Combinatorics and Reasoning (eBook)

Preface 7
Acknowledgements 8
Contents 9
Introduction 11
Contributors 15
Part I Introduction, Background, and Methodology 16
1 The Longitudinal Study 17
1.1 Theoretical View 17
1.2 Background of the Study 18
1.2.1 Teacher Development Component 19
1.2.2 Intervention Design 20
1.3 Longitudinal Study: Grades 1-3 20
1.4 Longitudinal Study, Grades 4-8 21
1.5 Longitudinal Study: High School Years 21
1.6 Longitudinal Study: Beyond High School 21
2 Methodology 23
2.1 Introduction 23
2.2 Theoretical Perspectives 24
2.3 Selected Problems 24
2.3.1 Shirts and Jeans 25
2.3.2 Towers 26
2.3.3 Pizzas 26
2.3.4 Taxicab 28
2.4 Concluding Remarks 28
Part II Foundations of Proof Building 1989-1996 29
3 Representations as Tools for Building Arguments 30
3.1 Introduction 30
3.2 Representation as a Tool for Problem Solving 31
3.3 Early Counting Task Strand Shirts and Jeans 31
3.3.1 Second-Grade Problem Solving 32
3.3.2 Third-Grade Problem Solving 34
3.4 Cognitive Implications and Differences Observed 36
3.5 Discussion 37
4 Towers: Schemes, Strategies, and Arguments 39
4.1 Introduction 39
4.2 Stephanie 40
4.2.1 Stephanie Grade 3, Class Session 40
4.2.2 Stephanie: Grade 4, Class Session 43
4.2.3 Stephanie: Grade 4, Interviews 45
4.3 Milin 47
4.3.1 Milin: Grade 4, Class Session 47
4.3.2 Milin: Grade 4, Interviews 49
4.3.3 Small Group Interview: March 10, 1992 -- Grade 4 50
4.4 Summary of Strategies and Justifications 54
4.5 Discussion 54
5 Building an Inductive Argument 56
5.1 Introduction 56
5.2 Early Ideas 56
5.2.1 Stephanie's Individual Interview: May 15, 1992 57
5.2.2 Written Assessments for Stephanie and Milin: June 15, 1992 57
5.2.3 Written Assessments for Stephanie and Milin: October 25, 1992 59
5.3 Investigating Inductive Reasoning 61
5.3.1 Stephanie and Matt's Beginning Exploration 61
5.3.2 Milin!s Explanation and Michelle!s Aha! 62
5.3.3 Matt!s Explanation and Stephanie!s Aha! 63
5.3.4 Stephanie's Sharing Milin's Family Tree 66
5.4 Discussion 67
6 Making Pizzas: Reasoning by Cases and by Recursion 69
6.1 Introduction 69
6.2 First Session: Initial Pizza Explorations 70
6.2.1 Group 1 70
6.2.2 Group 2 72
6.3 Second Session: Further Pizza Explorations 74
6.3.1 Group 1 74
6.3.2 Group 2 76
6.4 Third Session: Getting the Right Answer 77
6.5 Fourth Session: Giving the Solution 77
6.6 Fifth Session: Additional Justifications 78
6.6.1 Problem 1: Pizzas with Halves 78
6.6.2 Problem 2: The Four-Topping Pizza Problem 79
6.6.3 Problem 3: Another Pizza Problem 80
6.6.4 Problem 4: The Final Pizza Problem 80
6.7 Discussion 82
7 Block Towers: From Concrete Objects to Conceptual Imagination 83
7.1 Introduction 83
7.2 Theoretical Perspectives 84
7.3 Setting 85
7.4 Guiding Questions 85
7.5 Results 86
7.6 Discussion 95
Part III Making Connections, Extending, and Generalizing 1997-2000 97
8 Responding to Ankur's Challenge: Co-construction of Argument Leading to Proof 98
8.1 Introduction 98
8.2 Rominas Presentation of Proof 99
8.3 Discussion 103
9 Block Towers: Co-construction of Proof 105
