Encouraging Mathematical Thinking


 Abstract
 Introduction

 Discourse
 Interventions
 Decisions
  - Balance
  - Wait-time
  - Classroom Norms

 Cylinder Problem
 Lesson Reflections
 Student Predictions

 Project Reflections
 Conclusion

 References
 Acknowledgments
 Teacher Resources



Authors'
Biographies

Table of Contents


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Making Decisions: Wait-Time



In our discussions about teaching the cylinder problem, the conversation frequently turned to the importance of wait-time (Tobin, 1983a, 1983b). For us, wait-time includes providing time for students to generate their own solutions to problems, waiting for a student to find the words for an explanation, and listening patiently as students try to put their questions into words. In addition, we also realized the importance of giving ourselves wait-time -- time to weigh options before deciding what to do next.

In order to become proficient at self-directed inquiry, it helps for students to use other students as resources and to support each other through the frustrations of problem-solving. When we wait for several students to respond to a question, or when we accept different solutions to a problem before we respond to any one of them, we find that we are providing our students with the opportunity to evaluate and help each other with their responses.

We need to resist the temptation to answer all of our students' questions, or to tell them what to do next. If we intervene too much, they may become dependent on us and, more importantly, may not be able to cultivate the power of their own thinking. Waiting after posing a question sends a message that students can explore their ideas without help from the teacher, increasing their confidence. In this clip, Susan Stein encourages one student to help another:

  [view clip]


In this next clip, we see Susan Stein repeat a student's question in order to focus attention on the question asked and on the student's ownership of the question, rather than on herself [view clip].

A similar goal can be accomplished through a general review of the questions posed by the class. In the following clip, we see Art Mabbott reminding students of the questions they were trying to answer during class.

  [view clip]


When we provide students with time to think, and enable them to face the moments of uncertainty in problem-solving, we show our confidence in their abilities. Given enough time, students can develop the confidence to begin posing their own questions, and can seek resources to find answers for themselves. Here, for example, four students in Susan Boone's class work together to determine the relation between a cylinder's radius and its volume.

  [view clip]


In a supportive classroom students do learn to think together, helping each other to address their questions. In this clip from Susan Stein's class, we see one student correcting another as the teacher merely restates the proposed definition. This sort of experience provides students with evidence that thinking with others is an important tool in mathematics, and may give them confidence in their ability to guide each other.

  [view clip]


In John McKinstry's class, students are also encouraged to share their thoughts [view clip]. At this point in the lesson, students are making predictions about which cylinder will hold more. Notice that John does not tell the students whether or not their ideas are correct.

"I don't get it."

While we began our conversation by focusing on the concept of wait-time, we are also talking about teachers not jumping in with an answer, but knowing when to make room for students to figure out what is needed. Another form of this becomes visible when a student says, "I don't get it." We could assume that the student does not understand anything about the problem, and might launch into a re-explanation of the concept or procedure. There are alternatives, however, that can do more to support students' problem-solving abilities. In an anecdote from a different point in the year, Susan Stein describes how she learned what was causing her student difficulty:

One of my seventh graders did not complete his homework several times during the first two weeks of school. The first time he said he didn't understand the question, the next time he said he forgot, and the third time he again said he "didn't get it." I was becoming concerned and finally pinned him down to meet with me.

The two problems he "didn't get" were very similar. Both were pretty open-ended and involved four different, simple sketches of graphs showing only quadrant I, with no numbers or other identifiers on the axes. The graphs were all different shapes, a straight line going up, a curve that increased more at the start and then tapered off, a line with a negative slope, an arch-shaped curve. The two variables being compared were named in the body of the question. The first one compared age and height; the second compared selling price per T-shirt and profit. In each case the question asked which representation best showed the relation between the two variables, and expected an explanation/justification of the student's reasoning. Also in both cases, there were several "right" answers possible.

It took a while to get Chen to explain what he meant by "not getting it." I asked him to tell me what the different graphs showed. For the height and age one, he could describe pretty well how the straight (positive) line showed that height increased as age increased, the one that tapered off showed that you grew quickly when you were young, but slowed down, even stopped as you got older, etc. In fact, Chen understood each graph in both problems. With no specific prompting from me (except to ask, "what does this one mean?"), Chen seemed to me to "get it" very well.

So, what was the trouble? The trouble was that Chen understood very well. He was able to justify each of the four choices as being reasonable representations of the relation between age and height. "So," he asked, "how do I know which one is right?" What he didn't "get" was how to answer a question with more than one right answer. Helping Chen recognize that he understood the mathematical ideas allowed him to gain confidence in his own ability to evaluate when his ideas were "right" and learn how to demonstrate his knowledge.


Much can happen as a result of the mere expectation that the student has more to say. Whether we wait silently or ask questions, we create space for the student to figure out a way forward. We can ask for a description of what the problem asks. We can involve other students in explaining what they understand. We can ask a question like, "What part do you understand?" instead of, "What don't you understand?" We can help students articulate which parts of a problem make sense, in order to help them move from "I don't get it" to "I know this much, but I'm stuck on this part." When students start from what is already known, they are no longer totally lost.

Of course there are many important factors other than discourse strategies that enable students to have confidence that they can work from what they know, and can increasingly rely on themselves and each other. We would like to mention classroom norms as one area that interests us, and as a topic we hope is developed in the conversations that build from this paper.

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