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Manipulatives Enhance the Learn i n g of Mathematics

D r. Jean M. S h a w

and Professor Emerita of Elementary Education at University of Mississippi

Manipulatives take many forms in elementary and middle grades classro o m s w h e re students are learning mathematics by doing mathematics. Base-ten blocks, two-colored counters, fraction strips, beans, and geometric solids are a few of the many manufactured and teacher- m a d e manipulatives that stu- dents might use during their K–6 mathematics experience. M a n i p u l a t i v e s have been used over the long term for many years at the primary and early e l e m e n t a ry grades. However, as students pro g ress through their later ele- mentary and middle grades mathematics learning, it is i m p o rtant f o r i n s t ructional materials to continue including manipulatives. The National Council of Teachers of Mathematics (N C T M) Principles and Standards for School Mathematics emphasizes the importance of using manipulatives and visual re p resentations, as well as mathematical modeling, in each of its standards at all grade levels. This paper discusses some of the ways manipulatives can be used to enhance and deepen mathematical understanding for all students.

Building Understanding and Clarifying Concepts

Manipulatives help students develop conceptual understanding of mathe- matical ideas by re p resenting the ideas in multiple ways. For example, consider comparing unit fractions with unlike denominators such as 1/2 and 1/8. The symbol for halves looks like it re p resents a smaller number (with 2 in the denomi- nator) than eighths (with 8 in the denominator). At a symbolic level, many students would have difficulty understanding that the answer will be gre a t e r

Student Edition,Grade 2,page 315

than 1/2. However, when students use fraction models to re p resent each fraction in the algorithm and see that 1/2 is greater than 1/8, they begin to build mental images of the relative size of fractional parts of wholes. With this cleare r visual understanding, there is obviously less confu- sion in the students’ minds as to why s m a l l e r denominators suggest larger parts and larg e r denominators re p resent smaller part s .

When there is less confusion, deeper understanding can begin to take hold, develop, and g ro w, thereby lay- ing the gro u n d- work for future m a t h e m a t i c s l e a rning. It is also

i m p o rtant to recognize that when there is less con- fusion or conflict of mathematical ideas in a stu- d e n t ’s mind, then there are fewer meaningless ru l e s to remember or commit to memory. For example, t h rough visual and tactile work, students know and understand why visualizing and comparing frac- tions is somewhat diff e rent from working with whole numbers. Students can then deepen that understanding and explain why the symbolic re p re- sentation for fractions is not the same as the re l a- tive magnitude of whole numbers. Students do not need to rely on rules to understand what symbols re p re s e n t .

This “ownership” of knowledge has other benefits as well. Healthy attitudes bring with them intrinsic

re w a rds. If a stu- dent fully under- stands a concept or idea, then fear of the subject matter is lessened. In m a t h e m a t i c s this is part i c u l a r l y i m p o rtant since

“The wealth of knowledge that mathematics and science i m p a rt for understanding the world has such b re a d t h that it is easy to overlook the dimension of depth.” (A Report to the Nation, p. 14) Teachers are charged with the responsibility of ensuring that students grapple with pro b l e m s , thus building a deeper understanding of mathemat- ics and the roles it plays in all of our lives.

Engagement, Communication, and Multisensory Experiences

In addition to helping students build deeper math- ematics understanding and gain the benefits of healthy attitudes toward mathematics as a discipline, manipulatives and models are valuable resource tools for engaging students in the language and communication of mathematical ideas and con- cepts. “Students need opportunities to test their ideas on the basis of shared knowledge in the mathematical community of the classroom to see whether they can be understood and if they are s u fficiently convincing.” (N C T M, p. 61) When

s t u d e n t s

E S S E N T I A L C H A R AC T E R I S T I C S O F E F F E C T I V E M AT H E M AT I C S I N S T RU C T I O N

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can look at something that visually re p resents a mathematical concept, they have greater accessibili- ty to the language needed to describe it. Visual learning is exemplified. When students physically move manipulatives to show various relationships, their sense of touch is actively engaged. This enhances understanding and, in

t u rn, promotes communication of ideas.

When small or large learning groups explore new ideas or explain understandings, they have some- thing to talk about when they have manipulatives and models to work with. Discussions and learn i n g become more focused. Multisensory experiences p rovide access to ideas and concepts, and offer multiple entry points in discussions and re a s o n i n g , ensuring that all students in the group are active p a rt i c i p a n t s .

Manipulatives and models also aff o rd English language learners greater access to language and mathematical term i n o l o g y. A physical re p re s e n t a- tion of a mathematical idea or solution might p rovide an English language learner with gre a t e r confidence in his or her solution. Te rms in a new language are easier to learn when used in the context of a model. Just as student dictionaries provide illustrations of nouns, manipulative re p re s e n t a t i o n s of concepts and solutions provide illustrations of mathematical concepts and ideas.

Using mathematics manipulatives and models o ffers many benefits. Just as a picture can be wort h a thousand words, manipulatives can provide visual re p resentations of ideas, helping students to know and to understand mathematics. Manipulatives enhance the abilities of students at all levels to reason and communicate. Working with manipula- tives deepens understanding of concepts and relationships, makes skills practice meaningful, and leads to retention and application of information in new problem-solving situations. In turn, the valu- able time spent on manipulative- and m o d e l - based lessons has the sustained, long-term effect of building student con- fidence and deepening mathematics understanding. Indeed, it is time well spent!

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Student Edition,Grade 3,page 516

R e f e r e n c e s

Clements, D. H. and M. T. Battista. (1990). Constructivist learning and teaching. Arithmetic Te a c h e r, vol. 38, no. 1.

H i e b e rt, James. The struggle to link written symbols with understandings: An update. Arithmetic Te a c h e r, vol. 36, no. 7.

Suydam, Marilyn N. 1984. Manipulative materials and achievement. Arithmetic Te a c h e r, vol. 31, no. 27.

National Council of Teachers of Mathematics (N C T M). Principles and Standards for School Mathematics. Reston, VA: N C T M, 2000.

U. S. Department of Education. 2000. B e f o re It’s Too Late: A Report to the Nation from the Commission on Mathematics and Science Teaching for the 21st Century. Washington, D.C.: Education Publication Center.