Anticipated Date of Graduation

Summer 2019

Document Type

Thesis

Degree Name

Master of Science in Mathematical Sciences

Department

Mathematics

First Advisor

Douglas Darbro

Abstract

The subject of elementary algebra, hereafter referred to "algebra", is considered a fundamental skill in the civilized world. Equation solving in particular is one of the most basic skills that is taught in the subject of algebra. While the axioms and inference rules of algebra are well established and agreed upon, the methods of how to employ these axioms and rules are not taught using an explicit system [Bundy 1975]. It is know that when a person attempts to solve an equation they must be, to some degree, using some implicit system to select each new step. It is evident that they are not simply applying the axioms, theorems and inference rules of algebra randomly and indiscriminately, otherwise their “solutions” would be useless. Two questions then arises: what is the nature of these implicit systems, and how can more be learned about them? While it is clear that one or more systems for solving equations are in use by humans, and by computers that generate human like solutions, these systems are usually opaque or difficult to understand. Humans cannot explain in any great detail how they are able to select the correct steps while solving equations [Bundy 1975]. Software that is designed to mimic human equation solving is usually not inspectable because it is closed source. While research into the nature of systems that can be used both by humans and computers exists, it did not develop the concepts necessary to apply it to education research. [Bundy A 1983]. Therefore, this paper describes an initial conceptual framework for describing different aspects of these systems that can be used by educational researchers.

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