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. Enter your e-mail to be notified of discussion on this videopaper: Privacy Policy		Lesson Plans and Resources Elementary Level Lesson Plan Middle School Lesson Plan High School Level Lesson Plan Calculus Level Lesson Plan The Geometer's SketchpadTM: Sketches Constant Perimeter Project Charts and Graphs Download Excel Spreadsheet Fixed Area: Worksheet (blank) Fixed Area: Worksheet (filled) Fixed Area: Height to Volume Fixed Area: Radius to Volume Fixed Perimeter: Worksheet (blank) Fixed Perimeter: Worksheet (filled) Fixed Perimeter: Height to Volume Fixed Perimeter: Radius to Volume Useful and Related Links The Change Site (ENC): Resources to facilitate discussion and reflection on improving K-12 mathematics. Learning and Mathematics (Math Forum): Research synopses and discussions. PLEASE READ: You will need QuickTime 4 in order to view the video clips that are included in this article. To get QuickTime, go to the Download QuickTime 4 page, scroll down to the "Tell us who you are" section, tell them who you are, and then click on Download QuickTime. Video Clips (in order of appearance) Page      Video Clip / Transcript of Text Teacher Size (MB) Interventions: Approaches     A teacher introduces a tool [transcript] Teacher as confidence-builder [transcript] Mabbott Basden 2.9 1.6 Interventions: Leading Questions     A leading question [transcript] Several leading questions [transcript] Focusing the student on the question [transcript] Boone Koenig Stein 3.3 4.9 1.3 Interventions: Non-leading Questions     A non-leading question [transcript] McKinstry 2.5 Interventions: Paraphrasing     Paraphrasing, adding an example [transcript] Paraphrasing, using math vocabulary [transcript] Paraphrasing, using math vocabulary [transcript] Koenig Basden Mabbott 5.8 2.0 1.4 Interventions: Summarizing     Summarizing a conversation [transcript] McKinstry 2.5 Decisions: Balance     Moving a conversation [transcript] Silent encouragement [transcript] McKinstry McKinstry 2.5 2.3 Decisions: Wait Time     A student helps another student [transcript] Repeating a student's question [transcript] Reminding students of their questions [transcript] Four students work together [transcript] One student corrects another [transcript] Students share their thoughts [transcript] Stein Stein Mabbott Boone Stein McKinstry 1.2 1.8 2.8 4.2 1.5 4.8 Decisions: Classroom Norms     Not just one right way [transcript] Encouraging taking risks [transcript] Changing a prediction [transcript] Basden Mabbott Stein 6.7 1.1 5.9 The Cylinder Problem: Promoting Discourse   Which cylinder will have greater volume? [transcript] Stein 3.6   "Aha" moments [transcript] Basden, Stein, Koenig, McKinstry 7.4   Students interpret a graph [transcript] Students provide reasoning for predictions [transcript] Real life applications [transcript] Koenig Stein Basden 10.0 5.7 5.9 The Cylinder Problem: Lesson Reflections     Eliciting prior knowledge [transcript] Students write what they know [transcript] A class reviews what it's written [transcript] Struggling to find explanations [transcript] "You can't be wrong" [transcript] Forming opinions about a problem [transcript] Paraphrasing a student's prediction [transcript] Forming questions by making more cylinders [transcript] Keeping the area constant [transcript] Technology helps with number-crunching [transcript] Using graphing calculators [transcript] Connecting mathematical ideas [transcript] Reviewing the questions [transcript] Boone Stein Stein McKinstry Boone Stein Mabbott Basden Basden Mabbott Boone McKinstry Mabbott 2.9 1.3 6.5 2.3 0.8 5.9 1.4 8.0 6.7 2.9 3.6 5.4 1.8

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