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GPS Training Days 1, 2 and 3 Mathematics 1 Research and Resource Manual

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Reprinted with permission from MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

Copyright@ 2000 The National Council of Teachers of Mathematics, Inc. All rights reserved.

STEVEN C. REINHART

.478 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL Copyright@ 2000 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other, format without written permission from NCTM.

AFTER EXTENSIVE PLANNING, I PRESENTED

what should have been a masterpiece lesson. I worked

several examples on the overhead projector, answered

every student's question in great detail, and explained

the concept so clearly that surely my students

understood. The next day, however, it became

obvious that the students were totally confused. In my

early years of teaching, this situation happened all too

often. Even though observations by my principal

clearly pointed out that I was very good at explaining

mathematics to my students, knew my subject matter

well, and really seemed to be a dedicated and caring

teacher, something was wrong. My students were

capable of learning much more than they displayed.

Implementing Change over time THE LOW LEVELS OF ACHIEVEMENT ~ of many

students caused me to question ~ how I was teaching,

and my search for a ~ better approach began. Making

a commitment to change 10 percent of my if teaching

each year, I began to collect and use materials and

ideas gathered from supplements, workshops,

professional journals, and university classes. Each

year, my goal was simply to teach a single topic in a

better way than I had the year before.

STEVE REINHART, [email protected], teaches

mathematics at Chippewa Falls Middle School, ChiPpewa Falls,

WI 54729. He is interested in the teaching of algebraic thinking at

the middle school level and in the professional development of

teachers.

Before long, I noticed that the familiar teacher-

centered, direct-instruction model often did not fit

well with the more in-depth problems and tasks that I

was using. The information that I had gathered also

suggested teaching in nontraditional ways. It was not

enough to teach better mathematics; I also had to

teach mathematics better. Making changes in

instruction proved difficult because I had to learn to

teach in ways that I had never observed or

experienced, challenging many of the old teaching

paradigms. As I moved from traditional methods of

instruction to a more student-centered, problem-based

approach, many of my students enjoyed my classes

more. They really seemed to like working together,

discussing and sharing their ideas and solutions to the

interesting, often contextual, problems that I posed.

The small changes that I implemented each year

began to show results. In five years, I had almost

completely changed both what and how I was

teaching.

The Fundamental Flaw AT SOME POINT DURING THIS METAMORPHOSIS, I

concluded that a fundamental flaw existed in my

teaching methods. When I was in front of the class

demonstrating and explaining, I was learning a great

deal, but many of my students were not! Eventually, I

concluded that if my students were to ever really learn

mathematics, they would have to do the explaining,

and I, the listening. My definition of a good teacher

has since changed from "one who explains things so

well that students understand" to "one who gets

students to explain things so well that they can be

understood."

Getting middle school students to explain their

thinking and become actively involved in classroom

discussions can be a challenge. By nature, these

GPS Training Days 1, 2 and 3 Mathematics 1 Research and Resource Manual

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students are self-conscious and insecure. This inse-

curity and the effects of negative peer pressure tend

to discourage involvement. To get beyond these and

other roadblocks, I have learned to ask the best

possible questions and to apply strategies that require

all students to participate. Adopting the goals and

implementing the strategies and questioning

techniques that follow have helped me develop and

improve my questioning skills. At the same time,

these goals and strategies help me create a classroom

atmosphere in which students are actively engaged in

learning mathematics and feel comfortable in sharing

and discussing ideas, asking questions, and taking

risks.

Questioning Strategies That Work for Me

ALTHOUGH GOOD TEACHERS PLAN

DETAILED lessons that focus on the mathematical

content, few take the time to plan to use specific

questioning techniques on a regular basis. Improving

questioning skills is difficult and takes time, practice,

and planning. Strategies that work once will work

again and again. Making a list of good ideas and

strategies that work, revisiting the list regularly, and

planning to practice selected techniques in daily

lessons will make a difference.

Create a plan.

The following is a list of reminders that I have

accumulated from the many outstanding teachers

with whom I have worked over several years. I revisit

this list often. None of these ideas is new, and I can

claim none, except the first one, as my own.

Although implementing any single suggestion from

this list may not result in major change, used

together, these suggestions can help transform a

classroom. Attempting to change too much too fast

may result in frustration and failure. Changing a little

at a time by selecting, practicing, and refining one or

two strategies or skills before moving on to others

can result in continual, incremental growth.

Implementing one or two techniques at a time also

makes it easier for students to accept and adjust to the

new expectations and standards being established.

1. Never say anything a kid can say! This one goal

keeps me focused. Although I do not think that I have

ever met this goal completely in anyone day or even

in a given class period, it has forced me to develop

and improve my questioning skills. It also sends a

message to students that their participation is

essential. Every time I am tempted to tell students

something, I try to ask a question instead.

2. Ask good questions. Good questions require more

than recalling a fact or reproducing a skill. By asking

good questions, I encourage students to think about, and

reflect on, the mathematics they are learning. A student

should be able to learn from answering my question, and

I should be able to learn something about what the

student knows or does not know from her or his re-

sponse. Quite simply, I ask good questions to get

students to think and to inform me about what they

know. The best questions are open ended, those for

which more than one way to solve the problem or more

than one acceptable response may be possible.

