING CHAO S INT O CALCULU S COURSE S

2 T

24 7

FIGURE 6 . F(x) = x 2 ha s a repellin g fixed poin t a t 1 and a n attracting fixed poin t a t 0 .

As another example , conside r F(x) — — x3. Thi s functio n ha s a 2-cycle give n by ± 1 , since F ( l ) = - 1 an d F ( - l ) = 1 . W e hav e F'(x) = -3x 2 s o F'(l) = - 3 an d F ' ( - l ) - - 3 . Therefore , ( F 2 ) ' ( l ) - F'(F(1)) - F ' ( l ) = - 3 - -3 = 9 , so thi s cycl e i s repelling . Thi s i s readil y observe d usin g graphica l analysis . See Figure 7 . Indeed , analyzin g al l orbits o f a give n dynamica l syste m usin g both graphica l analysi s an d th e abov e technique s provide s a valuabl e (an d often challenging ) exercis e fo r students .

I A. 1-1

•

fc V W

/ 4 ^

/ \

—̂ k 11 1

F I G U R E 7 . Graphica l analysi s o f F(x) = —x 3

248 R O B E R T L . DEVANE Y

5. Chao s

One o f the mos t interestin g mathematica l discoverie s o f the pas t quarte r century i s the fac t tha t ver y simpl e dynamica l system s ma y behav e "chaoti - cally." Despit e th e fac t tha t thi s phenomeno n i s a topi c o f muc h contempo - rary interes t i n mathematics, i t ma y stil l be explained i n elementary calculu s courses. Moreover , a s w e wil l se e whe n w e discus s Newton' s method , chao s occurs i n a grea t man y simpl e dynamica l systems .

While ther e i s no t ye t a universall y accepte d definitio n o f chaos , on e definite ingredien t o f an y definitio n i s sensitive dependence on initial condi- tions. A n orbi t i s sai d t o hav e sensitiv e dependenc e i f nearb y orbit s behav e in a vastl y differen t manner . T o b e precise , th e orbi t o f # o ha s sensitiv e dependence i f ther e exist s K > 0 suc h that , fo r an y 6 > 0 , ther e exist s x\ with \x\ — xo | < 8 bu t \F n(x\) — F n(xo)\ > K fo r som e intege r n. Tha t is, arbitraril y nea r XQ there exis t initia l condition s x\ whos e orbi t eventu - ally separate s fro m tha t o f xo b y a t leas t K units . W e als o cal l suc h orbit s chaotic orbits.

According t o thi s definition , repellin g fixe d an d periodi c point s ar e al - ways chaotic . Ther e ar e man y othe r kind s o f chaoti c orbits . Her e i s a n example o f a dynamica l syste m wit h man y chaoti c orbits . W e will mee t thi s function i n a completel y differen t settin g i n th e nex t section .

Consider th e doublin g functio n

2x, 0 < x < \ D(x) •{ 2 a ; - 1 , h<x<

Note tha t D i s define d o n th e interva l [0,1) . Sinc e th e derivativ e o f D i s always 2 , eac h iteratio n o f D double s th e distanc e betwee n correspondin g points o n differen t orbits , a t leas t unti l thes e point s appea r o n differen t side s of 1/2 . Henc e al l orbit s i n th e uni t interva l ar e chaotic .

Another propert y o f D i s that ther e ar e infinitel y man y periodi c points . Figure 8 indicate s tha t th e grap h o f D n crosse s th e diagona l exactl y 2 n

times; eac h o f thes e fixe d point s o f D n i s a periodi c poin t o f D. Indeed , it ca n b e show n tha t i f p/q i s rationa l wit h (p , q) = 1 an d q odd , the n p/q lie s o n a cycl e fo r D. Fo r example , th e orbi t o f 1/ 7 unde r doublin g i s 1/7, 2 / 7 , 4 / 7 , 1 / 7 , . . . whic h i s a 3-cycle . Also , th e orbi t o f 1/ 5 i s a 4-cycle : 1 / 5 , 2 / 5 , 4 / 5 , 3 / 5 , 1 / 5 , . . . .

F I G U R E 8 . Th e graph s o f £> , D2, an d D 3.

PUTTING CHAO S INT O CALCULU S COURSE S 24 9

A thir d propert y o f D tha t i s typica l o f chaoti c system s i s th e fac t tha t most orbit s ar e no t periodic . Indeed , th e orbit s o f mos t point s ten d t o fil l out th e interva l [0,1 ) i n a dens e fashion . Thi s ca n b e prove d rigorousl y i n an elementar y setting . Le t XQ G [0,1) an d conside r th e orbi t o f XQ. Assig n an infinit e sequenc e 5(#o ) o f 0' s an d l' s t o xo accordin g t o th e followin g prescription.

S(xo) = 5 0 5 i S 2 . . .

where SJ = 0 i f D^(x 0) G [0,1/2), Sj = 1 if D j(xQ) G [1/2,1). Th e sequenc e

S(XQ) give s th e itinerary o f th e orbi t o f xo i n th e sens e tha t S(xo) tell s whether th e j th iterat e o f XQ lie s i n th e lef t o r righ t hal f o f th e uni t interval . But S(xo) ha s anothe r interpretation : S(xo) i s jus t th e binar y expansio n of XQ. Thu s ther e ar e clearl y many , man y orbit s whos e itinerarie s ar e non - periodic.

This interpretatio n o f th e orbit s o f D als o show s wh y al l orbit s ar e chaotic. Fo r a give n poin t xo , w e ca n typicall y "know " it s binar y represen - tation wit h onl y finit e precision , i.e. , u p to , say , I binary digits . Thi s means , then, tha t afte r I iteration s o f T w e wil l hav e obliterate d an y knowledg e whatsoever o f th e locatio n o f point s o n th e orbi t o f x$.

One other issu e that sometime s surface s regardin g th e doublin g functio n is the behavio r o f computed orbits . O n machine s tha t us e binary arithmetic , students alway s observe that eac h computed orbi t alway s ends up eventuall y fixed a t 0 . O f course , thi s stem s fro m th e fac t tha t th e typica l xo-valu e i s represented i n finit e binar y form . Eac h successiv e applicatio n o f D effec - tively remove s on e o f th e digit s i n thi s binar y representation , thu s leadin g to th e abov e behavior . Thi s i s a n excellen t lesso n fo r student s t o learn : th e computer ma y lie !

For example , th e binar y representatio n o f 1/ 3 i s .010101.. . sinc e

A - 2 _! A JL £ 3 ~ 2 + 2 2 + 2 3 + 2 4 + ¥ + ' "

1 1 1 ~~ 4 + 4 2 + 4 3 H

= J_ = 1

However, th e compute r store s only a finite numbe r o f these binary digits , and thes e ar e remove d on e b y on e a s w e iterat e th e doublin g function . Fo r more detail s o n th e dynamic s o f th e doublin g function , w e refe r t o [2] .

6. N e w t o n ' s M e t h o d

With iteration , graphica l analysis , attractin g fixe d points , an d chao s a s concepts in hand, th e introduction o f Newton's metho d i n the calculus cours e becomes a centra l topi c i n th e course , rathe r tha n a periphera l curiosity . Indeed, Newton' s metho d utilize s al l o f th e precedin g topic s i n a n essentia l fashion.

25 0 R O B E R T L . DEVANE Y

As i s wel l known , i f P(x) i s a polynomial , th e associate d Newto n itera - tion functio n

N(x) = x- P'(x)

has th e propert y tha t x o is a roo t o f P i f and onl y i f #o is an attractin g fixed point o f N. Indeed , N f(xo) = 0 i f x o i s a simpl e roo t o f P. Thus , t o find a roo t o f P , al l w e nee d d o i s selec t som e rando m initia l conditio n X Q an d compute th e orbi t o f xo unde r N. Hopefully , thi s orbi t wil l converg e t o on e of th e attractin g fixed point s o f N, i.e. , t o on e o f th e root s o f P. This , o f course, nee d no t b e th e case , sinc e th e orbi t o f XQ may b e attracte d t o a n attracting cycl e or , eve n worse , ma y behav e chaotically .

Newton's metho d togethe r wit h graphica l analysi s provid e a natura l an d ideal plac e i n th e curriculu m fo r student s t o manipulat e an d comprehen d the grap h o f "complicated " functions . Fo r example , th e Newto n iteratio n function correspondin g t o P(x) = x 2 — 1 is give n b y

N(x)=1-(x+~)

and th e correspondin g graphica l analysi s i s depicte d i n Figur e 9 .

F I G U R E 9 . Newto n iteratio n fo r P(x) = x 2 - 1 .

For th e cubi c functio n Q(x) = x(x 2 — 5) , th e correspondin g Newto n function i s mor e complicate d

2x 3 N(x)

3 x 2 - 5 ' Nonetheless, student s ca n b e expecte d t o understan d th e graph s o f suc h functions an d perfor m th e graphica l analysis , a s i n Figur e 10 .

This las t exampl e give s on e simpl e reaso n wh y Newton' s metho d some - times fail s t o converge : not e tha t th e point s + 1 an d — 1 lie o n a cycl e o f

P U T T I N G CHAO S INT O CALCULU S COURSE S 251

F I G U R E 10 . Newto n iteratio n fo r Q(x) = x(x 2 - 5) .

period 2 . Thu s the perfectly natura l initia l guess of 1 leads to no convergenc e of Newton' s method .

For polynomial s o f degre e fou r o r more , th e se t o f point s a t whic h New - ton's metho d fail s o n th e rea l lin e i s mor e complicated . I f th e polynomia l has real , distinc t roots , i t turn s ou t tha t th e se t o f initia l point s a t whic h Newton's metho d fail s i s a Canto r set . Thi s i s discusse d i n a n awar d win - ning paper i n the America n Mathematica l Monthl y b y Saar i an d Urenk o [7]. Here w e se e a simpl e fracta l appearin g a s th e "interesting " se t o f point s i n a dynamica l system , a fac t tha t occur s ove r an d ove r agai n i n dynamics .

Another manne r i n which Newton's method fail s to converge occurs when the origina l functio n i s not differen t iable at a root. Fo r example, th e Newto n iteration correspondin g t o F(x) = x 1 / 3 i s N(x) = —2x. Usin g graphica l analysis w e se e tha t 0 i s a repellin g fixe d poin t fo r N an d al l orbit s ten d away fro m 0 .

Finally, i t i s interestin g t o as k wha t happen s whe n w e appl y Newton' s method t o th e polynomia l P(x) = x 2 + 1 . Th e Newto n iteratio n functio n i n this cas e i s

N(x) = -(x ;)•

Graphical analysi s show s tha t orbit s o f thi s functio n ten d t o behav e quit e chaotically. Se e Figur e 11 . I n fact , thi s functio n ha s dynamic s tha t ar e exactly the same a s the doublin g function introduce d i n the previous section .

Recall tha t

2s, 0 < x < \ 2 x - l , ^ < x < l D(

*) = { Consider th e functio n

C : [ 0 , l ) - { l / 2 } - R

252 R O B E R T L . DEVANE Y

F I G U R E 11 . Newto n iteratio n fo r N(x) = \{x - \)

given b y C{x) = co t nx. W e hav e

CoD(x) = COt(7T'D(x))

— cot(27ro: )

__ cos 2(7rx) — sin 2(7rx)

2 sin(7rx) cos(7rx ) 1 ( t x 1 \ = - COt(7TX ) ; r

= NoC(x).

This fac t mean s tha t C carrie s orbit s o f D ont o orbit s o f iV , becaus e

NnoC(x) = CoD n{x).

So, i f x lie s o n a cycl e o f perio d n fo r £) , the n C(x ) lie s o n a simila r cycl e for iV ! Moreover , sinc e al l orbit s o f D ar e chaotic , th e sam e i s tru e fo r al l orbits o f A H Thu s w e se e tha t chao s occur s i n eve n th e mos t unexpecte d places: Newton' s metho d fo r a quadrati c polynomial !

These idea s ar e b y n o mean s new : the y g o bac k t o Cayle y i n th e nine - teenth century . Indeed , Cayle y showed that, fo r a complex quadratic polyno - mial, Newton' s metho d faile d t o converg e onl y on the perpendicula r bisecto r of th e roots . O n thi s line , Newton' s iteratio n behave d similarl y t o th e dou - bling function . Cayle y eve n wen t s o fa r a s t o announc e tha t h e planne d a similar solutio n fo r cubi c polynomials . Bu t thi s pape r neve r appeared , fo r reasons that ar e only nowadays becoming clear . Th e chaoti c set fo r Newton' s method fo r a comple x cubi c polynomia l i s hardl y a line ! I t i s ofte n a ver y complicated fractal—th e Juli a se t o f th e Newto n iteratio n function . Whil e this topi c i s to o advance d fo r a standar d calculu s course , i t nevertheles s makes a wonderfu l subjec t fo r compute r experimentatio n i n a lab . W e hav e found that , eve n i f student s d o no t hav e acces s t o a la b i n whic h t o perfor m

PUTTING CHAO S INT O CALCULU S COURSE S 253

such experiment s themselves , the y lov e t o se e th e computer-generate d im - ages o f th e Juli a set s fo r Newton' s method . The y readil y understan d wha t these picture s mea n an d ar e mos t intrigue d b y th e fac t tha t thes e picture s were firs t see n a mer e te n year s ago . I ofte n conclud e m y lectur e o n New - ton's metho d b y showin g student s a fe w slide s o f Newton' s metho d i n th e plane. I hav e foun d tha t thi s give s student s mor e tha n jus t a pee k a t prett y pictures. Indeed , thes e image s giv e student s a glimps e o f wha t goe s o n i n research mathematics . Man y student s fin d thi s quit e enticing : the y ofte n tell m e tha t the y neve r imagine d ho w beautifu l mathematic s coul d be .

References

[1] Blaine , L. , "Theor y vs . Computatio n i n Som e Ver y Simpl e Dynamica l Systems, " Coll. Math. J. 2 2 (1991) , 42-44 .

[2] Devaney , R . L. , A First Course in Chaotic Dynamical Systems, Addison-Wesle y Co. , Reading MA , 1992 .

[3] , Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics, Addison-Wesley Co. , Menl o Par k CA , 1989 .

[4] , "Puttin g Chao s int o th e Classroom, " Discrete Mathematics across the Cur- riculum K-12, 199 1 NCT M Yearboo k (Margare t J . Kenne y an d Christia n R . Hirsch , eds.), NCTM , Resto n VA , 1991 , pp. 184-194 .

[5] Parris , R. , "Th e Roo t Findin g Rout e t o Chaos, " Coll Math. J. 2 2 (1991) , 48-55 . [6] Strang , G. , " A Chaoti c Searc h fo r z, " Coll. Math. J. 2 2 (1991) , 3-12 . [7] Saari , D . G. , an d J . Urenko , "Newton' s Method , Circl e Maps , an d Chaoti c Motion, "

Amer. Math. Monthly. 9 1 (1984) , 3-17 .

D E P A R T M E N T O F MATHEMATICS , B O S T O N U N I V E R S I T Y , B O S T O N , M A 0221 5

E-mail address'. bobQmath.bu.ed u

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Making a Differenc e wit h Differenc e E q u a t i o n s

John A . Dosse y

The us e o f differenc e equation s i n th e modelin g o f processe s an d i n th e description o f chang e i s a topi c usuall y studie d b y advance d student s i n mathematics. However , man y example s o f arithmetica l growt h ar e foun d in elementar y mathematics , an d th e topi c o f differenc e equation s use d t o model suc h growt h alread y enter s th e presen t schoo l curriculu m i n man y hidden ways .

At th e colleg e level , differenc e equation s hav e bee n reserve d fo r a lead - in t o differentia l equation s o r fo r student s studyin g actuaria l mathematics . In recen t years , stud y o f th e topi c ha s move d somewha t earlie r i n th e col - lege curriculu m an d towar d a broade r spectru m o f mathematic s students . The inclusio n o f differenc e equation s i n introductor y course s o n discret e mathematics fo r undergraduate s i n th e firs t tw o year s o f the curriculu m ha s heightened th e visibilit y o f difference equations , thei r applications , an d thei r power. Th e nex t decad e shoul d se e th e increase d recognitio n o f differenc e equations a s a n importan t topi c fo r th e K-1 2 curriculum .

