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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Conclusion


I may never know when I am "right," but collaboration, communication, and reflection on teaching help to make me an effective teacher in the classroom.
-- Susan Boone, Project teacher



As we shared videos made in our classrooms and examined our teaching practice with one another, our confidence sometimes gave way to doubt, raising the question: "How do I know I'm right?" What we know is that the improvement of teaching is never finished, and we may never know we are "right," but collaboration, communication, and reflection on teaching help to make us effective teachers in the classroom.

We have learned in new ways that questioning, listening, writing, and reflecting are important tools, both for our students and for ourselves as teachers. We are examining carefully the ways in which students are actively engaged in learning when they are presented with mathematics that challenges their understanding and are given opportunities to think and discuss among themselves. Not only is this essential for student learning, we also found it invaluable in our own learning.

Our conversations and writing during this project led us to reflect on our lessons, consider possibilities for improvement, and identify potential difficulties. Through our open and extensive work together we have deepened our appreciation for the variety of effective approaches to teaching. Often these include focusing on meaningful problems that are within the students' reach and modeling approaches to problem-solving.

Focusing on the same problem in each of our classes enabled us to think about the variance that our own interactions with our students contributes to their abilities to think mathematically. Although we teach students at different levels of mathematical knowledge, we find that the issues we face as teachers are similar: how to help our students question, how to teach problem-solving, how to provide students with the tools they need. We have also learned that the solutions to these problems are many, and that they become clearer when we have the opportunity to think them through with our colleagues.

In order to talk with our colleagues we needed to make our self-assessments and reflections explicit, deliberate, and systematic. This process not only enhanced our abilities to converse, it also meant that we documented alternate solutions to similar problems. It has pushed us to try others' suggestions and to revise plans that we'd thought we knew worked, in turn encouraging us to push our thinking about mathematics teaching.


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