Abstract  Introduction  Discourse  Interventions  Decisions  Cylinder Problem  Lesson Reflections  Student Predictions  Project Reflections  Conclusion  References  Acknowledgments  Teacher Resources Authors' Biographies Table of Contents VIDEO CLIPS: Internet access via modem may mean very long download times for video clips. If you are not on a fast line, you may want to read this paper without viewing the clips. share your thoughts, ideas, questions, and experiences; read what others have to say.		Cynthia Lanius       [Biography] Since I am not currently in the classroom, my work on this paper has been somewhat different from that of the other teachers involved. I wasn't able to teach the lesson and reflect on it with everyone else. But I had developed the activity for a teacher workshop and have taught it to former students of both geometry and calculus, and so I'd like to reflect on how this BRAP experience has affected me. The most important factor that I will take from this experience is a reinforced respect for my teaching colleagues. The "wrappers" are intelligent, articulate, and caring. They all make me both proud and humbled to call myself a teacher. Our profession is often maligned, but there are thousands of us doing a fabulous job, and we should not be ashamed to acknowledge it and defend our profession. I believe that the experience of middle school and high school teachers working together is an effective model of professional development. I think we learned a lot from each other that we wouldn't have learned otherwise. I had worked the constant area problem abstractly, but I had never pictured it as taking a sheet of paper and cutting it up and pasting it together again, as the middle school teachers saw it. I very much liked our focus on discourse. If the opposite of discourse is silence, then discourse is going to occur in the classroom regardless of what we do. Teachers can, however, influence the quality of that discourse, both in content and tone. Students must be confident that the classroom is an emotionally safe place before they will participate fully. The rewards of effective discourse are many and they are outlined in the paper. Not long ago, I was observing students work out a problem together. When they finally "got" it, one of the students exclaimed, "We are so smart." Isn't that what we are striving for? Not "the teacher is so smart," not "they are so smart," but we are. In that statement, I heard both confidence and community. Healthy discourse promotes these things.

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