Discrete mathematics , an d differenc e equation s i n particular , ha s re - ceived a grea t dea l o f interes t i n recen t year s [5 , 10 , 12 , 1 3 , 14] . However , it i s tim e t o recogniz e tha t man y o f th e topic s relate d t o differenc e equa - tions hav e applicatio n muc h earlie r i n th e schoo l mathematic s curriculum , especially i f student s ar e t o com e t o se e mathematic s a s a disciplin e havin g connections t o pattern s the y observ e i n thei r everyda y worlds . Informa l in - troduction t o differenc e equation s a s way s o f counting , a s way s o f relatin g successive item s i n a pattern , a s way s o f thinkin g abou t processes , shoul d enter th e schoo l mathematic s curriculu m i n th e uppe r elementar y grades .

Such recommendation s wer e mad e i n 196 3 i n th e repor t o f th e Cam - bridge Conferenc e o n Schoo l Mathematic s [3] , but, unfortunately , thes e rec - ommendations neve r took hold . Wit h th e curren t interes t i n reform i n schoo l mathematics an d supportin g recommendation s fro m th e NCT M Standard s [12], differenc e equation s ma y onc e again hav e an opportunit y t o mak e a dif- ference i n students ' mathematica l experiences . Bu t befor e w e explor e tha t

1991 Mathematics Subject Classification. Primar y 39A10 , 00A05 , 00A35 .

255

https://doi.org/10.1090/dimacs/036/19

256 JOHN A . DOSSE Y

possibility, i t i s bes t t o develo p th e topi c o f differenc e equation s an d thei r applications, an d thei r relevanc e t o schoo l mathematics .

1. Differenc e equation s

Difference equation s have played a n important rol e in mathematics acros s time. The y appea r informall y i n th e wor k o f th e Greek s a s the y consid - ered th e value s o f term s i n sequence s tha t represente d convergen t processe s [2], i n th e wor k o f Fibonacc i [7 , 11 , 16 ] i n th e 13 t/l centur y a s h e mod - eled th e growt h o f rabbi t population s i n hi s tex t Liber Abaci, an d i n th e work o f 16t h an d 17t h centur y analyst s a s the y struggle d t o plac e calculu s on a fir m footin g [2] . I n eac h o f these , an d man y othe r settings , mathe - maticians hav e turne d t o examinin g th e value s o f successiv e term s i n som e sequence ao , a i, a2, as, a±,..., an d t o understandin g th e proces s o f chang e as on e move s fro m on e valu e i n th e sequenc e t o subsequen t values . Thi s study o f chang e i s usuall y capture d i n a n expressio n simila r t o tha t whic h describes th e growt h o f Fibonacci' s populatio n o f rabbi t pairs :

rn = r n-i + rn-2, fo r n = 2 , 3 , 4 , . . .

r0 = n = 1 .

This famou s differenc e equatio n set s th e initia l condition s fo r th e pair s o f rabbits i n year s 0 an d 1 an d the n describes , vi a th e differenc e equatio n rn = r n-i + r n _ 2 , th e wa y i n whic h tha t populatio n fro m yea r 2 forwar d grows,1 Th e proces s o f iteratin g th e differenc e equatio n gives :

n r n o r Populatio n Initial tim e 0 1 End Yea r 1 1 1 End Yea r 2 2 1 + 1 o r 2 End Yea r 3 3 2 + 1 o r 3 End Yea r 4 4 3 + 2 o r 5... .

While man y i n mathematic s ar e familia r wit h th e Fibonacc i sequence , 1,1 , 2, 3 , 5, 8, 13 , 21, . . ., an d it s man y famou s properties , the y d o no t kno w th e central rol e tha t othe r differenc e equation s hav e playe d i n th e mathematic s curriculum, eve n withi n th e las t tw o centuries . However , student s o f G . Chrystal's 188 6 work , Textbook of Algebra [4] , an d Hal l an d Knight' s 188 7 classic Higher Algebra [9 ] wer e wel l awar e o f differenc e equation s an d thei r power i n illuminatin g th e natur e o f chang e i n finit e processes .

1 Editors' note : Se e Kowalczyk' s articl e i n thi s volum e fo r a n elementar y approac h t o deriving thi s relationship .

MAKING A D I F F E R E N C E W I T H D I F F E R E N C E EQUATION S 25 7

2. Schoo l M a t h e m a t i c s an d Differenc e Equation s

Difference equation s first ente r th e presen t schoo l curriculum i n a hidde n way a s student s com e t o stud y arithmeti c an d geometri c sequence s i n up - per middl e schoo l an d i n secondar y schoo l mathematics . Thes e importan t concepts captur e mathematicall y th e essenc e o f tw o o f th e mos t powerfu l forms o f change . Unfortunately , thes e format s fo r modelin g chang e ar e no t exploited t o thei r maximu m potential . Instea d o f developing arithmeti c an d geometric sequence s a s symbolic numbe r patterns , curricul a shoul d focu s o n the type s o f chang e the y describe .

Many examples of arithmetic growt h ar e found i n elementary mathemat - ics, rangin g fro m th e multiple s o f 3 (3 , 6, 9 , 12 , . . . ) ; to th e effec t o f simpl e interest a t 5 % o n a $10 0 investmen t ove r year s ($100 , $105 , $110 , . . . ) ; t o the growt h i n value s o f a linea r functio n f(x) = Sx + 5 evaluated a t nonneg - ative intege r value s ( 5 , 8 , 1 1 , 1 4 , . . . ) . Formally , thi s arithmeti c growt h ca n be describe d b y th e differenc e equatio n

An = .A n_i + d fo r n — 1 , 2 , 3 , . . . ,

A0 = a.

Prom th e initia l valu e o f a , eac h successiv e ter m i n a n arithmeti c se - quence i s foun d b y addin g th e commo n differenc e o f d : a, a + d , a + 2d , a + 3 d , . . . . Th e centra l propert y signalin g th e underlyin g differenc e equa - tion mode l i n thes e sequence s i s th e fac t tha t th e subtractiv e compariso n An — A n _i betwee n successiv e term s i s a constan t differenc e d.

Many example s o f geometri c growt h ar e als o foun d i n elementar y math - ematics, fro m th e stud y o f th e power s o f 5 (5 , 25 , 125 , 625 , . . . ) ; t o th e effect o f compoun d interes t a t 5 % o n a $10 0 investmen t ove r year s ($100 , $105.00, $110.25 , $115.76 , . . . ) ; t o th e growt h i n value s o f th e exponentia l function g(x) = e x evaluate d a t nonnegativ e intege r value s (1 , e, e2, e 3 , . . . ) . Geometric growt h i s formall y describe d b y th e differenc e equatio n

An = rA n-i fo r n = 1 , 2 , 3 , . . . ,

A0 = a.

Prom th e initia l valu e o f a , eac h successiv e ter m i n a geometri c sequenc e is foun d b y multiplyin g th e precedin g ter m b y th e commo n facto r o f r : a, ar, ar2,ar3,.... Th e centra l propert y signalin g th e underlyin g differenc e equation mode l i n thes e sequence s i s th e fac t tha t th e compariso n rati o An/An-i betwee n successiv e term s i s the constan t r . Thes e example s reflec t the occurrenc e o f differenc e equations , eithe r formall y o r implicitly , i n a variety o f level s an d context s withi n th e K-1 2 curriculum . However , to o often, student s onl y se e th e beaut y an d powe r o f differenc e equation s i n th e second-year algebr a o r precalculu s curriculu m i n thei r stud y o f sequence s and series . Eve n i n thes e cases , th e actua l differenc e equatio n structur e i s often bypasse d t o mov e quickl y t o establishin g formula s fo r th e n th term s in arithmeti c an d geometri c sequences , an d sum s fo r th e first n term s i n

258 JOHN A . DOSSE Y

progressions involvin g term s fro m eithe r typ e o f sequence . I n doin g so , students ar e rushe d pas s th e centra l notion s o f change , rat e o f change , an d the powerfu l insigh t tha t differenc e equation s an d thei r stud y provid e fo r both modelin g commo n situation s an d preparin g a bas e fo r furthe r stud y i n mathematics.

Both o f th e differenc e equations , A n = A n-\ + d an d A n = rA n-\, are specia l case s o f th e broade r clas s o f differenc e equation s f n = Af n-\ + B, know n a s first-orde r linea r differenc e equations . Linea r becaus e o f th e general form , an d first-orde r becaus e th e valu e o f th e n th cas e i s dependen t on onl y th e precedin g valu e f n-\ i n th e sequence . Th e Fibonacc i differenc e equation observe d earlie r i s a second-orde r linea r differenc e equation , sinc e the valu e o f an y ter m depend s o n th e value s o f tw o precedin g terms .

3 . Modelin g w i t h First-Orde r Linea r Differenc e Equation s

The stud y o f differenc e equation s provide s valuabl e opportunitie s t o in - troduce student s t o th e richnes s o f applyin g mathematics . Conside r th e following example s reflectin g th e us e o f differenc e equation s t o explai n th e long-term effect s o f quantitativ e decision s i n real-worl d settings .

Forestry. Th e Clea r Lak e Pin e Compan y own s a timbe r stan d wit h 7000 pin e tree s [6] . Eac h yea r th e compan y harvest s 12 % o f it s tree s an d plants 60 0 seedlings . The y ar e particularl y intereste d i n th e pin e tre e pop - ulation i n thi s timbe r stan d i n 1 0 year s an d i n th e long-rang e future . Ex - amining thi s situatio n fo r th e natur e o f chang e fro m on e yea r t o another , one ca n buil d th e first-orde r linea r differenc e equatio n mode l fo r th e pin e population p n i n yea r n a s follows :

pn = 0.88p n_i + 60 0

p0 = 7000 .

One ca n iterat e thi s mode l o n a TI-8 1 wit h th e followin g keystrokes : 7000 , ENTER, 0.8 8 * 2n d AN S + 600 , an d the n ENTE R t o ge t th e valu e fo r p i , ENTER fo r th e valu e o f P2, an d ENTE R fo r successiv e value s o f pi, a s th e value o f i increases . Th e first 2 6 value s fo r th e sequenc e ar e show n o n th e calculator screen s i n Figur e 1 .

7000 7006

0.88*flns+6Q0 6766 6549 6363 6199

6059 5929 581^ 5719 5633 55571 5490

1 1

5431 5380 5334 5294 5259 5228 5206

5176 5155 5137 5120 5106 5093 5082

F I G U R E 1 .

Continuing t o iterat e th e differenc e equatio n forward , on e gets a patter n indicating tha t th e pin e tre e populatio n tend s t o stabiliz e a t 500 0 ove r th e long haul . A n examinatio n o f th e dat a abov e show s tha t afte r te n year s th e population woul d be 5557 . I n the nex t sectio n we will present an d discus s th e

MAKING A D I F F E R E N C E W I T H D I F F E R E N C E EQUATION S 259

general solutio n fo r a first-orde r linea r differenc e equatio n f n = A / n _ i + B\ this wil l revea l that , excep t i n th e cas e o f arithmeti c growt h whe n A i s 1, th e solutio n alway s model s exponentia l growth . Employin g th e solutio n form fo r first-order , linea r differenc e equation s (se e Sectio n 4) , on e arrive s at th e genera l valu e fo r th e n t h yea r followin g th e initiatio n o f th e process :

pn = 2000(0.88) n + 5000 .

Examining thi s a s a functio n describin g th e pin e tre e populatio n a t eac h year followin g th e firs t coun t o f 700 0 trees , i t i s eas y t o se e tha t thi s mode l predicts a n eventua l stead y stat e populatio n o f 500 0 tree s i n th e stand . When viewe d graphicall y i n Figur e 2 , using a continuou s grap h t o mak e th e trend mor e visible , on e see s thi s developin g ove r th e first 10 0 years ; eac h horizontal interva l i n Figur e 2 represents 1 0 years an d eac h vertica l interva l represents 100 0 trees .

F I G U R E 2 .

This limitin g proces s ca n b e viewed i n a step-wis e fashio n usin g th e cobweb approac h discusse d i n [15 ] an d [1] . Thi s approac h allow s fo r a n investigation o f th e relationshi p betwee n th e differenc e equation , th e initia l value, an d th e existenc e o f limitin g value s fo r a sequence . Fo r example , th e graph o f th e lin e y = 0.88 x + 60 0 considere d togethe r wit h th e grap h o f the auxiliar y lin e y = x, allow s on e t o establis h geometricall y th e long-ter m behavior o f th e proces s describe d b y th e differenc e equation .

In Figur e 3 we have entere d th e y- value o f 700 0 fo r a n initia l populatio n on the lin e y = x\ w e then dro p verticall y t o th e lin e y = 0.88 x + 600 definin g the differenc e equatio n t o ente r thi s a s a n x valu e fo r determinin g th e nex t y value , the n horizontall y t o th e y = x lin e t o ge t th e nex t x value , the n vertically t o th e differenc e equatio n lin e fo r th e nex t y value , transformin g back an d fort h fro m presen t valu e t o nex t valu e a s th e y — x lin e transfer s the outpu t valu e a t on e stag e t o th e inpu t valu e a t th e nex t stage . Th e resulting patter n provide s a graphica l pictur e o f th e convergenc e o f th e tre e population t o th e limitin g valu e of 5000 pine trees . Figur e 3 shows the TI-8 1 cobweb grap h o f thi s transformatio n o f values .

Comparisons betwee n th e variou s possible values of the "slope " o f a first - order linea r differenc e equatio n quickl y show s tha t th e proces s converge s fo r "slopes" wher e th e absolut e valu e o f A i s les s tha n 1 .

260 J O H N A . DOSSE Y

lK=S000 .7=5000 J F I G U R E 3 .

Medicine. Suppos e tha t a perso n take s a dos e o f medicin e containin g 16 units o f a particula r dru g ever y fou r hours . Further , suppos e that , o n av - erage, th e perso n eliminate s 2 5 percent o f that dru g throug h bod y function s every four hours . Ho w many unit s o f the dru g woul d b e i n the person' s bod y after th e fourt h dose ? afte r th e 10 th dose ? I f a leve l of 8 0 units i s considere d to b e a n overdos e warnin g stage , wil l thi s perso n eve r b e i n danger ? (Se e [15].)

Insect Control . I n a n attemp t t o reduc e th e sprea d o f a pesticide - resistant frui t fly, sterilize d mal e flies wer e release d t o mat e wit h fertil e females t o cu t th e growt h o f th e pes t population . I f th e effec t o f thi s effor t in a controlle d environmen t i s t o reduc e th e overal l populatio n b y 3 % pe r month, wha t reductio n i n th e populatio n coul d b e expecte d i n half-a-year ? in on e year ?

Investments. Whic h o f th e followin g i s a bette r investmen t schem e t o employ ove r a thirt y yea r period ? Inves t $50 0 pe r mont h ove r th e entir e period a t 5 % interest compounde d monthl y o r inves t $100 0 pe r mont h ove r the las t 1 5 years a t 6 % interes t compounde d monthly .

4. T h e Structur e o f Differenc e Equation s

First-order linea r differenc e equation s f n = Af n-\ + B hav e a rich theor y that i s carefull y explicate d i n a numbe r o f source s [8 , 15] . Th e searc h fo r a solutio n fo r th e equatio n lead s t o a ric h understandin g o f wha t "solution " means. Conside r th e differenc e equatio n f n = 2 / n _ i + 1 , wher e / 0 = 1 . Working backwar d fro m f n — 2(2/ n_2 + 1) + 1 wit h continue d resubstitutio n and consolidation , on e arrive s a t th e formul a

/ „ = 2 " + 1 - 1 .

This "solution " fo r th e differenc e equatio n i s a functio n o f n tha t satisfie s the origina l differenc e equatio n an d th e initia l conditio n / 0 = 1 . I f the initia l value / o i s differen t fro m 1 , then th e situatio n i s somewhat different . I n thi s setting, f n — 2

n/o + 2n —1 satisfie s th e differenc e equatio n fo r an y /Q . Hence , as in indefinite integration , wher e there i s a family o f solutions i n the absenc e of a fixed initia l condition , her e too ther e i s a famil y o f functions tha t satisf y

MAKING A DIFFERENCE WIT H DIFFERENC E EQUATION S 26 1

the differenc e equation ; an d whe n a n initia l conditio n i s specified, ther e i s in fact a uniqu e solutio n [8] . I n general , a first-order, linea r differenc e equatio n of th e for m

fn — Af n-i + B,

with a give n initia l valu e o f /o , ha s th e uniqu e solution :

= f A-Vo + ^ j ) - ^ ifA^l \ fo + nB i f A = l.

