Maher / Powell / Uptegrove Combinatorics and Reasoning

Representing, Justifying and Building Isomorphisms
1. Auflage 2010
ISBN: 978-0-387-98132-1
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 224 Seiten

Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms is based on the accomplishments of a cohort group of learners from first grade through high school and beyond, concentrating on their work on a set of combinatorics tasks. By studying these students, the editors gain insight into the foundations of proof building, the tools and environments necessary to make connections, activities to extend and generalize combinatoric learning, and even explore implications of this learning on the undergraduate level. This volume underscores the power of attending to basic ideas in building arguments; it shows the importance of providing opportunities for the co-construction of knowledge by groups of learners; and it demonstrates the value of careful construction of appropriate tasks. Moreover, it documents how reasoning that takes the form of proof evolves with young children and discusses the conditions for supporting student reasoning.

Autoren/Hrsg.

Maher, Carolyn A.

Powell, Arthur B.

Uptegrove, Elizabeth B.

Inhaltsverzeichnis

1;Preface;7
2;Acknowledgements;8
3;Contents;9
4;Introduction;11
5;Contributors;15
6;Part I Introduction, Background, and Methodology;16
6.1;1 The Longitudinal Study;17
6.1.1;1.1 Theoretical View;17
6.1.2;1.2 Background of the Study;18
6.1.2.1;1.2.1 Teacher Development Component;19
6.1.2.2;1.2.2 Intervention Design;20
6.1.3;1.3 Longitudinal Study: Grades 1-3;20
6.1.4;1.4 Longitudinal Study, Grades 4-8;21
6.1.5;1.5 Longitudinal Study: High School Years;21
6.1.6;1.6 Longitudinal Study: Beyond High School;21
6.2;2 Methodology;23
6.2.1;2.1 Introduction;23
6.2.2;2.2 Theoretical Perspectives;24
6.2.3;2.3 Selected Problems;24
6.2.3.1;2.3.1 Shirts and Jeans;25
6.2.3.2;2.3.2 Towers;26
6.2.3.3;2.3.3 Pizzas;26
6.2.3.4;2.3.4 Taxicab;28
6.2.4;2.4 Concluding Remarks;28
7;Part II Foundations of Proof Building 1989-1996;29
7.1;3 Representations as Tools for Building Arguments;30
7.1.1;3.1 Introduction;30
7.1.2;3.2 Representation as a Tool for Problem Solving;31
7.1.3;3.3 Early Counting Task Strand Shirts and Jeans;31
7.1.3.1;3.3.1 Second-Grade Problem Solving;32
7.1.3.2;3.3.2 Third-Grade Problem Solving;34
7.1.4;3.4 Cognitive Implications and Differences Observed;36
7.1.5;3.5 Discussion;37
7.2;4 Towers: Schemes, Strategies, and Arguments;39
7.2.1;4.1 Introduction;39
7.2.2;4.2 Stephanie;40
7.2.2.1;4.2.1 Stephanie Grade 3, Class Session;40
7.2.2.2;4.2.2 Stephanie: Grade 4, Class Session;43
7.2.2.3;4.2.3 Stephanie: Grade 4, Interviews;45
7.2.3;4.3 Milin;47
7.2.3.1;4.3.1 Milin: Grade 4, Class Session;47
7.2.3.2;4.3.2 Milin: Grade 4, Interviews;49
7.2.3.3;4.3.3 Small Group Interview: March 10, 1992 -- Grade 4;50
7.2.4;4.4 Summary of Strategies and Justifications;54
7.2.5;4.5 Discussion;54
7.3;5 Building an Inductive Argument;56
7.3.1;5.1 Introduction;56
7.3.2;5.2 Early Ideas;56
7.3.2.1;5.2.1 Stephanie's Individual Interview: May 15, 1992;57
7.3.2.2;5.2.2 Written Assessments for Stephanie and Milin: June 15, 1992;57
7.3.2.3;5.2.3 Written Assessments for Stephanie and Milin: October 25, 1992;59
7.3.3;5.3 Investigating Inductive Reasoning;61
7.3.3.1;5.3.1 Stephanie and Matt's Beginning Exploration;61
7.3.3.2;5.3.2 Milin!s Explanation and Michelle!s Aha!;62
7.3.3.3;5.3.3 Matt!s Explanation and Stephanie!s Aha!;63
7.3.3.4;5.3.4 Stephanie's Sharing Milin's Family Tree;66
7.3.4;5.4 Discussion;67
7.4;6 Making Pizzas: Reasoning by Cases and by Recursion;69
7.4.1;6.1 Introduction;69
7.4.2;6.2 First Session: Initial Pizza Explorations;70
7.4.2.1;6.2.1 Group 1;70
7.4.2.2;6.2.2 Group 2;72
7.4.3;6.3 Second Session: Further Pizza Explorations;74
7.4.3.1;6.3.1 Group 1;74
7.4.3.2;6.3.2 Group 2;76
7.4.4;6.4 Third Session: Getting the Right Answer;77
7.4.5;6.5 Fourth Session: Giving the Solution;77
7.4.6;6.6 Fifth Session: Additional Justifications;78
7.4.6.1;6.6.1 Problem 1: Pizzas with Halves;78
7.4.6.2;6.6.2 Problem 2: The Four-Topping Pizza Problem;79
7.4.6.3;6.6.3 Problem 3: Another Pizza Problem;80
