Inhibiting factors in generating examples by mathematics teachers and student teachers: The case of binary operation

The main objectives of this study were to identify difficulties encountered by mathematics teachers and student teachers associated with the concept of binary operation regarding the associative and commutative properties and to reveal possible sources for them. Thirty-six in-service mathematics teachers and 67 preservice mathematics teachers participated in the study. All participants were presented with a task calling for the generation of a counterexample, namely, a binary operation that is commutative but not associative. Responses to the task were analyzed according to four categories: correctness, productiveness, mathematical content, and underlying difficulties. The findings point to similarities and differences between the two groups. Both groups exhibited a weak concept by failing to produce a correct example and by using a limited content search-space. These findings suggest two main inhibiting factors: one related to the overgeneralization of the properties of basic binary operations and the other related to pseudo-similarities attributed to these properties, which seem to be created by the recurring theme of order. Teachers were superior to student teachers on the categories of correctness and productiveness.