Examination o f thi s solutio n show s tha t i n th e cas e tha t A = 1 an d B = d , th e situatio n describe d b y th e differenc e equatio n i s th e genera l arithmetic sequence . I f it s initia l ter m / o i s a , it s ( n + l)s t ter m f n i s give n by a + nd. If , o n th e othe r hand , A = r an d B = 0 , the n w e hav e th e general geometri c sequence , wit h th e exceptio n tha t whe n A = r = 1 th e situation represent s th e constan t arithmeti c sequenc e wit h eac h ter m equa l to r . Whe n A = r ^ 1 an d / o = a , the n th e ( n + l)s t ter m i s give n b y arn. Thus , th e abov e solutio n include s genera l form s fo r th e n th term s o f arithmetic an d geometri c sequences of a+(n—l)d an d ar n — 1 as we normall y see the m i n schoo l mathematics .

The instructo r wh o review s th e theor y o f differenc e equation s [8 , 15 ] will find tha t th e subjec t provide s a hos t o f opportunities fo r innovativ e an d explorative approache s i n course s a t a mor e advance d level . Spreadsheet s can b e use d t o calculat e th e firs t 1 5 values, fo r example , o f a homogeneou s first-order differenc e equation , tha t is , one wher e ever y ter m o f the equatio n involves th e recursiv e variabl e X{ for som e valu e o f i. Student s ca n b e aske d to conjectur e a genera l for m fo r th e genera l solution , a s wel l a s particula r solutions give n specifi c initia l values , fo r suc h equations . Usin g th e metho d of undetermined coefficient s [8] , forms ca n b e generalized t o describ e genera l and specifi c solution s t o non-homogeneous , first-order differenc e equations . Again, calculatin g th e first 1 5 value s o f th e non-homogeneou s differenc e equations, student s ca n us e thei r solution s fo r thei r homogeneou s portion s and th e "residuals " observe d betwee n thes e values and th e calculate d values , term-by-term, fo r th e first 1 5 terms o f the non-homogeneou s equations . Fit - ting model s t o thes e "residuals, " student s ca n develo p form s fo r th e genera l and particula r solution s t o non-homogeneous , linea r differenc e equations .

These procedure s involv e patter n recognition , modeling , an d functio n knowledge, an d exemplif y a connected approac h t o doin g mathematics. Stu - dents com e t o se e th e developmen t o f the theor y a s a unifie d process , rathe r than a s th e developmen t o f a serie s o f specifi c formula s fo r solutio n form s for differenc e equations . Th e proces s ca n b e furthe r extende d b y examinin g the solutio n o f th e secon d orde r differenc e equations , usin g th e character - istic equatio n an d system s o f first-order equation s wit h elementar y linea r algebra, employin g eigenvalue s an d eigenvectors . Eac h ste p u p th e ladde r of difference equation s i s only a sligh t generalizatio n o f the proces s fo r mod - eling an d solvin g problem s a t th e lowe r level . Student s ge t a chanc e t o se e

262 JOHN A . DOSSE Y

how mathematic s i s developed , wit h th e mathematic s a t on e leve l support - ing proble m solvin g a t th e nex t level ; thus , differenc e equation s provid e a strong mode l fo r illustratin g a number o f the issue s central t o a mathematic s program base d o n th e NCT M Standards .

5. Conclusio n

The stud y o f first-order, linea r differenc e equation s provide s a stron g thread throug h th e curriculum , pullin g man y divers e topic s togethe r int o a mor e cohesiv e whole . Th e natur e o f arithmeti c an d geometri c patterns , viewed a s sequences , ca n b e analyze d i n term s o f th e typ e o f chang e takin g place in each. A constant differenc e d signals an underlying arithmeti c mode l of the form y n = y n-\+d. Whe n the quotient o f consecutive terms y nIVn-\ i s a constan t rati o r , th e mode l signale d i s geometri c o f th e for m y n = ry n-\. In othe r settings , th e determinatio n o f th e typ e o f mode l t o describ e th e relationship betwee n successiv e state s o f th e situatio n require s considerabl y more effor t an d insigh t o n th e par t o f th e modeler . Th e examinatio n o f the term s o f eac h sequenc e throug h th e iteratio n o f th e differenc e equation s defining the m allow s th e consideratio n o f th e relationship s betwee n thes e two dominan t patterns . I n bot h case s student s hav e a n opportunit y t o loo k at situation s wher e tim e enter s a s a variabl e i n discret e setting s wher e i t i s generally eas y bot h t o comput e an d interpre t th e values .

The consideratio n o f the rat e o f change i n th e middl e grades , alon g wit h data an d it s graphi c representation , provide s a n earl y introductio n t o th e concepts whic h late r wil l generaliz e int o slop e i n linea r equation s i n algebr a and t o th e derivativ e i n th e stud y o f curve s i n calculus . Th e movemen t to finding close d for m functiona l representation s fo r f n a s a functio n o f n , starting wit h a differenc e equation , give s a ne w meanin g t o th e natur e o f a "solution " fo r mos t students . Thi s i s especiall y powerful , a s student s note th e differenc e betwee n specific , o r particular , an d genera l solutions . This set s a soli d bas e fo r late r wor k wit h indefinit e integral s an d differentia l equations i n th e calculus . Finally , th e cobwe b grap h approac h t o studyin g successive value s resultin g fro m differenc e equation s provide s stron g insigh t into functions , thei r representations , an d th e long-ter m behavio r o f discret e processes.

The study of first-order linea r difference equation s can be used t o connec t the stud y o f arithmetic an d geometri c sequences , an d t o se t th e stag e fo r th e concept o f genera l an d specifi c solution s t o bot h differenc e an d differentia l equations. Bu t mor e tha n this , thi s clas s o f differenc e equation s provide s a workplace fo r examinin g chang e an d th e natur e o f change .

R e f e r e n c e s

[1] Bannard , D . N. , "Makin g Connection s Throug h Iteration" , i n [10] . [2] Boyer , C . B. , A History of Mathematics, J . Wile y & Sons , Ne w Yor k NY , 1968 .

MAKING A D I F F E R E N C E W I T H D I F F E R E N C E E Q U A T I O N S 26 3

[3] Cambridg e Conferenc e o n Schoo l Mathematics . Goals for School Mathematics: the Report of the Cambridge Conference on School Mathematics, Houghto n Mifflin , Boston MA , 1963 .

[4] Chrystal , G. , Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges, 7t h ed. , 2 vols., Chelsea , Ne w Yor k NY , 196 4 (origina l wor k published, 1886) .

[5] Dossey , J . A. , "Discret e Mathematics : Th e Mat h fo r ou r Time" , i n [10] . [6] , A . D . Otto , L . E . Spence , an d C . Vande n Eynden , Discrete Mathematics,

(2nd ed.) . Harpe r Collins , Ne w Yor k NY , 1993 . [7] Garland , T . H. , Fascinating Fibonaccis, Dal e Seymou r Publications , Pal o Alt o CA ,

1987. [8] Goldberg , S. , Introduction to Difference Equations, J . Wile y & Sons , Ne w Yor k NY ,

1958. [9] Hall , H . S. , an d S . R . Knight , Higher Algebra: A Sequel to Elementary Algebra for

Schools, Macmilla n & Company , New Yor k NY , 196 4 (origina l wor k published , 1887) . [10] Hirsch , Christia n R. , an d Margare t J . Kenney , eds . Discrete Mathematics Across the

Curriculum, K-12, Yearboo k o f th e Nationa l Counci l o f Teacher s o f Mathematics , Reston VA , 1991 .

[11] Hoggatt , V . E. , Jr. , Fibonacci and Lucas Numbers, Houghto n Mifflin , Bosto n MA , 1969.

[12] Nationa l Counci l o f Teacher s o f Mathematics , Curriculum and Evaluation Standards for School Mathematics, NCTM , Resto n VA , 1989 .

[13] , Discrete Mathematics and the Secondary Mathematics Curriculum, NCTM , Reston VA , 1990 .

[14] Sandefur , J . T. , Jr. , "Discret e Mathematics : Th e Mathematic s w e al l Need, " I n C . Hirsch an d M . Zweng , eds. , The Secondary School Mathematics Curriculum, NCTM , Reston VA , 1985 .

[15] , Discrete Dynamical Systems: Theory and Applications, Oxfor d Universit y Press, Ne w Yor k NY , 1990 .

[16] Vorob'ev , N . N. , Fibonacci Numbers, Blaisdel , Ne w Yor k NY , 1951 .

ILLINOIS S T A T E U N I V E R S I T Y , N O R M A L , I L 6179 0

E-mail address: j d o s s e y @ m a t h . i l s t u . e d u

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 3 6 , 199 7

Discrete M a t h e m a t i c a l Modelin g I n T h e Secondary Curriculum : Rational e an d Example s

from T h e Core-Plu s M a t h e m a t i c s P r o j e c t

Eric W . Har t

Discrete Mathematica l Modelin g i n t h e Secondar y Curriculu m

Discrete mathematic s i s a n importan t branc h o f mathematic s tha t ha s been widel y recommende d fo r inclusio n i n th e secondar y curriculu m [13 , 2 , 6, 11] . Bu t whic h part s o f discret e mathematic s shoul d b e included , an d how shoul d the y b e incorporated ? Thi s articl e attempt s t o answe r thes e two questions . Th e answer s propose d hav e bee n applie d a s guideline s fo r weaving discrete mathematics int o a new integrated hig h school mathematic s curriculum: Th e Core-Plu s Mathematic s Projec t curriculum .

W h a t I s T h e Core-Plu s M a t h e m a t i c s P r o j e c t ?

The Core-Plu s Mathematic s Projec t (CPMP) , funde d b y severa l grant s from th e National Scienc e Foundation, i s developing, fiel d testing , an d evalu - ating a n integrate d four-yea r hig h schoo l mathematic s curriculum . Th e firs t three year s ar e designe d t o fulfil l th e mathematica l need s o f bot h college - bound an d employment-bound students , while the fourth-year cours e focuse s on th e transitio n t o colleg e mathematics . Th e progra m feature s mathemat - ical modeling , studen t investigation , integrate d content , an d appropriat e use o f th e numeric , graphic , programming , an d lin k capabilitie s o f moder n calculators.

Interwoven strand s o f algebr a an d functions , geometr y an d trigonome - try, probabilit y an d statistics , an d discret e mathematic s ar e develope d eac h year i n th e contex t o f realisti c applications . Connecte d 5-wee k unit s ar e comprised o f severa l multi-da y lesson s centere d o n cor e mathematica l ideas . Units ar e connecte d b y commo n theme s o f data , shape , change , chance , and representation ; b y commo n topic s suc h a s matrices , symmetry , an d

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 .

265

https://doi.org/10.1090/dimacs/036/20

266 ERIC W . HAR T

curve-fitting; an d b y mathematica l mode s o f though t suc h a s reasoning , vi - sualization, an d recursiv e thinking . Guide d studen t investigation s lea d t o construction o f importan t mathematic s tha t make s sens e t o student s an d i n turn enable s the m t o mak e sens e o f ne w situation s an d problems .

Each cours e i s developed throug h a 4-yea r cycl e of writing , fiel d testing , and revising. 1 Fiel d testin g i s carrie d ou t i n abou t 5 0 school s nationwide . As o f Fal l 1997 , Course s 1 an d 2 ar e complete d an d publishe d b y Every - day Learnin g Corporation , unde r th e titl e Contemporary Mathematics in Context Course s 3 an d 4 wil l b e complete d i n subsequen t years .

Extensive evaluatio n researc h i s bein g conducte d i n 3 6 school s aroun d the country . Result s fro m th e firs t tw o year s o f th e three-yea r nationa l fiel d test sho w tha t th e curriculu m i s well-receive d an d successful , a s measure d by surveys , cas e studies , studen t interviews , teache r reports , loca l tests , and nationa l norme d tests . Fo r example , dat a sho w tha t CPM P students ' growth o n th e mathematic s portio n o f th e Iow a Tes t o f Educationa l Devel - opment (ITED ) i s significantl y greate r tha n tha t o f comparabl e student s i n traditional curricul a [21] .

W h a t i s Discret e M a t h e m a t i c s ?

Before decidin g whic h part s o f discret e mathematic s t o includ e i n an y curriculum, w e nee d t o kno w wha t discret e mathematic s is. 2 Ther e hav e been a variet y o f definition s give n ove r th e years : "Discret e mathematic s i s the mathematic s o f decisio n makin g fo r finit e settings " ([2] , p . l ); "Discret e mathematics describe s processe s tha t consis t o f a sequenc e o f individua l steps" ([4] , p. xv)\ "Discret e mathematic s potentiall y involve s th e stud y o f objects an d idea s tha t ca n b e divide d u p int o 'separate ' o r 'discontinuous ' parts" ([3] , p . 1) ; " A goo d shor t answe r contrast s 'discrete ' topic s wit h those tha t ar e 'continuous. ' " ([1] , p. ix).

The workin g definition o f discrete mathematic s i n the CPM P curriculu m is th e following : Discret e mathematic s consist s o f concept s an d technique s for modelin g an d solvin g problem s involvin g finit e processe s an d discret e phenomena. Mor e specifically , discret e mathematic s deal s wit h problem s that involv e enumeration , decision-makin g i n finit e settings , relationship s among a finit e numbe r o f elements , an d sequentia l change . Centra l theme s of discret e mathematic s i n CPM P ar e existenc e (Doe s a solutio n exist?) , algorithmic proble m solvin g (Ca n yo u efficientl y construc t a solution?) , an d optimization (Whic h solutio n i s best?) .

d e v e l o p m e n t tea m member s ar e Christia n Hirsch , director , Wester n Michiga n Uni - versity; Gai l Burrill , Universit y o f Wisconsin , Madison ; Arthu r Coxford , Universit y o f Michigan; Jame s Fey , Universit y o f Maryland ; Eri c Hart , Wester n Michiga n Univer - sity; Harol d Schoen , Universit y o f Iowa ; an d An n Wa t kins, Californi a Stat e University , Northridge. Th e Core-Plu s Mathematic s Projec t i s supporte d b y NS F gran t no . MDR - 9255257.

2 Editors' note : Se e als o th e article s b y Maure r an d Rosenstei n i n Sectio n 3 o f thi s volume, whic h addres s th e issu e o f definin g discret e mathematics .

D I S C R E T E MATHEMATICA L MODELIN G I N T H E SECONDAR Y C U R R I C U L U M 26 7

W h i c h Discret e M a t h e m a t i c s Topic s Shoul d B e Include d i n t h e Secondary Curriculum ?

The lis t o f topics and area s that hav e been liste d unde r th e titl e "discret e mathematics" i s quit e lon g indeed , includin g grap h theory , gam e theory , difference equation s (als o calle d recurrenc e relations) , combinatorics , oper - ations research , managemen t science , logic , algorithms , matrices , applie d modern algebra , finite probability , codin g theory , linea r programming , an d so on. Wha t fro m thi s lis t shoul d b e include d i n th e hig h schoo l mathemat - ics curriculum ? Thi s questio n clearl y ha s man y possibl e answers , bu t ther e is som e consensus .

The NCT M Curriculu m Standard s [13 ] recommend s th e followin g top - ics: grap h theory , matrices , sequences , recurrenc e relations , algorithms , sys - tematic counting , finit e probability , an d linea r programming . A numbe r of NS F discrete-mathematic s teacher-enhancemen t project s o f th e 1990' s — fo r example , thos e directe d b y Har t [8] , Kenne y [10] , Rosenstei n [17] , and Sandefu r [18] , have ha d a conten t focu s o n grap h theory , iteratio n an d recursion, socia l choic e theory , matrices , an d combinatorics . Thes e sam e five majo r topic s mak e u p mos t o f th e first publishe d hig h schoo l discret e mathematics tex t [1] .

Roughly th e sam e topic s ar e develope d i n th e Core-Plu s Mathematic s Project curriculum . I n particular , th e CPM P curriculu m includes :

a: grap h theor y — usin g vertex-edg e model s t o stud y relationship s a - mong a finite numbe r o f elements , a s i n a transportatio n networ k o r a predator-pre y foo d web ;

b : socia l choic e theor y — suc h a s th e mathematic s o f voting , fai r divi - sion, apportionment , an d cooperatio n an d competition ;

c: combinatoric s — systemati c counting ; d: recursio n — the method o f describing sequential chang e by indicatin g

how th e nex t stag e o f a proces s i s determine d fro m previou s stages ; and

e: matrice s — use d t o represen t an d solv e problem s fro m a variet y o f real-world setting s whil e connectin g importan t idea s fro m differen t strands o f mathematics .

Before continuin g th e discussio n o f what discret e mathematics t o includ e and ho w t o includ e it , w e shoul d conside r a mor e fundamenta l question .

W h y Shoul d Discret e M a t h e m a t i c s B e Par t o f t h e Hig h Schoo l Curriculum?

