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478 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

A FTER EXTENSIVE PLANNING, I PRE-sented what should have been a master-piece lesson. I worked several exampleson the overhead projector, answered

every student’s question in great detail, and ex-plained the concept so clearly that surely my stu-dents understood. The next day, however, it be-came obvious that the students were totallyconfused. In my early years of teaching, this situa-

tion happened all too often. Eventhough observations by my principalclearly pointed out that I was very goodat explaining mathematics to my stu-dents, knew my subject matter well, andreally seemed to be a dedicated and car-ing teacher, something was wrong. Mystudents were capable of learning muchmore than they displayed.

Implementing Change over Time

THE LOW LEVELS OF ACHIEVEMENT of many students caused me to questionhow I was teaching, and my search for abetter approach began. Making a com-mitment to change 10 percent of myteaching each year, I began to collect

and use materials and ideas gathered from supple-ments, workshops, professional journals, and uni-versity classes. Each year, my goal was simply toteach a single topic in a better way than I had theyear before.

Before long, I noticed that the familiar teacher-centered, direct-instruction model often did not fitwell with the more in-depth problems and tasksthat I was using. The information that I had gath-ered also suggested teaching in nontraditionalways. It was not enough to teach better mathemat-ics; I also had to teach mathematics better. Makingchanges in instruction proved difficult because Ihad to learn to teach in ways that I had never ob-served or experienced, challenging many of the oldteaching paradigms. As I moved from traditionalmethods of instruction to a more student-centered,problem-based approach, many of my students en-joyed my classes more. They really seemed to likeworking together, discussing and sharing theirideas and solutions to the interesting, often contex-tual, problems that I posed. The small changes thatI implemented each year began to show results. Infive years, I had almost completely changed bothwhat and how I was teaching.

The Fundamental Flaw

AT SOME POINT DURING THIS METAMORPHOSIS,I concluded that a fundamental flaw existed in myteaching methods. When I was in front of the classdemonstrating and explaining, I was learning agreat deal, but many of my students were not! Even-tually, I concluded that if my students were to everreally learn mathematics, they would have to do theexplaining, and I, the listening. My definition of agood teacher has since changed from “one who ex-plains things so well that students understand” to“one who gets students to explain things so wellthat they can be understood.”

Getting middle school students to explain theirthinking and become actively involved in classroomdiscussions can be a challenge. By nature, these

STEVE REINHART, [email protected], teachesmathematics at Chippewa Falls Middle School, ChippewaFalls, WI 54729. He is interested in the teaching of alge-braic thinking at the middle school level and in the profes-sional development of teachers.

Never Say Anything

a Kid Can Say!a Kid Can Say!S T E V E N C. R E I N H A R T

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Copyright © 2000 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For use in EdTech Leaders Online Course Only. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

VOL. 5, NO. 8 . APRIL 2000 479

students are self-conscious and insecure. This inse-curity and the effects of negative peer pressuretend to discourage involvement. To get beyondthese and other roadblocks, I have learned to askthe best possible questions and to apply strategiesthat require all students to participate. Adopting thegoals and implementing the strategies and ques-tioning techniques that follow have helped me de-velop and improve my questioning skills. At thesame time, these goals and strategies help me cre-ate a classroom atmosphere in which students areactively engaged in learning mathematics and feelcomfortable in sharing and discussing ideas, askingquestions, and taking risks.

Questioning Strategies That Work for Me

ALTHOUGH GOOD TEACHERS PLAN DETAILEDlessons that focus on the mathematical content, fewtake the time to plan to use specific questioningtechniques on a regular basis. Improving question-

ing skills is difficult and takestime, practice, and planning.Strategies that work once willwork again and again. Making alist of good ideas and strategiesthat work, revisiting the list regu-larly, and planning to practice se-lected techniques in daily lessonswill make a difference.

Create a plan. The following is alist of reminders that I have accumulated from themany outstanding teachers with whom I haveworked 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 imple-menting any single suggestion from this list maynot result in major change, used together, thesesuggestions can help transform a classroom. At-tempting to change too much too fast may result infrustration and failure. Changing a little at a time by

Students feelcomfortablesharing anddiscussingideas

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selecting, practicing, and refining one or two strate-gies or skills before moving on to others can resultin continual, incremental growth. Implementingone or two techniques at a time also makes it easierfor students to accept and adjust to the new expec-tations and standards being established.

1. Never say anything a kid can say! This onegoal keeps me focused. Although I do not think thatI have ever met this goal completely in any one dayor even in a given class period, it has forced me todevelop and improve my questioning skills. It alsosends a message to students that their participation

is essential. Every time I am temptedto tell students something, I try toask a question instead.

2. Ask good questions. Goodquestions require more than recall-ing a fact or reproducing a skill. Byasking good questions, I encouragestudents to think about, and reflecton, the mathematics they are learn-ing. A student should be able to learnfrom answering my question, and Ishould be able to learn somethingabout what the student knows ordoes not know from her or his re-sponse. Quite simply, I ask goodquestions to get students to thinkand to inform me about what theyknow. The best questions are open-ended, those for which more thanone way to solve the problem ormore than one acceptable responsemay be possible.

