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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Interventions: Making Decisions


So I came back to my kids and talked to them about what it was that I wanted. If I ask a question and get no response, then I don't know what to do. If you answer the question correctly, I know where to go next. But if you answer the question incorrectly, then I know where to make a correction in what I am doing and know where to go next. I also talked about the risks that I was asking them to make in order to answer and possibly be wrong. To do that is very difficult for them as teenagers. Recognizing and validating this for the kids, while not completely solving the problem, went a long way to making my life simpler. Now, at least, they are willing to respond and are willing to take more risks.

This project allowed me the time and the safety to take the same kind of risk that I eventually asked of my kids.
-- Art Mabbott, Project teacher



While many interventions can be used to engage students in mathematical conversations, it is not always easy to determine which strategy to use, or when. How do I know when it would help to wait, rather than ask a question or volunteer more information? When should I summarize in order to move on, and when should I encourage my students to play out their thinking? How do I judge whether to probe for a misconception, or let classmates be responsible for sense-making and validation?

There are no easy answers to such questions. Here we will highlight several elements that influence our decision-making: balance, wait-time, and classroom norms.

Balance

Teachers must continually balance students' pursuit of their own questions with the imperatives of the curriculum. Consciously or not, we are always trying to accommodate competing needs: (a) individual thinking and learning with a collective search for knowledge, (b) a student's individual needs with the needs of the group, (c) allowing time for student discovery and conversation with covering a body of knowledge, (d) encouraging divergent ideas with moving toward a particular method or concept, and (e) leading the class while responding to student ideas about direction.

We are aware that we can easily interfere with the flow of discourse in the classroom. We can be too anxious to elicit a particular statement, and can jump in too quickly to turn students in that direction. As a result, we can miss critical contributions made by students who have questions or comments that are productive but different from our own. Often our missteps appear to result from losing the balance between covering the curriculum and supporting the development of students' understanding.

For example, when one of John McKinstry's students is explaining why the cylinder with the larger circumference will hold more because of the area of its base, a second student asks a question about height. John acknowledges that height is important, but instead of shifting the conversation to the second student's question, he summarizes the points discussed and moves to the next topic.

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Here John felt that his students understood the concepts well enough, and that it was time to move on. In another moment he might have placed greater value on the student's question than on his own summary statement.

In another situation, a student in John's class has difficulty explaining her answer to the question, "Why is the circumference important?" She struggles to find the right words to explain her prediction. John waits; rather than filling in the words she needs, he allows her to sort through and work on her understanding.

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He could have helped her by introducing vocabulary or asking questions; however, to paraphrase Zinsser (1988), he allowed her to learn what she was ready to figure out. In this situation, John believed that the student needed time and silent encouragement in order to find a way of verbalizing her understanding for herself.

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