Combinatorics and Reasoning (eBook)

Representing, Justifying and Building Isomorphisms

Carolyn A. Maher, Arthur B. Powell, Elizabeth B. Uptegrove (Herausgeber)

eBook Download: PDF

2010 | 2011
XVIII, 226 Seiten
Springer Netherland (Verlag)
978-94-007-0615-6 (ISBN)

Infos

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.

Inhalt

Preface 7
Acknowledgements 8
Contents 9
Introduction 11
Contributors 15
Introduction, Background, and Methodology 16
The Longitudinal Study 17
Methodology 23
Foundations of Proof Building (1989–1996) 29
Representations as Tools for Building Arguments 30
Towers: Schemes, Strategies, and Arguments 39
Building an Inductive Argument 56
Making Pizzas: Reasoning by Cases and by Recursion 69
Block Towers: From Concrete Objects to Conceptual Imagination 83
Making Connections, Extending, and Generalizing ( 1997– 2000) 97
Responding to Ankur’s Challenge: Co- construction of Argument Leading to Proof 98
Block Towers: Co-construction of Proof 105
Representations and Connections 113
Pizzas, Towers, and Binomials 129
Representations and Standard Notation 140
So Let’s Prove It! 152
Extending the Study, Conclusions, and Implications 162
“Doing Mathematics” from the Learners’ Perspectives 163
Adults Reasoning Combinatorially 176
Comparing the Problem Solving of College Students with Longitudinal Study Students 189
Closing Observations 205
Appendix A Combinatorics Problems 209
Appendix B Counting and Combinatorics Dissertations from the Longitudinal Study 217
References 218
Index 223

Produkt-Details

• Erscheint lt. Verlag: 25.11.2010
• Reihe/Serie: Mathematics Education Library
• Reihe/Serie: Mathematics Education Library
• Zusatzinfo: XVIII, 226 p.
• Verlagsort: Dordrecht
• Sprache: englisch
• Themenwelt: Schulbuch / Wörterbuch
• Themenwelt: Geisteswissenschaften
• Themenwelt: Mathematik / Informatik ► Mathematik ► Graphentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Sozialwissenschaften ► Pädagogik ► Erwachsenenbildung
• Themenwelt: Sozialwissenschaften ► Pädagogik ► Schulpädagogik / Grundschule
• Themenwelt: Technik
• Schlagworte: College Mathematics • combinatorics • Combinatorics Learning • isomorphism • Longitudinal Studies • Proof Building
• ISBN-10: 94-007-0615-4 / 9400706154
• ISBN-13: 978-94-007-0615-6 / 9789400706156