Surprises from Mathematics Education Research: Student (Mis)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. 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? 422 c © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 111 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 In addition to Definition 1, the following definitions were given in the second interview. The last two were new to the students. Definition 2. Suppose K is a subgroup of a group G and a is an element of G. The set {ay : y ∈ K } is called a left coset of K in G. It is denoted by aK . Definition 3. Suppose K is a subgroup of a group G. We define G/K to be the set of all left cosets of K in G. In other words, G/K = {aK : a ∈ G}. Definition 4. Suppose K is a subgroup of a group G. For left cosets bK and cK of K in G, we define (bK )(cK ) = bcK . Each student was given the following task to complete based on the definitions. The symmetry group of the square is denoted by D4, and F denotes a reflection across one of the diagonals. Consider the subgroup 〈F〉 of the group D4 under the operation of composition.