Making Sense of Abstract Algebra: Exploring Secondary Teachers’ Understandings of Inverse Functions in Relation to Its Group Structure

ABSTRACT

This article draws on semi-structured, task-based interviews to explore secondary teachers’ (N = 7) understandings of inverse functions in relation to abstract algebra. In particular, a concept map task is used to understand the degree to which participants, having recently taken an abstract algebra course, situated inverse functions within its group structure (i.e., the set of invertible functions under composition). In addition, their particular conceptions of functions and function composition throughout the interviews were then also considered as a means to explore further their responses during the interviews. Findings indicate that only two participants showed evidence of the desirable mathematically powerful understandings from abstract algebra in relation to inverse functions, and further analysis suggests a variety of challenges in terms of developing meaningful connections, which were more related to conceptions about secondary content than to the abstract algebra content. Implications for the mathematical preparation of secondary teachers are discussed.