The hig h schoo l mathematic s curriculu m i s alread y quit e full . Wh y should w e mak e roo m fo r discret e mathematics ? Ther e ar e a t leas t thre e compelling reasons : it' s goo d mathematics ; it' s usefu l mathematics ; an d it' s pedagogically powerfu l mathematics .

268 E R I C W . HAR T

D i s c r e t e mathematic s i s g o o d m a t h e m a t i c s . Th e five major topic s mentioned abov e ar e thriving , activ e researc h area s i n mathematics . Th e mathematics i s prett y an d profound . Studyin g discret e mathematic s wil l give student s a broade r vie w o f th e whol e field o f mathematics .

Discrete m a t h e m a t i c s i s usefu l m a t h e m a t i c s . Discret e mathemat - ics provide s ric h an d powerfu l mathematica l model s tha t ar e invaluabl e fo r making sens e o f th e worl d w e liv e in . Application s o f thes e five topics com e from a variet y o f settings , includin g projec t management , communicatio n networks, scheduling , routing , manufacturing , lotteries , voting, fai r division , finance, populatio n growth , inventor y control , wildlif e management , an d so - cial relations , t o nam e jus t a few . A s Dosse y states , "Discret e mathematic s is th e mat h o f ou r time " ([3] , p. 1) .

Discrete m a t h e m a t i c s i s pedagogicall y powerfu l m a t h e m a t i c s . Discrete mathematic s i s a poten t vehicl e fo r achievin g man y goal s o f math - ematics education . I n th e cours e o f investigatin g importan t concept s i n dis - crete mathematics , student s lear n an d appl y powerfu l habit s o f mind , lik e mathematical modeling , algorithmi c proble m solving , an d recursiv e think - ing. Man y students ' ol d belief s abou t mathematic s ar e challenge d an d changed. The y se e tha t mathematic s i s activ e an d alive , a s the y com e face - to-face wit h som e o f th e curren t frontier s o f mathematic s in , fo r example , graph theor y o r modelin g wit h recursion . The y se e that mathematic s i s use- ful an d modern , a s the y stud y th e ubiquitou s contemporar y application s o f discrete mathematics . Eve n th e belief s student s ma y hav e abou t thei r ow n ability t o lear n mathematic s ca n b e dramaticall y change d whe n the y find that man y discret e mathematic s topic s ar e ne w an d accessibl e an d d o no t have a plethor a o f technica l prerequisites .

How Shoul d Discret e M a t h e m a t i c s b e Incorporate d Int o t h e Secondary Curriculum ?

Two complementar y answer s ar e propose d here . Discret e mathematic s should b e wove n int o a n overal l integrate d mathematic s curriculum , an d th e emphasis shoul d b e o n discret e mathematica l modeling . Bot h approache s are implemente d i n th e CPM P curriculum , a s no w discussed .

Specific unit s focusin g o n discret e mathematic s ar e wove n int o eac h o f the integrate d CPM P courses . Fo r example , ther e ar e unit s entitle d Graph Models, Matrix Models, Network Optimization, Discrete Models of Change, Modeling Public Opinion, an d Counting Models, al l o f whic h ar e connecte d to eac h othe r a s wel l a s t o th e othe r CPM P units . I n addition , topic s and theme s o f discret e mathematics , suc h a s matrices , recursiv e thinking , optimization, an d algorithmi c proble m solving , permeat e th e entir e curricu - lum. Finally , th e discret e mathematic s stran d o f th e CPM P curriculu m is connecte d t o th e othe r strand s o f algebr a an d functions , geometr y an d trigonometry, an d statistic s an d probabilit y b y th e commo n theme s o f data , shape, representation , an d change .

DISCRETE MATHEMATICA L MODELIN G I N TH E SECONDAR Y CURRICULU M 26 9

The emphasi s throughou t th e CPM P discret e mathematic s stran d i s o n discrete mathematica l modeling . Student s ar e engaged , bot h i n group s an d individually, i n makin g sens e o f realisti c situation s b y constructing , operat - ing on , analyzing , an d interpretin g discret e mathematica l models . Th e res t of this articl e i s devoted t o tw o example s o f discret e mathematica l modelin g in th e CPM P curriculum .

Discrete Mathematica l Modelin g i n C P M P

Two example s ar e presente d here . On e exampl e involve s modelin g wit h recursion, take n fro m a uni t i n Cours e 3 entitled Discrete Models of Change, and th e othe r use s vertex-edge grap h models , fro m th e Graph Models unit i n Course 1 . Bot h ar e pulle d ou t o f contex t fro m th e complet e unit s i n whic h they appear , bu t the y giv e a flavor o f discrete mathematica l modelin g i n th e CPMP curriculum .

Each exampl e als o illustrate s th e styl e o f activ e learnin g an d teachin g that i s characteristi c o f th e CPM P curriculum . Lesson s ar e launche d wit h a brie f whole-clas s brainstormin g sessio n relate d t o a give n real-worl d situ - ation. Thi s i s buil t int o th e curriculu m materials , a s see n below , i n boxe s entitled "Thin k Abou t Thi s Situation. " Th e goa l i s t o motivat e th e lesso n and get som e important question s on the table. Next , student s wor k in team s on guide d investigation s relate d t o th e launchin g situation . Throug h thes e investigations student s explor e an d appl y importan t mathematica l concept s and methods . Th e investigation s ar e followe d b y a "Checkpoint " section . The checkpoin t consist s o f a fe w question s tha t summariz e th e lesso n s o far . Teachers typciall y lea d a whole-clas s discussio n o f th e checkpoin t questions . The goa l i s t o provid e a class-generate d summar y an d t o ensur e tha t stu - dents hav e indee d learne d th e targete d concept s an d methods . Finally , t o make sur e tha t student s ca n appl y wha t the y hav e learne d o n thei r own , there i s a n additiona l brie f investigatio n entitle d "O n You r Own. " Thes e features o f th e CPM P curriculu m ar e illustrate d i n th e followin g sampl e student investigation s (addresse d t o a studen t reader) .

E X A M P L E 1 : M o d e l i n g Sequentia l Chang e U s i n g Recursio n

We liv e i n a changin g world . Mathematic s ca n b e use d t o hel p describ e and understan d pattern s o f change . Example s yo u hav e alread y studie d in - clude usin g equations , tables , an d graph s t o investigat e linea r an d exponen - tial pattern s o f change , usin g coordinate s an d matrice s t o stud y geometri c transformations, an d usin g trigonometr y t o stud y periodi c change . Anothe r important patter n o f chang e i s sequentia l change , fo r example , chang e fro m year t o year . Yo u hav e alread y use d equation s involvin g th e word s NO W and NEX T t o stud y thi s typ e o f change . I n thi s uni t yo u wil l continu e th e study o f sequentia l change .

27 0 ERIC W . HAR T

Think A b o u t Thi s Situatio n Wildlife managemen t hat s become a n increasingl y importan t issu e a s modern civilizatio n put s greate r demand s o n wildlif e habitat . A s an example , conside r a fishing pon d tha t i s stocke d fro m a nearb y hatchery. Suppos e yo u ar e i n charg e o f managing th e fish populatio n in th e pond .

a: Wha t ar e som e factor s t o conside r i n managin g th e fish pop - ulation i n th e pond ? Lis t a s man y factor s a s yo u can .

b : Ho w coul d yo u figure ou t th e curren t siz e o f th e fish popula - tion?

c: Wh y woul d i t b e usefu l t o b e abl e t o predic t th e year-to - year change s i n th e fish population ? Wh y woul d knowledg e of long-term populatio n change s b e useful ?

I N V E S T I G A T I O N 1.1 : M o d e l i n g P o p u l a t i o n Change . I n thi s lesson yo u wil l buil d an d us e a mathematica l mode l t o hel p yo u predic t th e changing fish population .

1. S o fa r yo u hav e ver y littl e informatio n abou t th e fishing pon d tha t you ar e suppose d t o manage . Wha t additiona l informatio n d o yo u need i n orde r t o predic t change s i n th e siz e o f th e fish populatio n over time ? Mak e a list .

2. A typical first ste p i n mathematica l modelin g i s simplifying th e prob - lem an d decidin g o n som e reasonabl e assumptions . Thre e piece s o f information tha t yo u ma y hav e liste d abov e are : th e initia l siz e of th e fish population i n the pond , th e annua l growt h rat e o f the population , and th e annua l restockin g amount , tha t is , th e numbe r o f fish tha t are adde d t o th e pon d eac h year . Fo r th e res t o f thi s investigation , use jus t th e followin g thre e assumptions :

• Ther e ar e 300 0 fish currentl y i n th e pond . • Regardles s o f restocking, th e populatio n decrease s b y 20 % each

year du e t o th e combine d effec t o f al l causes , includin g natura l births an d death s an d fish bein g caught .

• 100 0 fish ar e adde d a t th e en d o f eac h year . Using thes e assumptions , yo u ca n buil d a mathematica l mode l t o analyze th e populatio n growt h i n th e pond .

(a) Estimat e th e fish populatio n i n th e pon d afte r 1 year. Afte r 2 years.

(b) Wha t i s th e populatio n afte r 3 years? Explai n ho w yo u figured it out .

(d) Us e th e equatio n fro m par t (c ) an d th e AN S ke y o n you r cal - culator t o find th e populatio n afte r 7 years . Explai n ho w th e

DISCRETE MATHEMATICA L MODELIN G I N TH E SECONDAR Y CURRICULU M 27 1

keystrokes yo u us e o n th e calculato r correspon d t o th e word s NOW an d NEX T i n th e equation .

3. Thin k abou t th e long-ter m populatio n o f fis h i n th e pond . (a) D o yo u thin k th e populatio n wil l gro w withou t bound ? Leve l

off? Di e out ? Mak e a conjectur e abou t th e long-ter m popula - tion.

(b) Comput e th e long-ter m population . Wa s you r conjectur e cor - rect? Explain , i n term s o f th e fishin g pon d ecology , wh y th e long-term populatio n yo u hav e compute d i s reasonable .

4. Doe s th e fis h populatio n chang e faste r aroun d yea r 5 or aroun d yea r 25? Ho w ca n yo u tell ?

I N V E S T I G A T I O N 1.2 : W h a t i f ... ? Thin k abou t wha t happen s t o the long-ter m populatio n i f certai n condition s change .

1. Wha t ar e th e thre e ke y condition s i n thi s problem ? 2. Wha t d o you think wil l happen t o th e long-ter m populatio n i f the ini -

tial populatio n i s differen t bu t al l othe r condition s remai n th e same ? Make a n educate d guess , an d the n chec k you r gues s b y findin g th e long-term populatio n whe n th e initia l populatio n i s 0 , 2000 , 4000 , and 10,000 . Describ e th e patter n o f chang e i n long-ter m populatio n as initia l populatio n varies .

3. Wha t happen s t o th e long-ter m populatio n i f th e annua l re-stockin g amount i s 500 , an d al l othe r condition s ar e a s i n th e origina l as - sumptions? Ho w abou t i f th e annua l re-stockin g amoun t i s 2000 ? 4000? Describ e th e relationshi p betwee n long-ter m populatio n an d re-stocking amount .

4. Wha t happen s t o th e long-ter m populatio n i f th e annua l decreas e rate i s 10% , an d al l othe r condition s ar e th e sam e a s i n th e origina l assumptions? Ho w abou t i f th e annua l decreas e rat e i s 40% ? 60% ? Describe an y pattern s tha t yo u see .

5. No w conside r th e cas e wher e condition s ar e suc h tha t th e fis h popu - lation show s a n annua l rat e o f increase.

a: Wha t d o yo u thin k wil l happe n t o th e long-ter m populatio n i f the populatio n increases a t a n annua l rate ? Mak e a conjectur e and the n tes t i t b y tryin g a t leas t tw o differen t annua l increas e rates.

b : Writ e equation s usin g NO W an d NEX T tha t represen t you r two tes t cases .

c: D o yo u thin k i t i s reasonable t o mode l th e populatio n o f a fis h in a pon d wit h a n annua l rat e o f increase ? Wh y o r wh y not ?

272 ERIC W . HAR T

y/ Checkpoin t Consider thi s equation : NEX T = 0. 6 NO W + 1500 .

a: Describ e a fish populatio n proble m tha t coul d b e modele d b y this equation .

b : Wha t additiona l informatio n i s neede d t o b e abl e t o us e thi s equation t o predic t long-ter m population ?

c: Wha t additiona l informatio n i s neede d t o b e abl e t o us e thi s equation t o predic t th e populatio n i n 5 years ?

d: Fo r a fish populatio n situatio n modele d b y a n equatio n lik e the on e above :

• I f th e initia l populatio n doubles , wha t wil l happe n t o the long-ter m population ?

• I f the annual re-stockin g amount doubles , what wil l hap- pen t o th e long-ter m population ?

• I f the annua l populatio n decreas e rate doubles , what wil l happen t o th e long-ter m population ?

e: Ho w would yo u modif y th e equatio n abov e i f it i s to represen t a situatio n wher e th e fish populatio n increase s annuall y a t a rate o f 15% ? Wha t effec t doe s suc h a n increas e rat e hav e o n the long-ter m population ?

Be prepare d t o shar e you r group' s thinkin g wit h th e entir e class .

The fish populatio n proble m yo u hav e investigate d involve s sequentia l change, since the change takes place sequentially or step-by-step. I n this cas e the step-by-ste p chang e i n populatio n i s recorde d year-by-year . Yo u hav e used th e term s NO W an d NEX T t o describ e th e sequentia l change . Thi s method o f describing the nex t ste p i n terms of previous step s is called recur - sion. Situation s involvin g sequentia l chang e ar e sometime s calle d discret e dynamical systems . A discret e dynamica l syste m i s a situatio n (system ) involving chang e (dynamical) , wher e th e natur e o f th e chang e i s step-by - step (discrete) . A n importan t par t o f analyzin g discret e dynamica l system s is determinin g long-ter m behavior , lik e wha t yo u di d whe n yo u foun d th e long-term populatio n o f fish.

On You r Own . A hospita l patien t i s takin g a n antibioti c t o trea t a n infection. H e initially takes a 30mg dose, and then take s anothe r lOm g at th e end o f ever y si x hou r perio d thereafter . Throug h natura l bod y metabolis m 20% o f th e antibioti c i s eliminate d fro m hi s syste m ever y si x hours .

a: Estimat e th e amoun t o f the antibioti c i n hi s syste m afte r th e first si x hours. Afte r th e secon d si x hours .

b : Writ e a n equatio n usin g th e word s NO W an d NEX T tha t model s this situation .

c: Fin d th e amoun t o f antibioti c i n hi s syste m afte r tw o weeks .

DISCRETE MATHEMATICA L MODELIN G I N TH E SECONDAR Y CURRICULU M 27 3

d: Hi s docto r decide s t o modif y th e prescriptio n s o tha t th e long-ter m amount o f antibioti c i n hi s syste m wil l b e abou t 2 5 mg . Ho w shoul d the prescriptio n b e modified ?

N o t e t o t h e Teacher : Som e subsequen t investigation s i n thi s uni t include th e followin g idea s an d topics 3:

• Continue d developmen t an d us e o f subscrip t notation ; • A discret e (recursive ) vie w o f linear , exponential , an d powe r func -

tions, throug h investigatio n o f arithmeti c sequences , geometri c se - quences, an d finite differences ;

• Functio n iteration , includin g som e analysi s o f fixed points an d cycles ; • Graphica l analysi s (includin g "cobweb " diagrams) ; • Muc h o f th e presen t wor k wit h recursio n i s summarize d an d formal -

ized i n term s o f affin e recurrenc e relations , A n = rA n-\ + b.

E X A M P L E 2 : Managin g Conflict s w i t h Vertex-Edg e Graph s

Have yo u eve r notice d ho w man y differen t radi o channel s ther e are ? Each radi o statio n ha s it s ow n transmitte r whic h broadcast s o n a particula r channel, o r frequency. Th e Federa l Communication s Commissio n (FCC ) makes sur e tha t th e broadcas t fro m on e radi o statio n doe s no t interfer e with th e broadcas t fro m an y othe r radi o station . Thi s i s don e b y assignin g an appropriat e frequenc y t o eac h station . Th e FC C require s tha t i f tw o stations ar e withi n transmittin g rang e o f each other , the y mus t us e differen t frequencies. Otherwise , yo u migh t tun e int o "ROC K 101.7 " an d ge t Mozar t instead!

