Combinatorics and Reasoning

Part I. Introduction, Background, and Methodology

1. The Longitudinal Study
Carolyn A. Maher

2. Methodology
Carolyn A. Maher, Elizabeth B. Uptegrove

Part II. Foundations of Proof Building (1989–1996)

3. Representations as Tools for Building Arguments
Carolyn A. Maher, Dina Yankelewitz

4. Towers: Schemes, Strategies, and Arguments
Carolyn A. Maher, Manjit K. Sran, Dina Yankelewitz

5. Building an Inductive Argument
Carolyn A. Maher, Manjit K. Sran, Dina Yankelewitz

6. Making Pizzas: Reasoning by Cases and by Recursion
Carolyn A. Maher, Manjit K. Sran, Dina Yankelewitz

7. Block Towers: From Concrete Objects to Conceptual Imagination
Robert Speiser

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

8. Responding to Ankur’s Challenge: Co-construction of Argument Leading to Proof
Carolyn A. Maher, Ethel M. Muter

9. Block Towers: Co-construction of Proof
Lynn D. Tarlow, Elizabeth B. Uptegrove

10. Representations and Connections
Ethel M. Muter, Elizabeth B. Uptegrove

11. Pizzas, Towers, and Binomials
Lynn D. Tarlow

12. Representations and Standard Notation
Elizabeth B. Uptegrove

13. So Let’s Prove It!
Arthur B. Powell

Part IV. Extending the Study, Conclusions, and Implications

14. “Doing Mathematics” from the Learners’ Perspectives
John M. Francisco

15. Adults Reasoning Combinatorially
Barbara Glass

16. Comparing the Problem Solving of College Students with Longitudinal Study Students
Barbara Glass

17. Closing Observations
Arthur B. Powell

Keywords: Education, Mathematics Education, Combinatorics