3. Use more process questions than product questions.

Product questions-those that require short answers or a

yes or no response or those that rely almost completely

on memory-provide little information about what a

student knows. To find out what a student understands, I

ask process questions that require the student to reflect,

analyze, and explain his or her thinking and reasoning.

Process questions require students to think at much

higher levels.

4. Replace lectures with sets of questions. When

tempted to present information in the form of a lecture, I

remind myself of this definition of a lecture: 'The transfer

of information from the notes of the lecturer to the notes

of the student without passing through the minds of

either." If I am still tempted, I ask myself the humbling

question ―What percent of my students will actually be

listening to me?‖

5. Be patient. Wait time is very important. Although

some students always seem to have their hands raised

immediately, most need more time to process their

thoughts. If I always call on one of the first students who

volunteers, I am cheating those who need more time to

think about, and process a response to, my question.

Even very capable students can begin to doubt their

abilities, and many eventually stop thinking about my

questions altogether. Increasing wait time to five seconds

or longer can result in more and better responses.

Good discussions take time; at first, I was un-

comfortable in taking so much time to discuss a single

question or problem. The urge to simply tell my students

and move on for the sake of expedience was

considerable. Eventually, I began to see the value in what

I now refer to as a "less is more" philosophy. I now

believe that all students learn more when I pose a high-

quality problem and give them the necessary time to

investigate, process their thoughts, and reflect on and

defend their findings.

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GPS Training Days 1, 2 and 3 Mathematics 1 Research and Resource Manual

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Share with students reasons for asking

questions. Students should understand that all their

statements are valuable to me, even if they are

incorrect or show misconceptions. I explain that I ask

them questions because I am continuously evaluating

what the class knows or does not know. Their

comments help me make decisions and. plan the next

activities.

Teach for success. If students are to value my

questions and be involved in discussions, I cannot use

questions to embarrass or punish. Such questions

accomplish little and can make it more difficult to

create an atmosphere in which students feel

comfortable sharing ideas and taking risks. If a

student is struggling to respond, I move on to another

student quickly. As I listen to student conversations

and observe their work, I also identify those who

have good ideas or comments to share. Asking a shy,

quiet student a question when I know that he or she

has a good response is a great strategy for building

confidence and self-esteem. Frequently, I alert the

student ahead of time: 'That's a great idea. I'd really

like you to share that with the class in a few

minutes."

Be nonjudgmental about a response or

comment. This goal is indispensable in encouraging

discourse. Imagine being in a classroom where the

teacher makes this comment: ''WOW! Brittni, that

was a terrific, insightful response! Who's next?" Not

many middle school students have the confidence to

follow a response that has been praised so highly by a

teacher. If a student's response reveals a miscon-

ception and the teacher replies in a negative way, the

student may be discouraged from volunteering again.

Instead, encourage more discussion and move on to

the next comment. Often, students disagree with one

another, discover their own errors, and correct their

thinking. Allowing students to listen to fellow

classmates is a far more positive way to deal with

misconceptions than announcing to the class that an

answer is incorrect. If several students remain

confused, I might say, "I'm hearing that we do not

agree on this issue. Your comments and ideas have

given me an idea for an activity that will help you

clarify your thinking." I then plan to revisit the

concept with another activity as soon as possible.

Try not to repeat students' answers. If students are

to listen to one another and value one another’s input,

I cannot repeat or try to improve on what they say. If

students realize that I will repeat or clarify what

another student says, they no longer have a reason to

listen. I must be patient and let students clarify their

own thinking and encourage them to speak to their

classmates, not just to me.

All students can speak louder - I have heard them in the

halls! Yet I must be careful not to embarrass someone

with a quiet voice. Because students know that I never

accept just one response, they think nothing of my asking

another student to paraphrase the soft-spoken comments

of a classmate.

Is this the right answer?” Students frequently ask

this question. My usual response to this question might

be that ―I’m not sure. Can you explain your thinking to

me?" As soon as I tell a student that the answer is

correct, thinking stops. If students explain their thinking

clearly, I ask a ''What if?" question to encourage them to

extend their thinking.

Participation is not optional! I remind my students

of this expectation regularly. Whether working in small

groups or discussing a problem with the whole class,

each student is expected to contribute his or her fair

share. Because reminding students of this expectation is

not enough, I also regularly apply several of the

following techniques:

1. Use the think-pair-share strategy. Whole-group

discussions are usually improved by using this technique.

When I pose a new problem; present a new project, task,

or activity; or simply ask a question, all students must

think and work independently first. In the past, letting

students begin working together on a task always

allowed a few students to sit back while others took over.

Requiring students to work alone first reduces this

problem by placing the responsibility for learning on

each student. This independent work time may vary from

a few minutes to the entire class period, depending on the

task.