Think A b o u t Thi s Situatio n Seven ne w radi o station s ar e plannin g t o star t broadcastin g i n th e same regio n o f th e country . Th e FC C want s t o assig n a frequenc y to eac h statio n s o tha t n o tw o station s interfer e wit h eac h other . The FC C als o want s t o assig n th e fewes t possibl e numbe r o f ne w frequencies.

a: Wha t factor s nee d t o b e considere d befor e th e frequencie s ca n be assigned ?

b : Wha t metho d ca n th e FC C us e t o assig n th e frequencies ?

I N V E S T I G A T I O N 2.1 : Buildin g a Mode L Suppos e tha t becaus e of geographi c condition s an d th e strengt h o f eac h station' s transmitter , th e FCC determine s tha t station s withi n 50 0 mile s o f eac h othe r mus t b e as - signed differen t frequencies , otherwis e thei r broadcast s wil l interfer e wit h each other . Th e locatio n o f the seve n station s i s shown o n th e gri d i n Figur e 1. A sid e o f eac h smal l squar e o n th e gri d represent s 10 0 miles .

3 Editors' note : Se e th e articl e b y Dossey , i n thi s volume , fo r a furthe r discussio n o f sequences, iteration , an d recurrenc e relations .

ERIC W . HAR T

Tl 1

1 w

\ 1

1 w

u , %w

^ f

\ %w

l r

z - I r

F I G U R E 1 . Scale : gri d line s ar e 10 0 miles apart .

Working o n you r own , figure ou t ho w man y differen t frequencie s ar e needed fo r th e seve n radi o stations . Remembe r tha t station s 50 0 miles o r less apart mus t hav e differen t frequencies , bu t station s mor e than 50 0 mile s apar t ca n us e th e sam e frequency . Try to use as few frequencies as possible. Compare you r answe r wit h other s i n you r group .

(a) Di d everyone us e the sam e number o f frequencies? Reac h agree - ment i n you r grou p abou t th e minimu m numbe r o f frequencie s needed fo r th e seve n radi o stations .

(b) Give n tw o particula r stations , i s i t possibl e tha t on e perso n as - signs them th e sam e frequency an d anothe r perso n assign s the m different frequencies , an d ye t bot h assignment s ar e acceptable ? Explain.

It i s possible i n thi s cas e t o find th e minimu m numbe r o f frequen - cies by tria l an d error . However , a mor e systemati c metho d i s neede d for mor e complicate d situations , suc h a s whe n ther e ar e man y mor e radio stations . On e wa y t o solv e thi s proble m mor e systematicall y i s to mode l th e proble m wit h a grap h simila r t o thos e i n th e previou s lesson. Remember , t o mode l a proble m wit h a graph , yo u mus t first decide wha t th e vertice s an d edge s represent . Working o n you r own , begi n modelin g thi s proble m wit h a graph .

(a) Wha t shoul d th e vertice s represent ? (b) Ho w will you decid e whether o r not t o connec t tw o vertices wit h

an edge ? Complet e thi s statement : Two vertices are connected by an edge if ...

DISCRETE MATHEMATICA L MODELIN G I N TH E SECONDAR Y CURRICULU M 27 5

4. Compar e you r grap h wit h other s i n you r group . (a) Di d everyon e i n your grou p defin e th e vertice s an d edge s i n th e

same way ? Discus s an y differences . (b) Fo r a give n situation , suppos e tha t tw o peopl e defin e th e ver -

tices an d edge s i n tw o differen t ways . I s i t possibl e tha t bot h ways accuratel y represen t th e situation ? Explai n you r reason - ing.

(c) Fo r a give n situation , suppos e tha t tw o peopl e defin e th e ver - tices an d edge s i n the sam e way . I s i t possibl e tha t thei r graph s could loo k differen t bu t bot h b e correct ? Explai n you r reason - ing.

5. A common choic e is to le t vertice s represen t th e radi o stations . Edge s might b e though t o f i n tw o ways , a s describe d i n part s (a ) an d (b ) below.

(a) Yo u migh t connec t tw o vertice s b y a n edg e wheneve r th e sta - tions the y represen t ar e 50 0 mile s o r less apart. Di d anyon e i n your grou p d o this ? I f not , dra w a grap h wher e tw o vertices ar e connected b y a n edg e wheneve r th e station s the y represen t ar e 500 mile s o r less apart .

(b) Yo u migh t connec t tw o vertice s b y a n edg e wheneve r th e sta - tions the y represen t ar e more tha n 50 0 miles apart . Di d anyon e in you r grou p d o this ? I f not , dra w a grap h wher e tw o vertice s are connecte d b y a n edg e wheneve r th e station s the y represen t are more tha n 50 0 mile s apart .

(c) Compar e th e graph s fro m part s (a ) an d (b) . • Ar e bot h graph s accurat e way s o f representin g th e situa -

tion? • Whic h grap h d o yo u thin k wil l b e mor e usefu l an d easie r

to us e a s a mathematica l mode l fo r thi s situation ? Why ? 6. Fo r th e res t o f thi s investigation , yo u wil l us e th e grap h wher e edge s

connect vertice s tha t ar e 50 0 miles o r less apart. Mak e sur e yo u hav e a nea t cop y o f thi s graph .

(a) Ar e vertice s (stations ) X an d W connecte d b y a n edge ? Ar e they 50 0 mile s o r les s apart ? Wil l thei r broadcast s interfer e with eac h other ?

(b) Ar e vertice s (stations ) Y an d Z connecte d b y a n edge ? Wil l their broadcast s interfer e wit h eac h other ?

(c) Compar e you r grap h t o th e grap h i n Figur e 2 . • Doe s thi s grap h als o represen t th e radio-statio n problem ? • Wha t criteri a ca n yo u us e t o decid e i f tw o graph s bot h

represent th e sam e situation ? 7. S o fa r yo u hav e a mode l tha t show s al l th e radi o station s an d whic h

stations ar e withi n 50 0 mile s o f eac h other . Th e goa l i s t o assig n frequencies s o tha t ther e wil l b e n o interferenc e betwee n stations . You stil l nee d t o buil d th e frequencie s int o th e model . So , a s th e las t

27 6 ERIC W . HAR T

13- x w

F I G U R E 2

step in building the grap h model , represent th e frequencie s a s colors. To color a grap h mean s t o assig n color s t o th e vertice s s o that tw o vertices connecte d b y a n edg e hav e differen t colors .

You can now think abou t th e problem in terms of coloring the ver- tices of a graph. Th e tabl e belo w contain s statement s abou t station s and frequencies i n the left-hand colum n and corresponding statement s about vertice s and color s in the right-han d column . Writ e statement s to complet e th e right-han d colum n o f th e table .

Statement s abou t station s an d frequencie s Tw o station s hav e differen t frequencies . Fin d a wa y t o assig n frequencie s so tha t station s withi n 50 0 mile s o f eac h othe r ge t differen t frequencies . Us e th e smalles t numbe r o f ne w frequencie s

Statement s abou t vertice s an d color s Tw o vertice s hav e differen t colors .

8. No w colo r you r grap h fo r th e radio-statio n problem . Tha t is , assig n a colo r t o eac h verte x s o tha t an y tw o vertice s tha t ar e connecte d b y an edg e hav e differen t colors . Tr y t o us e a s fe w color s a s possible . You ca n us e colore d pencil s o r jus t th e name s o f som e color s t o d o the coloring . Colo r o r writ e a colo r nam e nex t t o eac h vertex .

9. Compar e you r colorin g wit h tha t o f anothe r group . (a) Ar e bot h coloring s legitimate ? Tha t is , d o the y satisf y th e

condition that vertice s connected by an edge must hav e differen t colors?

(b) D o bot h coloring s us e th e sam e numbe r o f color s t o colo r th e vertices o f th e graph ?

(c) Reac h agreemen t abou t th e minimu m numbe r o f color s needed . Explain, i n writing, why the graph cannot b e colored with fewe r colors.

DISCRETE MATHEMATICA L MODELIN G I N TH E SECONDAR Y CURRICULU M 27 7

(d) Give n tw o particula r vertices , i s i t possibl e tha t on e grou p as - signs the m th e sam e colo r an d anothe r grou p assign s the m dif - ferent colors , an d ye t bot h assignment s ar e acceptable ? Wh y or wh y not ?

(e) Wha t i s th e connectio n betwee n grap h colorin g an d assignin g frequencies t o radi o stations ? A s yo u answe r thi s question , compare th e result s o f thi s activit y t o thos e i n Activit y 2 .

10. Thin k abou t th e strateg y yo u use d i n Activit y 8 t o colo r th e radio - station grap h wit h a s fe w color s a s possible .

(a) Writ e dow n a step-by-step descriptio n o f your colorin g strategy . Write th e descriptio n s o tha t you r strateg y ca n b e applie d t o graphs othe r tha n jus t th e radio-statio n graph .

(b) Us e the descriptio n o f your strateg y t o colo r a copy of the grap h in Figur e 3 .

F I G U R E 3

(c) Refin e th e direction s fo r you r colorin g strateg y s o that an y on e of you r classmate s coul d follo w th e directions .

11. Exchang e you r writte n colorin g direction s wit h anothe r group . The n do th e following :

(a) Us e th e othe r group' s direction s t o colo r a secon d cop y o f th e graph i n Figur e 3 . Th e othe r grou p wil l b e doin g th e sam e thing wit h you r directions .

(b) Compar e you r coloring s wit h th e othe r group' s colorings . (i) Ar e the y th e same ?

(ii) Ar e the y eac h legitimat e colorings ? (iii) D o the y eac h us e th e leas t numbe r o f color s possible ?

Reach agreemen t wit h th e othe r grou p abou t th e mini - mum numbe r o f color s neede d t o colo r th e graph .

(c) Discus s an y problem s tha t cam e u p wit h eithe r group' s colorin g directions. I f necessary , rewrit e you r direction s s o tha t the y work bette r an d ar e easie r t o follow .

As you sa w i n th e previou s lesson , a carefu l lis t o f directions fo r carryin g out a procedur e i s calle d a n algorithm. Designin g an d applyin g algorithm s is a n importan t metho d fo r solvin g problems . Ther e ar e man y possibl e

278 ERIC W . HAR T

algorithms fo r colorin g th e vertice s o f an y graph , includin g th e one s yo u developed.

y/ Checkpoin t a: Wha t d o the vertices , edges, an d color s represent i n the grap h

model that yo u have been usin g fo r th e radio-statio n problem ? b : Ho w doe s "colorin g a graph " hel p solv e th e radio-statio n

problem? c: I n wha t way s ca n tw o graph s diffe r an d ye t stil l bot h accu -

rately represen t a give n situation ? d: Wha t ar e som e strength s an d weaknesse s o f th e graph -

coloring algorith m create d b y you r group ? Be prepare d t o shar e you r group' s thinkin g an d colorin g algorith m with th e entir e class . Decid e a s a clas s on algorithm s tha t see m mos t efficient an d easil y understood .

Graph-coloring algorithm s continu e t o be an activ e area o f mathematica l research wit h man y applications . I t ha s prove n quit e difficul t t o find a n algorithm tha t color s the vertices of any graph usin g as few colors as possible. You ca n ofte n figure ou t ho w to d o thi s fo r a give n smal l graph , a s yo u hav e done i n thi s investigation , bu t n o on e know s a n efficien t algorith m tha t wil l color any grap h wit h th e fewest numbe r o f colors. Thi s i s a famou s unsolve d problem i n mathematics .

On You r Own : Suppos e thre e mor e radi o station s wan t t o mov e int o the sam e regio n wit h th e othe r seven . Ad d thre e mor e station s t o a cop y o f the gri d o n pag e 3 0 s o tha t a t leas t tw o o f the m ar e withi n 50 0 mile s o f on e of the existin g seve n stations . The n us e grap h colorin g t o assig n frequencie s to al l te n station s s o tha t thei r broadcast s d o no t interfer e wit h eac h other .

N o t e t o t h e Teacher : Afte r thi s initia l vertex-colorin g investigation , students the n appl y verte x colorin g to solv e a variety o f other problems , suc h as schedulin g meeting s withou t conflicts , settin g u p efficien t tou r schedules , coloring maps , an d settin g u p a n optima l emergenc y evacuatio n pla n fo r a hospital. 4 Severa l othe r grap h theor y topic s ar e studie d i n th e CPM P curriculum. Thes e topics , alon g wit h a sampl e application , ar e liste d below .

• Eule r path s (fin d efficien t sno w plo w routes ) • Critica l path s an d PER T chart s (schedul e larg e projects ) • Hamiltonia n path s (ran k player s i n a a tournament ) • Shortes t path s (measur e degre e o f influenc e withi n a socia l group ) • Minima l spannin g tree s (se t u p efficien t compute r networks ) • Travelin g salesperso n typ e problem s (manufactur e integrate d circui t

boards)

4 Editors' note : Fo r anothe r applicatio n o f grap h coloring , se e th e articl e b y Picke r i n this volume .

D I S C R E T E MATHEMATICA L MODELIN G I N T H E SECONDAR Y CURRICULU M 27 9

Conclusion

The ide a o f includin g discret e mathematic s i n th e secondar y curriculu m has bee n discusse d fo r man y years . Th e basi c question s o f what , when , an d how ar e stil l toug h question s tha t nee d t o b e carefull y considered . Thi s article ha s propose d on e se t o f answer s t o thes e questions . Th e answer s have been pu t int o practice by weaving discrete mathematic s int o a new hig h school curriculum , th e Core-Plu s Mathematic s Projec t curriculum . Discret e mathematics ca n an d shoul d b e i n th e sam e leagu e wit h ou r ol d friend s algebra, geometry , trigonometry , statistics , an d probability .

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, an d Hirsch , Christia n R. , eds. , Discrete Mathematics Across the Curriculum, K-12, Yearboo k o f th e Nationa l Counci l o f Teacher s o f Mathematics , NCTM , Resto n VA, 1991 . Malkevitch, Josep h an d Walte r Meyer , Graphs, Models, and Finite Mathematics, Prentice Hall , Englewoo d Cliff s NJ , 1974 . National Counci l o f Teacher s o f Mathematics , Curriculum and Evaluation Standards for School Mathematics, NCTM , Resto n VA , 1989 .

, Professional Standards for Teaching Mathematics, NCTM , Resto n VA , 1991. , Assessment Standards for School Mathematics, NCTM , Resto n VA , 1995 .

Roberts, Fred , Applied Combinatorics, Prentic e Hall , Englewoo d Cliff s NJ , 1984 . Rosenstein, Josep h G. , Directo r o f th e NS F Teache r Enhancemen t Project : "Leader - ship Progra m i n Discret e Mathematics, " 1989-97 . Se e als o "Th e Leadershi p Progra m in Discret e Mathematics, " b y Josep h Rosenstei n an d Valeri e DeBellis , i n thi s volume .

280 ERIC W . HAR T

[18] Sandefur , Jame s T. , Directo r o f the NS F Teache r Enhancemen t Project : "Leadershi p Training Institut e i n Dynamica l Modeling. " 1990-94 .

[19] "Drug s an d Pollutio n i n th e Algebr a Class, " The Mathematics Teacher, Feb - ruary, 1992 , pp . 139-145 .

[20] Discrete Dynamical Modeling, Oxfor d Universit y Press , Ne w York , 1993 . [21] Schoen , H . L. , an d S . W . Ziebarth , "Hig h Schoo l Mathematic s Curriculu m Re -

form: Rationale , Research , an d Recen t Developments. " i n Hiebowitsh , Pete r S. , an d William G . Wrag a (eds. ) Annual Review of Research for School Leaders, Macmillan , New York , (i n press) .

[22] Tannenbaum , Pete r an d Rober t Arnold , Excursions in Modern Mathematics, 2n d ed. , Prentice Hall , Englewoo d Cliff s NJ , 1995 .

[23] Tobt , B . an d T . Jensen , Graph Coloring Problems, Wiley , Ne w York , 1995 . [24] Wilson , Robi n J . an d Joh n J . Watkins , Graphs: An Introductory Approach, Wiley ,

New York , 1990 .