3. Use more process questionsthan product questions. Productquestions—those that require shortanswers or a yes or no response orthose that rely almost completely on

memory—provide little information about what astudent knows. To find out what a student under-stands, I ask process questions that require the stu-dent to reflect, analyze, and explain his or her think-ing and reasoning. Process questions requirestudents to think at much higher levels.

4. Replace lectures with sets of questions. Whentempted to present information in the form of a lec-ture, I remind myself of this definition of a lecture:“The transfer of information from the notes of thelecturer to the notes of the student without passingthrough the minds of either.” If I am still tempted, I

ask myself the humbling question “What percent ofmy students will actually be listening to me?”

5. Be patient. Wait time is very important. Al-though some students always seem to have theirhands raised immediately, most need more time toprocess their thoughts. If I always call on one of thefirst students who volunteers, I am cheating thosewho need more time to think about, and process aresponse to, my question. Even very capable stu-dents can begin to doubt their abilities, and manyeventually stop thinking about my questions alto-gether. Increasing wait time to five seconds orlonger 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 sin-gle question or problem. The urge to simply tell mystudents and move on for the sake of expediencewas considerable. Eventually, I began to see thevalue in what I now refer to as a “less is more” phi-losophy. I now believe that all students learn morewhen I pose a high-quality problem and give themthe necessary time to investigate, process theirthoughts, and reflect on and defend their findings.

Share with students reasons for askingquestions. Students should understand that alltheir statements are valuable to me, even if they areincorrect or show misconceptions. I explain that Iask them questions because I am continuously eval-uating what the class knows or does not know.Their comments help me make decisions and planthe next activities.

Teach for success. If students are to value myquestions and be involved in discussions, I cannotuse questions to embarrass or punish. Such ques-tions accomplish little and can make it more diffi-cult to create an atmosphere in which students feelcomfortable sharing ideas and taking risks. If a stu-dent is struggling to respond, I move on to anotherstudent quickly. As I listen to student conversationsand observe their work, I also identify those whohave good ideas or comments to share. Asking ashy, quiet student a question when I know that heor she has a good response is a great strategy forbuilding confidence and self-esteem. Frequently, Ialert the student ahead of time: “That’s a great idea.I’d really like you to share that with the class in afew minutes.”

Be nonjudgmental about a response or com-ment. This goal is indispensable in encouraging discourse. Imagine being in a classroom where the

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The best questions areopen-ended


VOL. 5, NO. 8 . APRIL 2000 481

teacher makes this comment: “Wow! Brittni, thatwas a terrific, insightful response! Who’s next?” Notmany middle school students have the confidence tofollow a response that has been praised so highly bya teacher. If a student’s response reveals a miscon-ception and the teacher replies in a negative way, thestudent may be discouraged from volunteeringagain. Instead, encourage more discussion andmove on to the next comment. Often, students dis-agree with one another, discover their own errors,and correct their thinking. Allowing students to lis-ten to fellow classmates is a far more positive way todeal with misconceptions than announcing to theclass that an answer is incorrect. If several studentsremain confused, I might say, “I’m hearing that wedo not agree on this issue. Your comments and ideashave given me an idea for an activity that will helpyou clarify your thinking.” I then plan to revisit theconcept with another activity as soon as possible.

Try not to repeat students’ answers. If stu-

dents are to listen to one anotherand value one another’s input, Icannot repeat or try to improveon what they say. If students real-ize that I will repeat or clarifywhat another student says, theyno longer have a reason to listen.I must be patient and let studentsclarify their own thinking and en-courage them to speak to theirclassmates, not just to me. All students can speaklouder—I have heard them in the halls! Yet I mustbe careful not to embarrass someone with a quietvoice. Because students know that I never acceptjust one response, they think nothing of my askinganother student to paraphrase the soft-spoken com-ments of a classmate.

“Is this the right answer?” Students fre-quently ask this question. My usual response tothis question might be that “I’m not sure. Can you

Let studentsclarify theirown thinking

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explain your thinking to me?” As soon as I tell a stu-dent that the answer is correct, thinking stops. Ifstudents explain their thinking clearly, I ask a“What if?” question to encourage them to extendtheir thinking.

Participation is not optional! I remind my stu-dents of this expectation regularly. Whether work-ing in small groups or discussing a problem withthe whole class, each student is expected to con-tribute his or her fair share. Because reminding stu-dents of this expectation is not enough, I also regu-larly apply several of the following techniques:

1. Use the think-pair-share strat-egy. Whole-group discussions areusually improved by using this tech-nique. When I pose a new problem;present a new project, task, or activ-ity; or simply ask a question, all stu-dents must think and work indepen-dently first. In the past, lettingstudents begin working together ona task always allowed a few studentsto sit back while others took over.Requiring students to work alonefirst reduces this problem by placingthe responsibility for learning oneach student. This independentwork time may vary from a few min-utes to the entire class period, de-pending on the task.

