Introduction

One of the earliest subjects of undergraduate mathematics education research was students' difficulties in writing formal mathematical proofs. Some research focused on the heuristics involved in proof-writing, but early attempts to show that the teaching of heuristics and strategies benefited students' proof-writing skills (Bittinger, 1968; Goldberg, 1973) failed to produce statistically significant results. Other difficulties have been identified, including students' weak understanding of logic and/or mathematical concepts and their definitions (cf. Hart, 1986; Moore, 1994). Several recent studies have looked further at students' proof-writing skills (cf. Dreyfus, 1999, Harel & Sowder, 1998; Selden & Selden, 2003); this topic is also addressed in this volume in chapters by Selden & Selden, Harel & Brown, and Zazkis. The purpose of this chapter is to look closely at one topic that arose from research on proof-writing, mathematical definitions, and most importantly, the role that these definitions play in the mathematical enterprise as well as in the teaching of undergraduate mathematics courses.

Mathematical definitions are of fundamental importance in the axiomatic structure that characterizes mathematics. The enculturation of college mathematics students into the field of mathematics includes their acceptance and understanding of the role of mathematical definitions, that the words of the formal definition embody the essence of and completely specify the concept being defined. But definitions also play a role in the students' experiences in mathematics courses themselves, in the sense that definitions are often used as a vehicle toward a more robust understanding of a given concept.