After students have had adequate time to work

independently, they are paired with partners or join small

groups. In these groups, each student is required to report

his or her findings or summarize his or her solution

process. When teams have had the chance to share their

thoughts in small groups, we come together as a class to

share our findings. I do not call for volunteers but simply

ask one student to report on a significant point discussed

in the group. I might say, ''Tanya, will you share with the

class one important discovery your group made?" or

"James, please summarize for us what Adam shared with

you." Students generally feel much more confident in

stating ideas when the responsibility for the response is

being shared with a partner or group. Using the think-

pair-share strategy helps me send the message that

participation is not optional.

A modified version of this strategy also works in

whole-group discussions. If I do not get the responses

that I expect, either in quantity or quality, I give students

a chance to discuss the question in small groups. On the

basis of the difficulty of the question, they may have as

little as fifteen seconds or as long as

480 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

GPS Training Days 1, 2 and 3 Mathematics 1 Research and Resource Manual

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several minutes to discuss the question with their

partners. This strategy has helped improve discus-

sions more than any others that I have adopted.

2. If students or groups cannot answer a question

or contribute to the discussion in a positive way, they

must ask a question of the class. I explain that it is all

right to be confused, but students are responsible for

asking questions that might help them understand.

3. Always require students to ask a question when

they need help. When a student says, "I don't get it,"

he or she may really be saying, "Show me an easy

way to do this so I don't have to think." Initially,

getting students to ask a question is a big im-

provement over "I don't get it." Students soon realize

that my standards require them to think about the

problem in enough depth to ask a question.

4. Require several responses to the same question.

Never accept only one response to a question.

Always ask for other comments, additions, clarifica-

tions, solutions, or methods. This request is difficult

for students at first because they have been condi-

tioned to believe that only one answer is correct and

that only one correct way is possible to solve a prob-

lem. I explain that for them to become better thinkers,

they need to investigate the many possible ways of

thinking about a problem. Even if two students use

the same method to solve a problem, they rarely

explain their thinking in exactly the same way.

Multiple explanations help other students understand

and clarify their thinking. One goal is to create a

student-centered classroom in which students are

responsible for the conversation. To accomplish this

goal, I try not to comment after each response. I

simply pause and wait for the next student to offer

comments. If the pause alone does not generate

further discussion, I may ask, "Next?" or ''What do

you think about _____________________________________________'s idea?"

5. No one in a group is finished until everyone in

the group can explain and defend the solution. This

rule forces students to work together, communicate,

and be responsible for the learning of everyone in the

group. The learning of any one person is of little

value unless it can be communicated to others, and

those who would rather work on their own often need

encouragement to develop valuable communication

skills.

6. Use hand signals often. Using hand signals -

thumbs up or thumbs down (a horizontal thumb

means "I'm not sure") - accomplishes two things.

First, by requiring all students to respond with hand

signals, I ensure that all students are on task. Second,

by observing the responses, I can find out how many

students are having difficulty or do not understand.

Watching students' faces as they think about how to

respond is very revealing.

7. Never carry a pencil. If I carry a pencil with me or

pick up a student's pencil, I am tempted to do the work

for the student. Instead, I must take time to ask thought-

provoking questions that will lead to understanding.

8. Avoid answering my own questions. Answering my

own questions only confuses students because it requires

them to guess which questions I really want them to

think about, and I want them to think about all my

questions. I also avoid rhetorical questions.

9. Ask questions of the whole group. As soon as I

direct a question to an individual, I suggest to the rest of

the students that they are no longer required to think.

10. Limit the use of group responses. Group responses

lower the level of concern and allow some students to

hide and not think about my questions.

11. Do not allow students to blurt out answers. A

student's blurted out answer is a signal to the rest of the

class to stop thinking. Students who develop this habit

must realize that they are cheating other students of the

right to think about the question.

Summary LIKE MOST TEACHERS, I ENTERED THE TEACHING

profession because I care about children. It is only

natural for me to want them to be successful, but by

merely telling them answers, doing things for them, or

showing them shortcuts, I relieve students of their

responsibilities and cheat them of the opportunity to

make sense of the mathematics that they are learning. To

help students engage in real learning, I must ask good

questions, allow students to struggle, and place the

responsibility for learning directly on their shoulders. I

am convinced that children learn in more ways than I

know how to teach. By listening to them, I not only give

them the opportunity to develop deep understanding but

also am able to develop true insights into what they

know and how they think.

Making extensive changes in curriculum and instruction

is a challenging process. Much can be learned about how

children think and learn, from recent publications about

learning styles, multiple intelligences, and brain research.

Also, several reform curriculum projects funded by the

National Science Foundation are now available from

publishers. The Connected Mathematics Project,

Mathematics in Context, and Math Scape, to name a few,

artfully address issues of content and pedagogy.

Bibliography Burns, Marilyn. Mathematics: For Middle School. New

Rochelle, N.Y.: Cuisenaire Co. of America, 1989.

Johnson, David R. Every Minute Counts. Palo Alto, Calif.:

Dale Seymour Publications, 1982.

National Council of Teachers of Mathematics (NCTM).

Professional Standards for Teaching Mathematics.

Reston, Va.: NCTM, 1991.

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