After students have had adequatetime to work independently, theyare paired with partners or joinsmall groups. In these groups, eachstudent is required to report his orher findings or summarize his orher solution process. When teamshave had the chance to share theirthoughts in small groups, we cometogether as a class to share our find-ings. I do not call for volunteers butsimply ask one student to report on

a significant point discussed in the group. I mightsay, “Tanya, will you share with the class one im-portant discovery your group made?” or “James,please summarize for us what Adam shared withyou.” Students generally feel much more confidentin stating ideas when the responsibility for the re-sponse is being shared with a partner or group.Using the think-pair-share strategy helps me sendthe message that participation is not optional.

A modified version of this strategy also works inwhole-group discussions. If I do not get the responses

that I expect, either in quantity or quality, I give stu-dents a chance to discuss the question in smallgroups. On the basis of the difficulty of the question,they may have as little as fifteen seconds or as long asseveral minutes to discuss the question with theirpartners. This strategy has helped improve discus-sions more than any others that I have adopted.

2. If students or groups cannot answer a ques-tion or contribute to the discussion in a positiveway, they must ask a question of the class. I explainthat it is all right to be confused, but students areresponsible for asking questions that might helpthem understand.

3. Always require students to ask a questionwhen they need help. When a student says, “I don’tget it,” he or she may really be saying, “Show me aneasy way to do this so I don’t have to think.” Ini-tially, getting students to ask a question is a big im-provement over “I don’t get it.” Students soon real-ize that my standards require them to think aboutthe problem in enough depth to ask a question.

4. Require several responses to the same ques-tion. Never accept only one response to a question.Always ask for other comments, additions, clarifica-tions, solutions, or methods. This request is difficultfor students at first because they have been condi-tioned to believe that only one answer is correct andthat only one correct way is possible to solve a prob-lem. I explain that for them to become betterthinkers, they need to investigate the many possibleways of thinking about a problem. Even if two stu-dents use the same method to solve a problem, theyrarely explain their thinking in exactly the sameway. Multiple explanations help other students un-derstand and clarify their thinking. One goal is tocreate a student-centered classroom in which stu-dents are responsible for the conversation. To ac-complish this goal, I try not to comment after eachresponse. I simply pause and wait for the next stu-dent to offer comments. If the pause alone does notgenerate further discussion, I may ask, “Next?” or“What do you think about __________’s idea?”

5. No one in a group is finished until everyone inthe group can explain and defend the solution. Thisrule forces students to work together, communi-cate, and be responsible for the learning of every-one in the group. The learning of any one person isof little value unless it can be communicated to oth-ers, and those who would rather work on their ownoften need encouragement to develop valuablecommunication skills.

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No one is finished untilall can explainthe solution


VOL. 5, NO. 8 . APRIL 2000 483

6. Use hand signals often. Using hand signals—thumbs up or thumbs down (a horizontal thumbmeans “I’m not sure”)—accomplishes two things.First, by requiring all students to respond withhand signals, I ensure that all students are on task.Second, by observing the responses, I can find outhow many students are having difficulty or do notunderstand. Watching students’ faces as they thinkabout how to respond is very revealing.

7. Never carry a pencil. If I carry a pencil with meor pick up a student’s pencil, I am tempted to do thework for the student. Instead, I must take time toask thought-provoking questions that will lead tounderstanding.

8. Avoid answering my own questions. Answeringmy own questions only confuses students because itrequires them to guess which questions I really wantthem to think about, and I want them to think aboutall my questions. I also avoid rhetorical questions.

9. Ask questions of the whole group. As soon as Idirect a question to an individual, I suggest to therest of the students that they are no longer requiredto think.

10. Limit the use of group responses. Group re-sponses lower the level of concern and allow somestudents to hide and not think about my questions.

11. Do not allow students to blurt out answers. Astudent’s blurted out answer is a signal to the restof the class to stop thinking. Students who developthis habit must realize that they are cheating otherstudents of the right to think about the question.

Summary

LIKE MOST TEACHERS, I ENTERED THE TEACHINGprofession because I care about children. It is onlynatural for me to want them to be successful, but bymerely telling them answers, doing things forthem, or showing them shortcuts, I relieve studentsof their responsibilities and cheat them of the op-portunity to make sense of the mathematics thatthey are learning. To help students engage in reallearning, I must ask good questions, allow studentsto struggle, and place the responsibility for learningdirectly on their shoulders. I am convinced thatchildren learn in more ways than I know how toteach. By listening to them, I not only give them theopportunity to develop deep understanding but alsoam able to develop true insights into what theyknow and how they think.

Making extensive changes in curriculum andinstruction is a challenging process. Much can belearned about how children think and learn, fromrecent publications about learning styles, multipleintelligences, and brain research. Also, several re-form curriculum projects funded by the NationalScience Foundation are now available from pub-lishers. The Connected Mathematics Project,Mathematics in Context, and Math Scape, toname a few, artfully address issues of content andpedagogy.

Bibliography

Burns, Marilyn. Mathematics: For Middle School. NewRochelle, 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. C

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