9.1 Introduction 105
9.2 Building Towers 105
9.2.1 Angela and Magda 106
9.2.2 Sherly and Ali 107
9.2.3 Michelle and Robert 107
9.2.4 Group Work 111
9.3 Discussion 112
10 Representations and Connections 113
10.1 Introduction 113
10.2 Session 1: A Common Notation 113
10.3 Session 2: Towers and Pizzas 115
10.4 Session 3: Towers and the Binomial Expansion 117
10.5 Session 4: Pizzas, Towers, and Pascals Triangle 119
10.6 Session 5: Towers, Pizzas, and Pascals Triangle 122
10.7 Discussion 126
11 Pizzas, Towers, and Binomials 129
11.1 Introduction 129
11.2 Table A: A Connection Between Pizzas and Towers 130
11.3 Table B: Connection Between Pizzas and Pascals Identity 135
11.4 Discussion 138
12 Representations and Standard Notation 140
12.1 Introduction 140
12.2 Summary of Earlier Student Work 141
12.3 The Night Session 142
12.4 Durability of Understanding 147
12.5 Discussion 149
13 So Let's Prove It! 152
13.1 Introduction 152
13.1.1 The Task 152
13.2 Justifying Claims 153
13.2.1 Generalizations, Isomorphisms, and Transitivity 154
13.2.2 Reasoning and Justifying 154
13.2.2.1 Realizing the Need to Discursively Build a Justification 154
13.2.2.2 Generalizing to Specialize 158
13.2.2.3 Building Isomorphisms to Justify 158
13.3 Conclusion 160
Part IV Extending the Study, Conclusions, and Implications 162
14 Doing Mathematics from the Learners Perspectives 163
14.1 Introduction 163
14.2 Findings 165
14.2.1 Personal Success/Failure in Mathematics 165
14.2.2 Knowing Mathematics as Sense Making 166
14.2.3 Mathematics as a Discovery Activity 168
14.2.4 Mathematics as an Activity Involving Discourse 170
14.2.5 Mathematics and Other Disciplines 171
14.3 Conclusions 173
15 Adults Reasoning Combinatorially 176
15.1 Introduction 176
15.2 The Study 177
15.3 Student Solutions 178
15.3.1 Towers Problems 179
15.3.2 Pizza Problems 182
15.3.3 Connections Between Problems 183
15.3.4 Connections with Pascal's Triangle 186
15.4 Discussion 187
16 Comparing the Problem Solving of College Students with Longitudinal Study Students 189
16.1 Introduction 189
16.2 Justifications by Cases 189
16.2.1 Romina, Jeff, and Brian's Solution (H) 190
16.2.2 Joanne and Donna's Solution (U) 190
16.2.3 Rob and Jessica's Solution (U) 191
16.2.4 Marie's Solution (U) 192
16.2.5 Bob's Solution by Cases (U) 192
16.2.6 April's Solution (U) 192
16.2.7 Bernadette's Solution (U) 193
16.2.8 Tim's Solution (G) 194
16.2.9 Traci's Solution (G) 195
16.3 Inductive Arguments 195
16.3.1 Errol's Solution (U) 195
16.3.2 Christina's Solution (U) 196
16.3.3 Bob's Inductive Solution (U) 198
16.3.4 Frances' Solution (G) 198
16.4 Elimination Arguments 199
16.4.1 Penny's Solution (U) 199
16.4.2 Robert's Solution (U) 200
16.4.3 Liz's Solution (G) 201
16.4.4 Mary's Solution (G) 201
16.5 Analytic Method 201
16.5.1 Leana's Solution (G) 202
16.6 Discussion 202
17 Closing Observations 205
Appendix A Combinatorics Problems 209
Appendix B Counting and Combinatorics Dissertations from the Longitudinal Study 217
References 218
Index 223