Inhibiting Factors in Generating Examples by Mathematics Teachers and Student Teachers: The Case of Binary Operation

Abstract

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. This study is part of a larger study the aim of which was to investigate ways in which mathematics teachers and student teachers generate counterexamples in mathematics. Two main goals of the study were, first, to identify difficulties encountered in generating examples related to a number of mathematical topics that teachers and student teachers are expected to teach and, second, to reveal possible sources of difficulty for them. The possible sources of difficulty in generating such examples were presumed to include the following: incomplete knowledge, inability to process existing knowledge, misconceptions, and insufficient logical knowledge. The case of binary operation represents a case in which the relevant knowledge is assumed to be available to secondary mathematics teachers and student teachers; however, that knowledge requires processing in order to produce such examples. The current work is designed to probe for factors inhibiting the processing involved in generating counterexamples, factors that in turn may shed light on the limited binary operation concept. The search for these limiting factors was carried out by an extensive analysis of the processes and difficulties involved in generating the required examples. As indicated by Bratina (1986), the task of generating an example is considered powerful in terms of revealing strengths and weaknesses. The notion of a binary operation is dealt with in various stages and courses at different levels and contexts throughout the mathematics education that prospective mathematics teachers experience. However, in many cases prospective teachers do not manage to integrate their various encounters into one comprehensive, abstract