Encouraging Mathematical Thinking


 Abstract
 Introduction

 Discourse
 Interventions
 Decisions

 Cylinder Problem
 Lesson Reflections
 Student Predictions

 Project Reflections
 Conclusion

 References
 Acknowledgments
 Teacher Resources



Authors'
Biographies

Table of Contents


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Reflections

Susan Stein       [Biography]



One example of the way in which reflecting on my teaching practice influences how I teach happened during our work on this paper. I had been thinking for quite a while about how difficult it was to get my students to become more self-reliant and interact more with one another when we had whole-class discussions. There was little conversation between students in that setting, although they were comfortable collaborating in small groups.

When I saw the tape of Judy K's class I realized that I was part of the problem. I realized that I tended to react with a head nod or a verbal "good" or "what about this...?" to most student comments. What I observed Judy doing was eliciting all the possible student responses before she made any comment other than, "Any more ideas?" It was clear that the expectation (culture) in her classroom was that all student ideas would be put on the table before any were discussed. It was powerful. As soon as I started to do the same thing, the discussion dynamic shifted. The dynamics still weren't great (that is, they were more teacher-centered than I thought they ought to/could be, because this was happening in April and the classroom expectations had been set long ago.

However, this year (the following) I started that way in September, and this has made an enormous difference. Students expect that several ideas will be explored, several methods discussed, and they are responsible for making sense of each one. In fact, I've even refined this "technique," because I discovered that the kids all wanted to get their two cents in even if the idea had already been said. Now I say, "Any ideas that are not the same as what we've heard?" I've found that kids don't always realize that they are being repetitious until they start saying their piece. Then I often hear, "Oh, well I guess it is really the same as Eli's..." It is interesting to me that I'm asking kids to analyze their own methods and ideas on a deeper level by having them look for similarities and differences.

Working on this project was full of pleasures. The gifts I received benefitted me personally, my teaching and my students. They fall into three groups. The largest gift was finding new colleagues with whom to learn. Another gift was the structure, which "forced" me to take the time to reflect consciously on my teaching practice. This included writing weekly reflections, being videotaped, and looking at my own tapes as well as tapes of others' classes. Finally, I had the opportunity to connect with some of the research on mathematics teaching that I have neglected for years. And, in addition to all of those momentous gifts, I had great fun.

Working on this article gave me new opportunities to reflect on my own practice along with colleagues. Somehow, when I reflected "in the presence" of others who were also thinking about what they did and how it was working, I learned more and deepened my understanding of what I do, and what my students need, than when I did it alone in the course of everyday teaching. Part of the reason for that was the wonderful opportunity (privilege, really) to "listen in" through the videotapes and conversation -- both real and virtual -- as those talented teachers thought aloud about issues that have on-going implications for my own classroom.

I learned that expecting my students to move from a cultural view of school mathematics in which right answers are regularly provided or validated by "outside authorities," such as textbooks and teachers, to one in which students play a substantial role in deciding when their problem-solving is "right," or at least "on the right track," was a challenging and complex goal. As I realized that the teachers in this group struggled with similar issues, I gained confidence in my direction as well as broadening my teaching repertoire of ways to achieve it. Reaffirming that while my teaching will never be perfect, it is always perfectible, was comforting and energizing.

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