7.4.6.4;6.6.4 Problem 4: The Final Pizza Problem;80
7.4.7;6.7 Discussion;82
7.5;7 Block Towers: From Concrete Objects to Conceptual Imagination;83
7.5.1;7.1 Introduction;83
7.5.2;7.2 Theoretical Perspectives;84
7.5.3;7.3 Setting;85
7.5.4;7.4 Guiding Questions;85
7.5.5;7.5 Results;86
7.5.6;7.6 Discussion;95
8;Part III Making Connections, Extending, and Generalizing 1997-2000;97
8.1;8 Responding to Ankur's Challenge: Co-construction of Argument Leading to Proof;98
8.1.1;8.1 Introduction;98
8.1.2;8.2 Rominas Presentation of Proof;99
8.1.3;8.3 Discussion;103
8.2;9 Block Towers: Co-construction of Proof;105
8.2.1;9.1 Introduction;105
8.2.2;9.2 Building Towers;105
8.2.2.1;9.2.1 Angela and Magda;106
8.2.2.2;9.2.2 Sherly and Ali;107
8.2.2.3;9.2.3 Michelle and Robert;107
8.2.2.4;9.2.4 Group Work;111
8.2.3;9.3 Discussion;112
8.3;10 Representations and Connections;113
8.3.1;10.1 Introduction;113
8.3.2;10.2 Session 1: A Common Notation;113
8.3.3;10.3 Session 2: Towers and Pizzas;115
8.3.4;10.4 Session 3: Towers and the Binomial Expansion;117
8.3.5;10.5 Session 4: Pizzas, Towers, and Pascals Triangle;119
8.3.6;10.6 Session 5: Towers, Pizzas, and Pascals Triangle;122
8.3.7;10.7 Discussion;126
8.4;11 Pizzas, Towers, and Binomials;129
8.4.1;11.1 Introduction;129
8.4.2;11.2 Table A: A Connection Between Pizzas and Towers;130
8.4.3;11.3 Table B: Connection Between Pizzas and Pascals Identity;135
8.4.4;11.4 Discussion;138
8.5;12 Representations and Standard Notation;140
8.5.1;12.1 Introduction;140
8.5.2;12.2 Summary of Earlier Student Work;141
8.5.3;12.3 The Night Session;142
8.5.4;12.4 Durability of Understanding;147
8.5.5;12.5 Discussion;149
8.6;13 So Let's Prove It!;152
8.6.1;13.1 Introduction;152
8.6.1.1;13.1.1 The Task;152
8.6.2;13.2 Justifying Claims;153
8.6.2.1;13.2.1 Generalizations, Isomorphisms, and Transitivity;154
8.6.2.2;13.2.2 Reasoning and Justifying;154
8.6.2.2.1;13.2.2.1 Realizing the Need to Discursively Build a Justification;154
8.6.2.2.2;13.2.2.2 Generalizing to Specialize;158
8.6.2.2.3;13.2.2.3 Building Isomorphisms to Justify;158
8.6.3;13.3 Conclusion;160
9;Part IV Extending the Study, Conclusions, and Implications;162
9.1;14 Doing Mathematics from the Learners Perspectives;163
9.1.1;14.1 Introduction;163
9.1.2;14.2 Findings;165
9.1.2.1;14.2.1 Personal Success/Failure in Mathematics;165
9.1.2.2;14.2.2 Knowing Mathematics as Sense Making;166
9.1.2.3;14.2.3 Mathematics as a Discovery Activity;168
9.1.2.4;14.2.4 Mathematics as an Activity Involving Discourse;170
9.1.2.5;14.2.5 Mathematics and Other Disciplines;171
9.1.3;14.3 Conclusions;173
9.2;15 Adults Reasoning Combinatorially;176
9.2.1;15.1 Introduction;176
9.2.2;15.2 The Study;177
9.2.3;15.3 Student Solutions;178
9.2.3.1;15.3.1 Towers Problems;179
9.2.3.2;15.3.2 Pizza Problems;182
9.2.3.3;15.3.3 Connections Between Problems;183
9.2.3.4;15.3.4 Connections with Pascal's Triangle;186
9.2.4;15.4 Discussion;187
9.3;16 Comparing the Problem Solving of College Students with Longitudinal Study Students;189
9.3.1;16.1 Introduction;189
9.3.2;16.2 Justifications by Cases;189
9.3.2.1;16.2.1 Romina, Jeff, and Brian's Solution (H);190
9.3.2.2;16.2.2 Joanne and Donna's Solution (U);190
9.3.2.3;16.2.3 Rob and Jessica's Solution (U);191
9.3.2.4;16.2.4 Marie's Solution (U);192
9.3.2.5;16.2.5 Bob's Solution by Cases (U);192
9.3.2.6;16.2.6 April's Solution (U);192
9.3.2.7;16.2.7 Bernadette's Solution (U);193
9.3.2.8;16.2.8 Tim's Solution (G);194
9.3.2.9;16.2.9 Traci's Solution (G);195
9.3.3;16.3 Inductive Arguments;195
9.3.3.1;16.3.1 Errol's Solution (U);195
9.3.3.2;16.3.2 Christina's Solution (U);196
9.3.3.3;16.3.3 Bob's Inductive Solution (U);198
9.3.3.4;16.3.4 Frances' Solution (G);198
9.3.4;16.4 Elimination Arguments;199
9.3.4.1;16.4.1 Penny's Solution (U);199
9.3.4.2;16.4.2 Robert's Solution (U);200
9.3.4.3;16.4.3 Liz's Solution (G);201
9.3.4.4;16.4.4 Mary's Solution (G);201
9.3.5;16.5 Analytic Method;201
9.3.5.1;16.5.1 Leana's Solution (G);202
9.3.6;16.6 Discussion;202
9.4;17 Closing Observations;205
9.5;Appendix A Combinatorics Problems;209
9.6;Appendix B Counting and Combinatorics Dissertations from the Longitudinal Study;217
10;References;218
11;Index;223