W E S T E R N MICHIGA N U N I V E R S I T Y , D E P A R T M E N T O F MATHEMATIC S AN D STATISTIC S

Current address: 61 3 S . 2n d St. , Fairfield , I A 5255 6 E-mail address: ehartQmum.ed u

DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

A Discret e M a t h e m a t i c s Experienc e wi t h Genera l M a t h e m a t i c s S t u d e n t s

Bret Hoye r

1. T h e N e e d

• "Whe n ar e w e eve r gonn a us e this? " • "Thi s i s dumb ; wh y d o w e hav e t o d o this? " • "Mat h i s hard! " • " I hat e math! " • "Math! ? Yuck!! "

These ar e som e o f th e many , man y comment s I hav e hear d student s and students ' parent s mak e wit h regar d t o mathematics . Ou r mathematic s department ha s struggle d eac h yea r t o ge t student s excite d enoug h t o tak e three o r fou r year s o f mathematics . Sustainin g studen t interes t i s a terrifi c challenge. Student s aspirin g t o atten d colleg e would tak e a t leas t thre e year s of mathematic s — u p t o Algebr a II , whic h woul d b e take n typicall y i n th e Junior year . Afte r thi s point , man y student s woul d choos e either no t t o tak e a mathematic s cours e durin g thei r Senio r year , o r t o tak e Senio r Advance d Math, onl y t o dro p afte r th e firs t semester . W e wante d t o addres s thi s problem, a s wel l a s anothe r problem : wha t t o d o wit h student s wh o don' t feel comfortable wit h Algebr a I I or Geometry ? Thes e student s don' t wan t t o take Consumer/Applie d Mat h becaus e the y don' t fee l challenged , an d the y fear th e stereotyp e tha t usuall y accompanie s tha t course . W e were searchin g for idea s and/o r material s t o addres s bot h o f thes e issues .

2. Goal s an d Objective s

We took a step towards solvin g this proble m afte r w e learned o f the text - book For All Practical Purposes: An Introduction to Contemporary Math- ematics [1] . I wa s introduce d t o thi s materia l i n th e Universit y o f Iowa' s

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 , 05C45 .

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282 B R E T H O Y E R

"Teachers a s Leaders " projec t durin g th e summe r o f 1990. 1 M y projec t group focuse d o n th e For All Practical Purposes materials . M y grou p an d the member s o f ou r mathematic s departmen t wer e ver y impresse d wit h the twenty-si x half-hou r videos 2 tha t complemen t th e text . Ou r student s thought th e ide a o f watching video s i n a mathematic s clas s wa s bizarre , bu t they didn' t object . W e fel t tha t th e materia l allowe d fo r a wid e rang e o f student abilit y levels . W e als o fel t tha t th e materia l woul d b e accessibl e t o students wh o ha d som e experienc e wit h hig h schoo l geometry . I bega n b y teaching on e o f th e units , "Stree t Networks " (Eule r circuits/paths) , t o m y Algebra I I an d Advance d Mat h classes . M y goal s wer e a s follows .

1. Engag e th e student s i n problem-solvin g activitie s tha t the y woul d find "fun" .

2. Introduc e a discret e mathematic s topi c an d it s applicatio n t o th e rea l world.

3. Discus s th e concep t o f a n optima l solutio n whe n multipl e solution s are possible .

4. Incorporat e cooperativ e learnin g activitie s int o a mathematic s class . 5. Giv e student s experienc e wit h non-routin e problem-solvin g withi n a

structured unit . 6. Engag e thos e student s wh o ar e startin g t o slid e int o th e " I just wan t

to ge t throug h Algebr a I I s o I don' t hav e t o tak e an y mor e math " mode.

7. Tak e a brea k fro m th e traditiona l textboo k durin g mid-February , when schoo l day s i n Iow a becom e ver y long .

The student s thoroughl y enjoye d th e materia l and , t o my surprise, foun d it ver y easy . Student s wer e actuall y doin g mathematics durin g othe r classes ! One particula r studen t ha d covere d hi s Englis h note s wit h graph s suc h a s that i n Figur e 1 , i n hi s attempt s t o fin d a n Eule r circuit .

F I G U R E 1 . A grap h withou t a n Eule r circui t

I share d m y classroo m experience s wit h m y colleagues . Ou r Consumer / Applied Mat h teache r wa s frustrated wit h her materials because her student s were "bored" . I suggeste d tha t sh e giv e thes e material s a try . Sh e enjoye d

E d i t o r s ' note : Thi s wa s a n NS F Teache r Enhancemen t Progra m directe d b y Eri c W. Har t an d Harol d Schoen , "Teacher s a s Leaders : Launchin g Mat h Educatio n int o th e Nineties", 1989-93 .

2 T h e video s wer e pu t togethe r b y COMA P wit h fundin g fro m th e Annenberg/CP B Project an d th e Carnegi e Corporatio n o f Ne w York .

D I S C R E T E MATHEMATIC S E X P E R I E N C E W I T H G E N E R A L MAT H S T U D E N T S 28 3

tremendous success . Th e student s wer e excited ! The y di d thei r homewor k and participate d i n class ! The y learne d th e materia l becaus e i t wa s fu n and the y sa w th e connectio n betwee n th e theor y an d th e rea l world . Th e videos playe d a majo r rol e i n helpin g th e student s se e th e connection . Eac h chapter ha s numerou s exercises , an d th e teacher' s manua l give s additiona l exercises tha t ar e accessibl e t o student s wit h lo w abilit y levels . Th e tex t also includes exercises (an d proofs ) tha t wil l challenge the brightes t Calculu s students. W e convince d ou r administratio n t o offe r a ne w cours e (w e calle d it Contemporar y Mathematics ) durin g th e 1992-9 3 school year . W e crosse d our fingers, hopin g there would be enough interes t t o sustain th e course. Th e Consumer/Applied Mat h teache r mentione d earlie r ha s thoroughl y enjoye d teaching th e material . I hav e ha d th e opportunit y t o weav e th e uni t o n "Street Networks " int o m y Algebr a I , Geometry , an d Genera l Mat h courses . I hav e als o enjoye d a grea t dea l o f success ; man y parent s a t ou r ope n hous e last fal l share d wit h m e comment s abou t thei r children' s excitement .

3. Teachin g Strategie s

The onl y rea l difficult y w e have face d i s the readin g leve l of the material . The textboo k wa s designe d fo r a cours e tha t woul d mee t a mathematic s re - quirement i n a libera l art s college . I n thi s section , I presen t cooperativ e learning activitie s whic h hel p addres s thi s issue . T o facilitate thes e coopera - tive learnin g activities , I hav e roun d table s i n m y classroom . Her e ar e som e of th e technique s I use .

Jigsawing. I us e jigsawin g t o giv e al l th e student s a n opportunit y t o practice teaching . I firs t divid e th e student s int o bas e group s o f thre e o r four students , an d assig n differen t exercise s o r activitie s t o th e member s o f each bas e group . Then , I for m ne w solutio n group s consistin g o f student s who hav e th e sam e exercise . Onc e th e solutio n group s hav e complete d thei r exercises, th e student s al l retur n t o thei r bas e groups , wher e the y teac h th e other member s o f thei r bas e grou p wha t the y hav e learned .

K W L . I n th e KW L strategy , th e "K " stand s fo r "know" , th e "W " stands fo r "want" , an d th e "L " stand s fo r "learned" . Eac h bas e grou p identifies wha t the y "know " abou t a n assignmen t o r topic . Afte r a fe w minutes, the y identif y wha t the y "want " t o kno w abou t th e homewor k o r topic. Afte r a coupl e mor e minute s tw o peopl e fro m eac h bas e grou p ar e selected t o trave l t o anothe r group , fo r compariso n o f "know " an d "want " lists. Afte r comparin g notes , mos t o f th e "want " lis t question s shoul d b e answered. Student s the n retur n t o thei r bas e group s t o summariz e wha t they hav e "learned" .

Think-pair-share. I n thi s strategy , th e student s wor k individuall y o n an exercis e (think) , the n partne r u p (pair ) t o shar e thei r solutions .

284 BRET HOYE R

4. A Sampl e Lesson—Stree t Network s

I decide d tha t th e bes t wa y t o approac h th e stree t network s materia l with m y freshma n Genera l Mat h student s wa s to forge t abou t th e textbook . I als o approache d i t a littl e bi t differentl y tha n th e tex t i n tha t I starte d with a puzzle . I chos e no t t o star t wit h a real-worl d proble m an d mode l it wit h a grap h becaus e thes e particula r student s respon d ver y positivel y to puzzle s an d games . Also , thes e student s foun d i t difficul t (initially ) t o model rea l problem s a s graphs . I wanted t o eas e them int o th e mathematic s so tha t thei r curiosit y an d excitemen t remaine d intact .

I starte d ou t b y challengin g m y student s t o trac e th e edge s o f a grap h like th e on e i n Figur e 1 without liftin g thei r pencil s o r retracin g an y edges , but returnin g t o thei r startin g point . I tol d the m tha t suc h a tracin g wa s called a n Euler circuit Man y o f m y student s though t the y ha d a solutio n to th e puzzle , bu t discovere d tha t the y didn' t whe n the y trie d t o displa y i t on th e board . Afte r a fe w attempts , a fe w o f the m announce d tha t ther e was n o solution . Afte r abou t five minutes , I tol d the m tha t ther e wa s i n fact n o solutio n t o thi s puzzl e (n o Eule r circui t fo r th e graph) . Next , w e tried t o find Eule r circuit s i n th e graph s i n Figur e 2 . I hav e found , t o a sur - prising degree , tha t th e student s develo p excellen t informa l definition s an d conjectures i n thei r groups . Al l I hav e t o d o i s giv e th e student s "enough " concrete examples .

l O A f f l H (a) (b ) (c ) (d ) (e ) (f ) (g )

F I G U R E 2 . Som e graph s fo r Eule r circui t exercises : onl y graphs (a) , (b) , (d) , an d (f ) hav e Eule r circuits .

I showed the m wha t th e term s edge , vertex (vertices) , an d degre e meant . We liste d th e degree s o f the vertice s o n eac h grap h an d classifie d th e graph s as eithe r havin g o r no t havin g a n Eule r circuit . I aske d th e student s t o independently writ e dow n a rul e fo r determinin g whethe r a grap h ha s a n Euler circuit . Man y cam e u p wit h vali d ideas , bu t di d no t stat e a complet e rule, fo r example , "I f i t i s even, i t wil l work, " o r "I f th e degre e i s a n od d # it i s not a n Eule r circuit. " Student s the n discusse d al l of the individua l rule s in group s o f fou r o r five. Eac h grou p the n cam e u p wit h on e rule . Differen t groups cam e u p wit h on e o f th e followin g tw o rules :

• Al l o f th e vertice s mus t hav e a n eve n degre e fo r th e grap h t o hav e a n Euler circuit .

• I f th e grap h ha s an y vertice s wit h od d degree , the n i t doesn' t hav e an Eule r circuit .

DISCRETE MATHEMATIC S EXPERIENC E WIT H GENERA L MAT H STUDENT S 28 5

I explaine d tha t th e tw o rule s wer e equivalent . I use d th e sam e metho d t o presen t Eule r paths (tracing s whic h cove r

each edg e exactl y onc e bu t ma y no t retur n t o thei r startin g points) . W e listed th e sam e graphs , an d foun d tha t onl y on e ha d a n Eule r pat h whic h was no t als o a circuit . Eventuall y the y discovere d a rul e fo r a grap h wit h no Eule r circui t t o hav e a n Eule r path—exactl y tw o vertice s o f od d degree . I the n offere d th e graph s i n Figur e 3 and pointe d ou t tha t non e o f the m ha s an Eule r path , althoug h eac h ha s exactl y tw o odd-degre e vertices .

F I G U R E 3 . Thre e disconnecte d graph s

The student s objecte d tha t thes e example s weren' t graphs , becaus e the y weren't "connected" ! I explained t o the m th e concep t o f connectedness, an d we revise d ou r rule s fo r th e existenc e o f Eule r circuit s an d paths .

The nex t ste p i n th e uni t wa s t o appl y Eule r path s an d circuit s t o a real-world situation . Th e followin g exampl e i s from pilo t material s fro m th e University o f Iowa' s Cor e Plu s Mathematic s Projec t [2 , 3] . Th e diagra m i n Figure 4 represents a schoo l floor plan . Th e blac k square s represen t lockers .

F I G U R E 4 . Floo r pla n o f a schoo l

The tas k i s to decid e whethe r yo u ca n pain t al l the locker s without retracin g your step s with th e heav y equipment. Th e challeng e i s to model th e proble m using a graph . Th e suggeste d answe r wa s t o trea t portion s o f th e hallway s as vertices , an d trea t rows o f locker s a s edge s (Figur e 5b) . Som e o f th e students place d vertice s a t each hallwa y intersection , bu t stil l represente d rows o f locker s b y edge s (Figur e 5c) . Othe r student s fel t tha t th e hallways , rather tha n th e rows of lockers, should becom e the edges of the graph (Figur e 5d), perhap s assumin g tha t on e coul d pain t locker s o n bot h side s o f a hal l at once .

We decide d tha t determinin g wha t th e edge s represen t i s vita l i n mod - eling a rea l worl d situatio n wit h a graph .

286 BRET HOYE R

•

• •

•

• •

F E # -

(a) (b) (c) (d)

F I G U R E 5 . (a ) Labele d problem , (b ) suggeste d answer , (c)-(d) studen t answer s

I hav e foun d tha t thi s uni t me t man y o f th e need s o f m y students , a s discussed i n Section s 1 and 2 . Generally , m y student s finished th e uni t wit h confidence i n thei r abilitie s t o d o mathematics . The y wer e challenge d t o reason an d communicat e i n mathematica l language . The y recognize d tha t what the y wer e doin g wa s importan t an d o f value .

I aske d student s t o compar e th e unit s the y complete d o n street network s and votin g theor y t o previou s wor k the y ha d don e i n Genera l Math . Her e are a fe w o f thei r responses .

• I lik e votin g theor y mor e tha n a lo t o f thing s tha t I hav e don e i n th e past. I thin k i t wa s fun . Stree t network s i s fun too . I lik e working o n these type s o f thing s becaus e the y mak e m e think . I lik e thes e unit s more tha n a lo t o f th e others . Thes e unit s wer e easie r fo r m e tha n the others , the y wen t a lo t slowe r s o I coul d kee p u p easier . (9th grad e boy )

• I lik e th e grap h unit . I lik e tryin g t o figure ou t i f i t i s Eule r circui t or not . Thi s uni t i s a change . I lik e th e grap h uni t bette r th e votin g because i t i s muc h mor e interestin g an d muc h fu n an d challenging . Pre Algebr a wa s nothing lik e the votin g an d stree t networks . I' m gla d I foun d somethin g new . (9 th grad e boy )

• I d o enjo y doin g th e theory s bette r tha n doin g ou r books . I t seem s to hel p m e mor e wit h m y math . I a m als o learnin g mor e s o I woul d encourage peopl e t o lear n fro m th e theory s othe r tha n jus t th e math - ematics book . (10 t/l grad e girl )

5. A s s e s s m e n t P r o c e d u r e s

Some of the procedures I use for assessmen t includ e paragraph responses , daily homework , individua l projects , grou p project s an d presentations , indi - vidual exam s an d quizze s i n class , tak e hom e exam s an d quizzes , an d grou p quizzes3. I n th e street-networ k unit , I use d th e paragrap h respons e give n

3 In a grou p quiz , student s wor k o n th e qui z independentl y fo r 10-1 5 minutes , the n i n groups, wit h eac h grou p turnin g i n on e paper . Th e student s don' t kno w whe n a grou p qui z

DISCRETE MATHEMATIC S EXPERIENC E WIT H GENERA L MAT H STUDENT S 28 7

above i n th e previou s section , take-hom e quizze s an d exams , in-clas s exams , and dail y homewor k activities .

6. Conclusio n

This summar y o f th e developmen t an d implementatio n o f th e uni t o n street network s illustrate s som e o f th e thing s tha t I tr y i n m y classroom . I feel i t ha s bee n successfu l i n meetin g th e goal s I se t an d th e need s o f m y students.

References

[1] Garfunkel , Solomon , e t al . For All Practical Purposes: Introduction to Contemporary Mathematics, 3r d edition. , W . H . Freema n an d Company , Ne w York , 1994 .

[2] Hart , Eri c W. , "Discret e Mathematica l Modelin g I n Th e Secondar y Curriculum : Ra - tionale an d Example s fro m Th e Core-Plu s Mathematic s Project, " thi s volume .

[3] Hirsch , Christia n R. , Arthu r F . Coxford , Jame s T . Fey , an d Harol d L . Schoen , "Core - plus mathematics : Teachin g sensibl e mathematic s i n sense-makin g ways, " The Math- ematics Teacher, Nov . 1995 , pp . 694-700 .

J O H N F . K E N N E D Y H I G H SCHOOL , 454 5 W E N I G R D . NE , C E D A R R A P I D S , IOW A

52402 E-mail address: b h o y e r @ c e d a r - r a p i d s . k l 2 . i a . u s

is goin g t o b e given , s o the y mus t assum e eac h qui z wil l b e individual . Thi s encourage s maximum effor t o n al l quizze s al l o f th e time .

