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


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.



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Resources



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


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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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