Investigation of students' reasoning with examples and non-examples of function in abstract algebra

Abstract

The focus of this dissertation is on students' thinking regarding the function concept in the context of abstract algebra, focusing on the properties of well- and everywhere-definedness. This dissertation follows a three-paper format, where each paper has a different yet related focus.

In the first paper, I analyze (non-)examples found in textbooks and those used during instruction related to function in abstract algebra to gain insight into how students are expected to reason about functions in this context. I investigate experts' thinking about functions by conducting a textbook analysis and semi-structured clinical interviews with mathematicians. I then elaborate the essential properties of well- and everywhere-definedness into four categories of non-examples that students are expected to successfully reason about in an introductory abstract algebra course.

In the second paper, I explore students' reasoning with examples and non-examples of function related to the four categories by conducting task-based clinical interviews. I provide characteristics of a coordinated way of understanding functions in abstract algebra and illustrate how such a coordinated way of understanding functions enables students to reason productively with function tasks and the properties of well- and everywhere-definedness. This paper addresses what is entailed in reasoning productively about well-definedness and everywhere-definedness.

In the last paper, I present a functions activity focused on recovering an example from a non-example of a function by prompting students to modify the domain, the codomain, and/or the rule. I argue that such an activity can help instructors have a clearer image of how they might support their students in developing a coordinated view of function and is a tool that abstract algebra instructors can use to help students attend to well-definedness and everywhere-definedness.