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Algorithms, Algebra , an d t h e C o m p u t e r La b

Philip G . Lewi s

One thin g lead s t o another . First I agree d t o hel p teac h th e introductor y programmin g cours e i n ou r

new hig h schoo l compute r department . Tha t mean t I ha d t o lear n som e Logo. Meanwhile , bac k i n th e mat h department , I wa s teachin g a littl e bi t of linea r algebra—vectors , vector-value d functions , linea r transformations , matrices. I noticed the n tha t a Logo 1 lis t an d a vecto r ar e on e an d th e sam e thing. O f cours e I neede d programmin g practice , so , fo r a mont h I spen t my fre e tim e gettin g Log o t o kno w everythin g I kne w abou t linea r algebra . I ha d s o muc h fu n doin g this , tha t i t seeme d natura l t o propos e t o teac h a mathematics cours e meetin g i n ou r ne w compute r lab , a a cours e tha t reall y uses a compute r languag e t o expres s an d develo p th e mathematics. " I did ; the cours e wa s successful ; i t resulte d i n a boo k [6] , Approaching Precalculus Mathematics Discretely] th e boo k le d t o m y participatio n a t th e DIMAC S conference2 an d th e conferenc e le d t o thi s article .

So here ar e som e question s I'l l tr y t o answer . What' s discrete , different , and wort h preservin g abou t th e cours e I taugh t an d trie d t o encapsulat e i n the book ? An d ar e ther e genera l principle s tha t migh t b e applie d t o othe r courses?

Let's tak e not e firs t o f th e rol e o f discret e mathematics . Th e stud y of algorithm s an d algorithmic s permeate s discret e mathematics . Student s learned mos t o f th e mathematic s b y writin g algorithm s an d runnin g the m on th e computer . I'l l argu e her e tha t concentratin g o n th e proces s o f con - structing an d analyzin g algorithm s i s a productiv e wa y t o teac h algebra , provided tha t certai n condition s ar e satisfied .

What wer e th e salien t characteristic s o f th e course ?

1991 Mathematics Subject Classification. Primar y 00A05 , 00A35 . ^ o g o i s a dialec t o f Lis p tha t avoid s it s insistenc e o n parentheses . Logo' s primar y

d a t a object s ar e numbers , word s (number s ar e als o words) , an d lists . List s o f number s can, o f course , represen t vectors , list s o f list s matrices .

2 Editors' note : Thi s conference , hel d i n 1992 , i s describe d i n th e Prefac e an d Intro - duction t o thi s volume .

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29 0 PHILIP G . LEWI S

1. Al l classes, about twenty-tw o student s t o a class, met i n the compute r room. The y were primarily college-boun d juniors at a n upper (bu t no t honors) achievemen t level . Mos t o f th e tim e ther e wa s on e compute r to a student , althoug h a fe w student s worke d closel y enoug h togethe r so tha t thei r effort s coul d b e considere d a tea m effort . Classe s wer e scheduled fou r time s a week , bu t th e roo m wa s fre e fo r la b wor k o n the fifth day .

2. O n a da y tha t require d ne w material , I usuall y bega n b y introducin g the basi c concep t togethe r wit h relevan t Log o notatio n an d the n se t a tas k tha t require d student s t o construc t th e algorithm s necessar y to develo p th e concept . Fo r example , a n introductio n o f th e concep t of vecto r a s ordere d pai r (trivia l t o thes e students ) le d the m t o writ e Logo programs first t o establish a n axis system given two scalar input s (AXES 2 0 20) , then t o dra w a given vector i n the appropriat e positio n to th e scal e o f th e axi s syste m (VECTDRA W [ 5 7 ] ).3

The concep t o f a workspac e save d t o dis k allowe d student s t o build an d sav e a n increasingl y comple x algorithmi c structur e havin g the flavor, i f no t th e logica l structure , o f a n axiomati c mathematica l system. Fo r example , give n definition s o f vecto r additio n (VSUM ) an d scalar multiplicatio n (SCALAR) , the studen t coul d defin e a n algorith m to perfor m a linea r combinatio n o f tw o vector s (LC ) b y invokin g th e two algorithm s VSU M and SCALAR: 4

TO L C :N 1 :N 2 :VECT 1 :VECT 2

OUTPUT VSU M (SCALA R :N 1 :VECT1 )

(SCALAR :N 2 :VECT2 )

END

3. I trie d t o se t a sufficien t numbe r o f task s t o occup y th e da y followin g a class . Thi s le t m e spen d a t leas t hal f o f th e clas s period s walkin g around, makin g suggestion s an d debuggin g programs . A t regula r intervals I' d hav e a n interactiv e class— a codificatio n an d catch-u p period durin g whic h student s woul d mak e sur e tha t everyon e ha d a n updated se t o f procedures . I regularl y aske d student s wh o cam e u p with version s o f algorithm s tha t wer e noteworth y t o presen t the m for clas s discussio n an d possibl e inclusio n i n th e "official " algorith m structure tha t constitute d a recor d o f th e mathematic s tha t th e clas s had accomplished . Thi s se t o f curren t procedure s wa s rathe r lik e a class notebook , excep t tha t th e note s wer e a compilatio n o f th e bes t of th e class-generate d algorithms . A s th e cours e progressed , som e o f these woul d ge t reworke d t o reflec t th e increasin g sophisticatio n o f

3Logo i s a prefi x language—tha t is , th e nam e o f th e procedur e precede s it s inputs . Notice tha t AXE S takes tw o numerical input s whil e VECTDRAW takes on e numerica l lis t input .

4For a procedur e t o provid e a n inpu t fo r anothe r procedure , Log o require s th e pro - grammer t o declar e th e output . Notic e t h a t th e outpu t i s th e vecto r su m (VSUM ) o f tw o vectors, eac h obtaine d fro m a scala r produc t o f a numbe r an d a vector .

A L G O R I T H M S , ALGEBRA , AN D T H E C O M P U T E R LA B 29 1

the student s an d th e mathematics . Student s kep t thei r ow n up-to - date versio n o f the curren t recor d o n disk. Often , however , thes e disk s would diffe r significantl y fro m th e "official " recor d becaus e student s tended t o develo p a stron g fondnes s fo r includin g thei r ow n workin g procedures rathe r tha n th e "official " version .

4. Th e goa l fro m th e star t wa s alway s t o develo p th e mathematic s tha t could b e applie d t o compute r graphics , s o durin g th e mos t successfu l part o f the course, the need to "ge t somewhere" kep t th e class together as a class . Thi s cohesivenes s diminishe d somewha t towar d th e en d o f the cours e a s som e students , afte r completin g th e compute r graphic s work, move d o n t o othe r algebrai c topic s (e.g . polynomia l operation s from a vecto r poin t o f view , pola r graphing ) whil e thei r colleague s completed th e compute r graphic s section .

Both th e student s an d I ha d ver y positiv e feeling s abou t th e course . Some of us talked the n abou t wh y it worke d an d I have done a lot o f thinkin g about i t since . Here , i n n o particula r order , ar e th e result s o f som e o f thes e reflections.

First, i t mus t b e sai d tha t th e compute r i s a wonderfu l critic . I t i s ruthlessly impartia l an d relentlessl y logical—lik e a forc e o f nature . Whe n a student write s a progra m tha t doesn' t work , it doesn' t wor k lik e a machin e doesn't work , an d i t ca n b e mad e t o wor k b y a combinatio n o f technique s ranging fro m logica l analysi s t o rando m tinkering . Furthermore , a studen t who fixed a progra m b y rando m tinkerin g wa s almos t neve r completel y sat - isfied. Eventuall y h e o r sh e woul d com e bac k t o th e questio n "Now , wh y the hel l di d tha t work? "

Some o f th e characteristic s o f a clas s conducte d i n a compute r la b ar e unique. Th e teache r tend s t o b e move d ou t o f th e feedbac k loop , takin g on mor e th e rol e o f mento r a s h e o r sh e analyze s an d debug s programs . I t turns ou t t o b e muc h mor e satisfyin g t o answe r th e questio n "Wh y doesn' t this work? " tha n t o evaluat e th e solutio n t o tes t problems . I f th e goa l o f a class i s t o hav e it s member s construc t origina l programs , student s com e t o class i n th e middl e o f ongoin g work . The y don' t wan t t o b e interrupte d b y instruction, s o a teache r finds i t muc h mor e difficul t t o teach . I t amount s to a bad-new s good-new s joke : the y won' t liste n s o yo u can' t teach ; o n th e other hand , they'r e learnin g s o yo u don' t have t o teach .

If th e tas k i s t o writ e a progra m t o teac h a compute r ho w t o d o some - thing, th e proces s o f learnin g th e mathematic s i s different . Firs t yo u mus t comprehend th e idea ; the n yo u mus t writ e a n algorith m t o expres s it . Whil e it i s possible t o comprehen d a concep t withou t understandin g i t an d stil l de- sign th e algorith m t o expres s it , onc e th e algorith m works , a t leas t yo u fee l better abou t understandin g th e concep t o n whic h th e algorith m i s based . While thi s feelin g ma y hav e bee n illusor y fo r som e students , th e psycho - logical impac t seeme d stron g enough . Afte r all , yo u ow n somethin g tha t you construc t yourself . An d eve n thoug h mos t student s don' t ge t a chanc e to ow n origina l concepts , the y see m t o b e abl e t o acquir e ownershi p b y

292 P H I L I P G . LEWI S

constructing algorithm s t o implemen t th e concepts . Th e effec t o f s o doin g makes a bi g difference .

We tal k a lo t abou t th e heuristic s o f proble m solvin g withou t teachin g them. Th e act o f solving the problems inherent i n constructing a hierarchica l system o f algorithm s naturall y focuse s o n thes e heuristics . I mentione d th e problem o f definin g L C to outpu t a linea r combinatio n o f tw o vector s an d two scalars . Man y student s initiall y solv e th e tas k b y rebuildin g VSU M and SCALAR inside LC , producing procedure s lik e this :

TO LC :N 1 :N 2 :VECT 1 :VECT 2 OUTPUT LIS T SU M (PRODUC T :N 1 FIRS T :VECT1 )

(PRODUCT :N 2 FIRS T :VECT2 ) SUM (PRODUC T :N 1 LAS T :VECT1 )

(PRODUCT :N 2 LAS T :VECT2 ) END

instead of : TO LC :N1 :N2 :VECT1 :VECT2

OUTPUT VSUM (SCALAR :N1 :VECT1) (SCALAR :N2 :VECT2)

END

After seein g a fe w example s o f thi s sort , the y no t onl y se e th e powe r of buildin g comple x procedure s ou t o f simple r ones , bu t ten d t o us e th e principle b y dividin g a comple x proble m int o simple r parts , solvin g these , and the n incorporatin g th e solution s int o the solutio n o f the large r task . Th e resulting structur e i s analogous to the axiomati c structure o f a mathematica l system: th e algorith m fo r L C depends completel y o n th e statu s o f th e tw o algorithms fo r vecto r additio n an d scala r multiplication . I t i s possibl e fo r students t o construc t algorithm s fro m th e "to p down, " saying , i n effect , " I don't kno w ho w to ad d vector s an d I don' t kno w ho w to multipl y b y scalars , but i f I did , the n here' s th e correc t algorith m fo r LC . I n suc h cases , th e definitions o f VSU M and SCALA R hav e a statu s simila r t o bypasse d lemmas . LC can b e defined an d Log o will accept th e definition , bu t i t won' t wor k unti l the tw o algorithm s o n whic h i t depend s ar e themselve s successfull y defined .

Once yo u hav e solve d a lo t o f algorithmi c problems , yo u hav e i n ef - fect develope d a se t o f tool s fo r gettin g th e compute r t o d o mathematics . This provide s a stron g impetu s o n th e on e han d t o exten d th e mathemat - ical concept s int o ne w territorie s (w e ca n handl e matrice s i n tw o space , why no t matrice s i n n-space? ) an d o n th e othe r t o se e on e thin g a s lik e another—adding polynomial s i s jus t lik e addin g vectors ! Consequentl y th e student's investmen t i n constructin g a n algorithmi c syste m ha s grea t poten - tial mathematica l payoffs . Ther e i s n o intellectua l satisfactio n comparabl e to discoverin g tha t one' s solutio n t o on e proble m ca n b e mad e t o appl y t o another proble m i n a totall y differen t environment .

The choic e of computer languag e i s important. Sudent s wh o follow a n al- gorithmic approach hav e a sense of the dependent structur e o f mathematics — provided th e languag e i s modula r lik e Logo . The y hav e a sens e o f functio n in terms o f input an d output—provide d tha t th e languag e take s a functiona l

ALGORITHMS, ALGEBRA , AN D TH E COMPUTE R LA B 29 3

form lik e Logo . An d the y wil l b e comfortabl e wit h recursiv e definition s an d the foundation s o f mathematica l induction—provide d tha t th e languag e i s as dependen t o n recursio n a s Logo . Th e latte r poin t i s bot h positiv e an d negative: yo u can' t d o anythin g powerfu l withou t recursio n i n Logo , bu t because i t i s s o dependen t o n recursion , yo u can' t easil y loo k a t th e worl d iteratively. Therefor e m y students wer e good a t formulatin g recursiv e defini - tons, bu t no t ver y goo d a t formulatin g o r analyzin g iterativ e definitions . O n the other hand , bein g able to look at a definition lik e SCALAR and stat e that i t must wor k o n an y n-dimensiona l vecto r require s tha t yo u kno w wha t math - ematical inductio n i s al l about , an d tha t i s a bi g plu s i n th e hig h school. 5

TO SCALAR :NUM :VECT IF EMPTYP :VECT [OUTPUT :VECT] OUTPUT FPUT (PRODUCT :NUM FIRST :VECT)

SCALAR :NUM BUTFIRST :VECT END

Many (I' d lik e to think most ) student s woul d analyz e th e definitio n thi s way, saying somethin g lik e "Well , i f i t work s fo r a vecto r o f dimensio n k, the n it wil l wor k fo r on e o f dimensio n k + 1 . S o let' s loo k a t th e first case . I t outputs th e empt y lis t whe n : VECT is empty , s o i t has t o work. "

Teaching student s t o construc t algorithm s i s hard ; gettin g the m t o ana - lyze th e structur e o f thos e algorithm s i s harder . If , however , th e algorithm s are implemente d o n a computer , analysi s i s a natura l consequenc e o f cre - ation. Som e algorithm s clearl y ru n mor e slowl y tha n others . Som e ar e clearly mor e elegantl y expresse d tha n others . I n n o othe r cours e hav e I found student s t o b e s o awar e o f efficienc y an d eleganc e a s desirabl e char - acteristics. Her e i s on e example . A studen t wrot e a progra m t o inver t a two-by-two matri x (i n Log o thi s i s a lis t o f tw o lists—e.g . [[ 1 2 ] [ 3 4 ] ] ) by first writin g a procedur e DE T to obtai n th e determinan t o f th e matrix :

TO MATINV :MAT OUTPUT MATRI X VEC T (LAS T LASTVEC T :MAT)/DE T :MA T

(MINUS LAS T FIRSTVEC T :MAT)/DE T :MA T VECT (MINU S FIRS T LASTVEC T :MAT)/DE T :MA T

(FIRST FIRSTVEC T :MAT)/DE T :MA T END

where the procedure VECT makes a vector out o f two numbers, the procedure s FIRSTVECT an d LASTVEC T retriev e th e appropriat e vector s fro m a matrix , and MATRI X makes a matri x ou t o f tw o vectors . I wa s particularl y please d that th e studen t ha d rename d familia r procedure s lik e FIRS T t o mak e th e algorithm mathematicall y readable , an d complimente d him , bu t wa s take n

5SCALAR provide s a goo d indicatio n o f Logo' s recursiv e power . I t depend s o n severa l list operatin g procedures : FPU T insert s a n objec t int o th e fron t o f a list . FIRS T output s the first elemen t i n a lis t an d BUTFIRS T output s a lis t comprise d o f al l bu t th e firs t o f a list. Th e procedur e SCALA R tests it s inpu t vecto r t o se e i f i t i s empt y (EMPTYP) . I f i t is , i t outputs th e empt y list . Otherwise , i t multiplie s th e numbe r (:NUM ) b y th e firs t elemen t of th e lis t an d insert s thi s int o th e lis t t h a t result s fro m callin g SCALA R recursively o n th e BUTFIRST o f th e list . Whe n th e lis t i s empty , SCALA R outputs th e lis t comprise d o f al l o f the product s o f :NU M and th e element s o f :VECT .

