Surprises from a Piece of Mathematics Education Research: Student Use of Mathematical Definitions

Algebra Interview 1 The first interview began with Edwards asking “What is mathematics?” By asking follow-up questions as needed, Edwards made sure each student said something about the nature and role of definitions in mathematics. After that introductory discussion, each student was handed the following definition as it appears below. Binary operations were being studied in class, but the definition of group had not yet been presented. Definition 1 Let G be a nonempty set together with a binary operation that assigns to each ordered pair (a, b) of elements of G an element of G denoted by a b. We say G is a group under if the following three properties are satisfied. 1. The operation is associative; that is, (a b) c = a (b c) for all a, b, c in G. 2. There is an element e (called the identity) in G, such that a e = e a = a for all a in G. 3. For each element a in G, there is an element b in G (called the inverse of a) such that a b = b a = e. After reading the definition, each student was given these three tasks to complete. Set: R, the set of real numbers Binary Operation: Ordinary subtraction, Is R a group under subtraction? Set: R, the set of real numbers Binary Operation: The operation ⊕ where a⊕ b = a+ b+ 3 Is R a group under ⊕? Set: R, the set of real numbers Binary Operation: Ordinary multiplication Is R a group under multiplication? Abstract Algebra Interview 2Algebra Interview 2