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

Judith Koenig       [Biography]



A few years ago a friend asked how I would spend the money if I received a large grant. I thought about materials that would make my students' learnings richer and deeper, but couldn't think of a thing that we didn't already have. However, I wouldn't turn down the grant. I would spend the money to give myself time to reflect on my practice at the end of each class period. Somehow I would hire someone to keep the students occupied so that I could have 10-20 minutes to reflect.

Well, I never received such a grant, but I have given myself this valuable experience of reflection weekly through my participation in the Bridging Research and Practice group. There are always a million things a teacher can be doing, so it really takes diligence to take the time to reflect. I have been able to look more at the "big picture" of what is going on in my classroom, whereas previously I was always too focused on individual students. Overall, I get the picture that my students are learning and achieving mastery of difficult concepts. They do this by grappling with engaging topics individually, in small groups, and as a class. My students communicate with one another and with me and increase their understandings of mathematics and my understandings of what they know and don't know. In the past few years, my teaching hasn't changed much, so what has contributed to my increased understanding of my practice?

Through my work on this project, I have been forced to videotape my class and watch the videos. In the past, this has not been an easy thing for me to do. However, it has become less painful and much more rewarding through my individual efforts and the collaboration of this supportive group of educators. By viewing the videos I have had time to see students both on task and not, and have heard side conversations that weren't available to me as I conducted the class. Often these side conversations led me to further understand my students' conceptual development. I have viewed student presentations in a new light and have come up with ideas for improving them. One idea was to have different groups answer different questions about the same assignment in order to make the class presentations more engaging.

I realize that it is not feasible to videotape every lesson and then have time to view it and critique it, but wouldn't that process be enlightening? So even when I'm not forced to videotape my class, I will do it occasionally for my own personal viewing and the benefit of my students.

Another aspect of education that I've neglected is incorporating research into my practice. I just don't take the time to seek out related research or to read what comes my way in journals or via the Internet. Participating in the project has enabled me to see that I need to take the time to read educational articles and that the benefits far outweigh the time taken from other endeavors. For example, reading a summary of practice in Japan vs. the United States gave me the confidence to continue allowing students to explore and develop conceptual understandings before incorporating an algorithm or formula. I wouldn't have known that this is the approach in Japan without reading Stigler's analysis of TIMSS. My confidence has increased, as has my ability to communicate with parents and peers.

Being a professional educator takes time -- time to plan, time to practice, time to grade, time to communicate -- and I never have enough time. However, I now realize that adding reflection and research to my agenda have made my life as a teacher easier, not more difficult. I hope to be able to continue to enrich my practice by participating in Bridging Research and Practice and deepening my understandings by having ongoing discourse with my peers in this program and new peers that we meet online.

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