Combinatorics and Reasoning

Representing, Justifying and Building Isomorphisms

ISBN: 978-94-007-0614-9 (Print) 978-94-007-0615-6 (Online)

Table of contents (17 chapters)

  1. Front Matter

    Pages I-XVII

  2. Introduction, Background, and Methodology

    1. Front Matter

      Pages 1-1

    2. No Access

      Book Chapter

      Pages 3-8

      The Longitudinal Study

    3. No Access

      Book Chapter

      Pages 9-14

      Methodology

  3. Foundations of Proof Building (1989–1996)

    1. Front Matter

      Pages 15-15

    2. No Access

      Book Chapter

      Pages 17-25

      Representations as Tools for Building Arguments

    3. No Access

      Book Chapter

      Pages 27-43

      Towers: Schemes, Strategies, and Arguments

    4. No Access

      Book Chapter

      Pages 45-57

      Building an Inductive Argument

    5. No Access

      Book Chapter

      Pages 59-72

      Making Pizzas: Reasoning by Cases and by Recursion

    6. No Access

      Book Chapter

      Pages 73-86

      Block Towers: From Concrete Objects to Conceptual Imagination

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

    1. Front Matter

      Pages 87-87

    2. No Access

      Book Chapter

      Pages 89-95

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

    3. No Access

      Book Chapter

      Pages 97-104

      Block Towers: Co-construction of Proof

    4. No Access

      Book Chapter

      Pages 105-120

      Representations and Connections

    5. No Access

      Book Chapter

      Pages 121-131

      Pizzas, Towers, and Binomials

    6. No Access

      Book Chapter

      Pages 133-144

      Representations and Standard Notation

    7. No Access

      Book Chapter

      Pages 145-154

      So Let’s Prove It!

  5. Extending the Study, Conclusions, and Implications

    1. Front Matter

      Pages 155-155

    2. No Access

      Book Chapter

      Pages 157-169

      “Doing Mathematics” from the Learners’ Perspectives

    3. No Access

      Book Chapter

      Pages 171-183

      Adults Reasoning Combinatorially

    4. No Access

      Book Chapter

      Pages 185-200

      Comparing the Problem Solving of College Students with Longitudinal Study Students

    5. No Access

      Book Chapter

      Pages 201-204

      Closing Observations

  6. Back Matter

    Pages 205-224