294 P H I L I P G . LEWI S

aback b y anothe r student' s comment , "That' s reall y inefficient! " O n bein g pressed, sh e observe d devastatingl y "Look , DE T is bein g calle d fou r time s when onc e wil l do. " I t i s hard t o imagin e tha t kin d o f algorithmic analysi s occurring i n the absence o f the computer .

There is one other componen t o f the cours e that shoul d b e mentioned. A student wit h a clever solution to a problem invariabl y had the opportunity t o present i t t o the class. Becaus e an y program wa s one that classmate s coul d use, th e audienc e tende d t o b e enthusiasti c i n thei r appreciation . Havin g a potentia l audienc e fo r thei r wor k le d some student s int o ne w intellectua l territory.

I hav e mad e a cas e fo r som e o f the virtues o f taking a n algorithmi c ap- proach t o vector algebra . I t is reasonable t o ask if the approach make s sens e in mor e elementar y contexts . Tw o of u s hav e taugh t a first-yea r algebr a course tha t take s a n algorithmi c approac h t o teachin g functions . W e hav e observed som e of the same sort s of instructional dividend s I have mentione d and, a s a result o f working with younger an d more mathematically unsophis - ticated students , ar e developin g th e theor y tha t student s initiall y com e t o algebra wit h a well developed vie w of function a s algorithm. Unfortunately , our curriculu m force s the m t o a set-theoretic vie w a t th e start an d only re- turns t o algorithms whe n the y ar e so thoroughly brainwashe d tha t thinkin g algorithmically ha s becom e difficult . Wouldn' t i t b e logica l t o revers e th e curriculum structur e an d begin wit h a n algorithmi c vie w o f function? Thi s argument ha s led two of us to produce a proposal fo r a computer-based be - ginning algebr a cours e i n which th e algorithmi c componen t i s significant. I hope t o be reporting o n that som e day.

References

N O T E : Followin g thi s not e i s a lis t o f book s fo r thos e intereste d i n usin g Log o t o teac h mathematics i n a compute r la b setting. Followin g ar e two appropriate version s o f Logo :

A : Th e most accessibl e vanill a (i.e. , standard , no-frills ) Log o for the IBM PC a s well as th e Macintos h wa s developed b y Bria n Harve y an d is free fo r the downloading . The UR L i s h t t p : / / h t t p . c s . b e r k e l e y . e d u / ~ b h . Th e material s wer e originall y developed o n the BBC Acorn Computer , whic h ha d a residen t versio n o f Logo .

B : Paradig m Softwar e ha s a powerfu l object-oriente d versio n o f the language fo r the Macintosh. Thi s come s i n a studen t versio n o r a full-fledge d developmen t version . Either versio n o f Objec t Log o i s capabl e o f bein g ru n i n a "vanilla " mode . Th e address i s Paradigm Softwar e P.O . Box 2995, Cambridge , Massachusett s 02238 .

[1] Abelson , Harold , an d Andre a H . deSesa, Turtle Geometry, MI T Press, 1980. [2] Abelson , Harold , an d Amand a Abelson , Logo for the Macintosh, and Introduction

Through Object Logo, Paradig m Software , Cambridg e MA , 1992 . [3] Cuocco , Albert , Investigations in Algebra, MI T Press, Cambridg e MA , 1989 . [4] Clayson , James , Visual Modeling with Logo, MI T Press, Cambridg e MA , 1988 . [5] Harvey , Brian , Computer Science Logo Style, Volume 1, Intermediate Programming,

MIT Press , Cambridg e MA , 1986 . [6] Lewis , Phili p G. , Approaching Precalc. Math. Discretely, MI T Press, 1990.

LINCOLN-SUDBURY (MA ) R E G I O N A L H I G H SCHOO L ( R E T I R E D )

E-mail address: pglQworld . s t d. com

DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Discrete M a t h e m a t i c s i s Alread y i n t h e Classroo m — B u t I t ' s Hidin g

Joan Reinthale r

The las t severa l years have presented hig h school teachers with a n almos t irresistible buffe t o f mathematica l goodie s t o ad d t o th e curriculum—real - world applications , recursion , fractal s an d chaos , dat a analysis , an d applica - tions o f matrices , t o nam e just a few—an d o f course , a workin g competenc y with th e technolog y tha t ha s mad e al l thi s possible . Discret e mathematic s is th e umbrell a tha t cover s a lo t o f thes e topic s and , despit e th e excitemen t these idea s hav e engendere d i n man y teacher s an d thei r students , whe n the realit y o f closel y prescribe d curricul a rear s it s ugl y head , th e inevitabl e question w e as k i s "Ho w ca n I find roo m t o fit thi s in? " Thi s i s no t a naiv e question an d i t deserve s mor e attentio n tha n i t generall y get s fro m a math - ematics communit y o f scholar s an d universit y teacher s whos e curricul a ar e not similarl y mandated .

As a first ste p i n answerin g thi s question , I sugges t tha t aspect s o f dis - crete mathematic s ar e alread y i n th e curriculu m bu t tha t w e a s teachers , and som e o f th e textbook s w e use, ten d t o ignor e them . Man y introductor y algebra book s glos s ove r th e differenc e betwee n a continuou s domain , a s i n the relationshi p betwee n distanc e an d time , an d a discret e domain , a s i n the relationshi p betwee n th e pric e o f a carto n o f mil k an d th e amoun t o f milk th e carto n holds , o r i n th e relationshi p betwee n frequenc y o f cricke t chirps an d th e temperature . Thes e ar e wonderfu l problems , bu t w e los e a n opportunity t o mak e distinction s betwee n discret e an d continuou s domain s and range s whe n w e trea t the m al l identically , o r when , automatically , w e connect th e point s o n thei r graphs . W e nee d instea d t o recogniz e problem s that involv e discret e domain s (problem s involvin g money , fo r instance , o r numbers o f thing s lik e pencil s o r peopl e o r rafH e tickets) , indee d t o b e o n the lookou t fo r them , t o rejoic e i n the m an d t o us e the m a s a jumping-of f place fo r creativ e an d productiv e investigations .

Here i s a n exampl e o f th e sor t o f materia l foun d i n man y introductor y algebra course s an d frequentl y treate d a s i f th e domai n wer e continuous .

1991 Mathematics Subject Classification. Primar y 00A35 , 00A05 .

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https://doi.org/10.1090/dimacs/036/23

29 6 JOAN REINTHALE R

A shoemaker makes moccasins and boots and needs 2 square feet of leather for each moccasin and 3 square feet of leather for each boot. 20 square feet of leather are available.

At thi s poin t student s ar e use d t o bein g aske d a question , th e obviou s one bein g somethin g abou t ho w man y boot s an d moccasin s ar e made , o r they ma y b e aske d t o dra w a grap h tha t represent s th e informatio n given , or t o writ e a n inequalit y tha t model s th e information . Th e usua l response s to thes e request s ar e

2M + SB < 20,

or a grap h tha t look s somethin g lik e tha t i n Figur e 1(a) . Th e assumptio n is mad e tha t M > 0 an d B > 0 an d tha t bot h rang e an d domai n ar e continuous (an d usuall y th e studen t i s unawar e tha t h e o r sh e ha s mad e such assumptions) .

20/ 3

w i o

F I G U R E 1 . (a ) Grap h o f 2M+3B < 2 0 ; (b) Grap h o f intege r lattice point s satisfyin g 2 M + SB < 20.

However, th e situatio n describe d abov e doesn't conclud e with a question . My experienc e ha s bee n tha t thi s ofte n lead s student s t o creativ e investiga - tions an d mor e widel y rangin g discussions . Student s ma y wel l as k th e ver y questions liste d abov e an d ma y provid e th e sam e answers , bu t ever y tim e I have use d thi s approach , som e student s hav e begu n t o questio n th e validit y of these conventiona l answer s when the y begi n to as k other question s a s well. How many differen t way s ca n a shoemake r us e he r resources ? Ca n yo u mak e a fractio n o f a boot ? D o yo u hav e t o manufactur e boot s an d moccasin s i n pairs? Ar e th e relation s an d graph s show n abov e appropriat e model s o f th e situation? Ar e ther e bette r models ?

At thi s point, w e can begin to explore the differenc e betwee n th e graph of a region and tha t o f a lattice, a s shown i n Figure 1(b). ) W e can as k abou t th e number o f solutions. W e can discus s th e natur e o f the boundar y an d loo k a t the numbe r o f solutions tha t li e on th e boundary . Al l of these investigation s lie entirel y withi n th e scop e o f materia l tha t i s include d i n eve n th e mos t conservative algebr a curriculum . Thi s sor t o f discussio n ca n begi n t o giv e students insigh t int o wha t i s mean t b y "Discret e Mathematics" .

DISCRETE MATHEMATIC S I S ALREADY I N TH E CLASSROO M 29 7

Here i s anothe r exampl e o f th e kin d o f materia l tha t i s alread y a t ou r fingertips jus t waitin g t o b e investigate d i n the contex t o f a discret e domain . It i s a proble m fro m a popula r precalculu s textbook .

Sally's office has a system to let people know when the depart- ment will have a meeting. Sally calls three people. Then, those three people each call three other people, and so on, until the whole department is notified. If it takes ten minutes for a per- son to call three people, and all calls are completed within 30 minutes, how many people will be called in the last round?

The assumptio n usuall y mad e i s tha t th e appropriat e mode l tha t wil l be discovere d fo r thi s proble m i s th e exponentia l functio n y = 3 X, with n o mention o f th e fac t tha t i n thi s cas e x (th e numbe r o f rounds ) an d y (th e number o f call s mad e i n a round ) ar e discret e quantities . Th e answe r give n in th e book , 3 3 = 27 , arises fro m simpl y substitutin g int o th e equation . Th e problem ca n als o b e investigate d b y examinin g a tre e diagra m a s i n Figur e 2, an d an y teache r wh o wishe s t o fin d way s t o mak e us e o f th e standar d tools o f discret e mat h shoul d tak e suc h a n opportunit y t o d o this . However , if th e proble m i s discusse d explicitl y a s on e involvin g discret e mathematics , a ver y differen t solutio n ca n b e foun d and , alon g the way , student s ca n hav e further experienc e wit h th e behavio r o f discret e systems .

FIGURE 2 . Standar d phon e tree: eac h person call s three peo - ple, waitin g unti l th e previou s roun d i s complet e befor e be - ginning thei r ow n calls .

The answer , 27 , arises onl y i f it i s assumed tha t th e secon d generatio n o f calls ar e no t begu n unti l all three o f th e firs t call s ar e made , an d i n general , that th e n + 1s t generatio n o f call s ar e no t mad e unti l th e n t h generatio n i s completed. Suppose , instead , tha t w e assum e tha t eac h cal l take s on e tim e slot (3 ^ minute s i n thi s proble m a s originall y stated ) an d tha t a perso n called i n tim e slo t t place s he r firs t cal l i n tim e slo t t + 1 instead o f waitin g for th e nex t generation . Then , a s explaine d below , th e sequenc e o f number s of peopl e calle d i n eac h generatio n i s a ter m i n a Fibonacci-lik e sequence . (The numbe r o f call s stil l grow s exponentially. ) I f eac h perso n make s onl y two calls, an d Gt i s the numbe r o f calls mad e i n tim e slo t £ , the term s follo w the standar d Fibonacc i pattern ,

Gt+2 = G t+i + Gt ( f o r t > l ) ,

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JOAN R E I N T H A L E R

F I G U R E 3 . Phon e tre e in which each person call s two people, but eac h perso n calle d ca n star t thei r first cal l i n th e nex t time slot .

u 2 M

4 R

7 F

13 L L

2 4 S

F I G U R E 4 . Phon e tre e i n which eac h perso n call s thre e peo - ple i n succession .

where G\ = 1 and G<2 = 2. (Se e Figure 3. ) I f each perso n make s thre e calls , as i n the origina l problem , the n th e term s gro w accordin g t o th e recursiv e relation

Gt+3 = Gt+ 2 + G t+i + G t, ( f o r t > l ) ,

where G\ — 1, G^ — 2, and G 3 = 4 . (Se e Figure 4. ) I n thi s case , th e las t round occur s a t t = 9 , and G9, the numbe r o f call s made , i s 14 9 instead of 27!

Given thi s model , question s can be asked abou t th e most efficien t callin g system fo r a given numbe r o f people. Or , if each perso n call s m othe r peopl e (m = 2 , 3 , 4 , . . . ) , wha t happen s t o the number o f calls in the final time slot ?

One las t exampl e o f th e opportunitie s offere d b y standar d problem s i s an extensio n o f this one , found i n an intermediat e algebr a book :

The length and the width of a rectangle are in the ratio of 3:2. If each dimension is increased by 4 inches, the new length and width are in the ratio of 7:5. Find the original dimensions.

DISCRETE MATHEMATIC S I S ALREADY I N TH E CLASSROO M 29 9

Since th e domai n fo r length s i s continuous , th e solutio n o f thi s proble m is easil y foun d b y standar d algebrai c methods . Howeve r her e i s a proble m that, a t first glance , look s ver y similar :

As Michael Jordan steps to the foul line for two shots, the announcer reports that his foul-shooting percentage for this year is 78%. Jordan makes one and misses one. The next time he steps to the line, the announcer reports that he is shooting at 76%. How many foul shots has he attempted this year?

In thi s proble m th e domain , th e numbe r o f shots , i s discrete , an d th e percents reporte d ar e th e ratio s o f shot s mad e t o shot s attempted , rounde d to th e neares t hundredth. 1 Thi s mean s that , i f m stand s fo r th e numbe r o f shots mad e an d a stand s fo r th e numbe r o f shot s attempted , the n

m m ~\~ 1 .775 < — < .78 5 an d .75 5 < — — - < .765 .

a a + 2 There i s mor e tha n on e possibl e answe r t o thi s proble m (fo r example , a — 36, m = 2 8 or a = 40 , m = 3 1 are solutions) an d finding the m al l is not easy . (You ca n star t wit h a diagra m lik e tha t use d i n Figur e 1(b). ) However , th e analysis tha t lead s t o th e syste m o f inequalities is , in itself , a usefu l exercis e and on e tha t arise s onl y becaus e o f th e discret e natur e o f th e situation .

Investigations suc h a s th e one s outline d abov e d o no t represen t digres - sions fro m th e standar d curriculum . The y aris e fro m problem s tha t aboun d in traditiona l text s an d provid e teacher s a n opportunit y t o brin g discret e mathematics int o th e classroo m naturall y an d withi n th e constraint s o f a n existing curriculum . Wha t i t take s t o d o thi s ar e teacher s wh o ca n recog - nize the problem s i n their text s tha t ar e goo d jumping-off point s fo r discret e investigations, an d wh o ar e intereste d i n pursuin g th e mathematic s tha t re - veals itsel f whe n the y an d thei r student s jump .

T H E SIDWEL L F R I E N D S SCHOOL , 382 5 W I S C O N S I N AVENUE , N . W. , W A S H I N G T O N ,

D. C . 2001 6 E-mail address: joanrQumd5.umd.ed u

1 Editors' comment : Se e als o Pollak' s articl e i n thi s volume , Sectio n 3 , exampl e (c) , for furthe r discussio n o f thi s problem .

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DIMACS Serie s i n Discret e Mathematic s and Theoretica l Compute r Scienc e Volume 36 , 199 7

Integrating Discret e M a t h e m a t i c s int o t h e Curriculum: A n Exampl e

James T . Sandefu r

Many teacher s believ e tha t the y ar e bein g aske d t o teac h discret e math - ematics i n additio n t o th e mathematic s tha t i s alread y par t o f thei r cur - riculum. Sinc e ther e isn' t adequat e tim e t o cove r th e existin g curriculum , i t seems impossibl e t o ad d discret e mathematic s excep t o n enrichmen t days , such a s th e da y befor e Christma s break . Bu t on e ca n effectivel y cove r a lo t of mathematics b y spreadin g a n appropriat e structure d activit y ove r severa l classes. Thi s approac h integrate s discret e mathematic s int o the existin g cur - riculum, result s i n deepe r studen t understanding , an d ca n b e accomplishe d in abou t th e sam e amoun t o f tim e a s i s presently devote d t o existin g topics . This approach , whic h involve s usin g algebraic , geometric , an d discret e top - ics to stud y comple x problems , i s the approac h recommende d b y th e NCTM Curriculum and Evaluation Standards [3] .

Let m e relat e a n exampl e tha t illustrate s thi s point . A t firs t glanc e th