How Can Mathematics Students Learn to Play?

When we teach mathematics, we strive to teach students to think like mathematicians. In this paper we discuss one particular mathematical habit of mind that students do not naturally display. More specifically our study of voting patterns in data collected from classroom voting questions indicates that the undergraduate students who were in the classes using these questions did not understand the significance of counterexamples to statements, or lacked the ability to construct them, or both. Searching for counterexamples to disprove statements is a natural habit of mind for professional mathematicians. In this paper we give examples, and make some recommendations. We believe that if our students get used to routinely seeking out counterexamples, as they play with various mathematical ideas, they may also end up enjoying their mathematical experiences more

Feasible Flows In Multicommodity Graphs

This thesis establishes the minimal representation of the necessary conditions for feasible supplies and demands for a given multicommodity network. The fundamental theorem is an extension of the WallaceWets connectivity result for both directed and undirected graphs. A system of absolute value inequalities is developed for the undirected case, and special properties of this system are explored. Additional results include counting the number of nonredundant inequalities for specific classes of graphs. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. Signed Harvey Greenberg iii CONTENTS Chapter 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Technical Background : : : : : : : : : : : : : : : : : : : : 3 2.1 Terms and Concepts : : : : : : : : : : : : : : : : : : 3 2.2 The Literature : : : : : : : : : : : : : : : : : : : : : 6 3. The Wallace-Wets Theorem : : : : : : : : : : : : : : : : : 13 4. Feasibility in D..