Discussion 1

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478 M A T H E M A T I C S T E A C H I N G I N T H E M I D D L E S C H O O L

A FTER EXTENSIVE PLANNING, I PRE- sented what should have been a master- piece lesson. I worked several examples on 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 totally confused. In my early years of teaching, this situa-

tion happened all too often. Even though observations by my principal clearly pointed out that I was very good at explaining mathematics to my stu- dents, knew my subject matter well, and really seemed to be a dedicated and car- ing 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 com- mitment to change 10 percent of my teaching 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 to teach a single topic in a better way than I had the year before.

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 gath- ered also suggested teaching in nontraditional ways. It was not enough to teach better mathemat- ics; 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 ob- served 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 en- joyed my classes more. They really seemed to like working together, discussing and sharing their ideas and solutions to the interesting, often contex- tual, 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! Even- tually, 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 ex- plains 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

STEVE REINHART, [email protected], teaches mathematics at Chippewa Falls Middle School, Chippewa Falls, 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.

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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 ques- tioning techniques that follow have helped me de- velop and improve my questioning skills. At the same time, these goals and strategies help me cre- ate 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 question-

ing 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 regu- larly, and planning to practice se- lected 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 imple- menting any single suggestion from this list may not result in major change, used together, these suggestions can help transform a classroom. At- tempting to change too much too fast may result in frustration and failure. Changing a little at a time by

Students feel comfortable sharing and discussing ideas

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480 M A T H E M A T I C S T E A C H I N G I N T H E M I D D L E S C H O O L

selecting, practicing, and refining one or two strate- gies 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 expec- tations 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 any one 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 recall- ing a fact or reproducing a skill. By asking good questions, I encourage students to think about, and reflect on, the mathematics they are learn- ing. 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 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 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 lec- ture, 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. Al- though 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 stu- dents can begin to doubt their abilities, and many eventually stop thinking about my questions alto- gether. 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 sin- gle 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” phi- losophy. 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.

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 eval- uating 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 ques- tions accomplish little and can make it more diffi- cult to create an atmosphere in which students feel comfortable sharing ideas and taking risks. If a stu- dent 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 com- ment. This goal is indispensable in encouraging discourse. Imagine being in a classroom where the

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The best questions are open-ended

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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 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 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 stu-

dents 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 real- ize 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 en- courage 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 com- ments of a classmate.

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

Let students clarify their own thinking

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482 M A T H E M A T I C S T E A C H I N G I N T H E M I D D L E S C H O O L

explain your thinking to me?” As soon as I tell a stu- dent 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 stu- dents of this expectation regularly. Whether work- ing in small groups or discussing a problem with the 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 are usually 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, 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 min- utes to the entire class period, de- pending 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 find- ings. 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 im- portant 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 re- sponse 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 stu- dents 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 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 ques- tion 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.” 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 about the 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 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 stu- dents use the same method to solve a problem, they rarely explain their thinking in exactly the same way. Multiple explanations help other students un- derstand and clarify their thinking. One goal is to create a student-centered classroom in which stu- dents are responsible for the conversation. To ac- complish this goal, I try not to comment after each response. I simply pause and wait for the next stu- dent 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, communi- cate, and be responsible for the learning of every- one in the group. The learning of any one person is of little value unless it can be communicated to oth- ers, and those who would rather work on their own often need encouragement to develop valuable communication skills.

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No one is finished until all can explain the solution

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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 re- sponses 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 op- portunity 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 re- form curriculum projects funded by the National Science Foundation are now available from pub- lishers. 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. C

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