Combinatorics and Reasoning

1. Front Matter

   Pages I-XVII

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2. Introduction, Background, and Methodology

   1. Front Matter

      Pages 1-1

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   2. The Longitudinal Study

      • Carolyn A. Maher

      Pages 3-8

   3. Methodology

      • Carolyn A. Maher, Elizabeth B. Uptegrove

      Pages 9-14

3. Foundations of Proof Building (1989–1996)

   1. Front Matter

      Pages 15-15

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   2. Representations as Tools for Building Arguments

      • Carolyn A. Maher, Dina Yankelewitz

      Pages 17-25

   3. Towers: Schemes, Strategies, and Arguments

      • Carolyn A. Maher, Manjit K. Sran, Dina Yankelewitz

      Pages 27-43

   4. Building an Inductive Argument

      • Carolyn A. Maher, Manjit K. Sran, Dina Yankelewitz

      Pages 45-57

   5. Making Pizzas: Reasoning by Cases and by Recursion

      • Carolyn A. Maher, Manjit K. Sran, Dina Yankelewitz

      Pages 59-72

   6. Block Towers: From Concrete Objects to Conceptual Imagination

      • Robert Speiser

      Pages 73-86

4. Making Connections, Extending, and Generalizing (1997–2000)

   1. Front Matter

      Pages 87-87

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   2. Responding to Ankur’s Challenge: Co-construction of Argument Leading to Proof

      • Carolyn A. Maher, Ethel M. Muter

      Pages 89-95

   3. Block Towers: Co-construction of Proof

      • Lynn D. Tarlow, Elizabeth B. Uptegrove

      Pages 97-104

   4. Representations and Connections

      • Ethel M. Muter, Elizabeth B. Uptegrove

      Pages 105-120

   5. Pizzas, Towers, and Binomials

      • Lynn D. Tarlow

      Pages 121-131

   6. Representations and Standard Notation

      • Elizabeth B. Uptegrove

      Pages 133-144

   7. So Let’s Prove It!

      • Arthur B. Powell

      Pages 145-154

5. Extending the Study, Conclusions, and Implications

   1. Front Matter

      Pages 155-155

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   2. “Doing Mathematics” from the Learners’ Perspectives

      • John M. Francisco

      Pages 157-169

   3. Adults Reasoning Combinatorially

      • Barbara Glass

      Pages 171-183

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.

• College Mathematics
• Combinatorics Learning
• Isomorphism
• Longitudinal Studies
• Proof Building
• combinatorics

• Graduate School of Education, Rutgers University, New Brunswick, USA

  Carolyn A. Maher

• Department of Urban Education, Rutgers University, Newark, USA

  Arthur B. Powell

• Department of Mathematical Sciences, Felician College, Rutherford, USA

  Elizabeth B. Uptegrove

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