Roles of Aesthetics and Affect in Mathematical Problem-Solving

Abstract

The purpose of this chapter is to tease out some of the significant aspects of both aesthetic sense and affective issues (two under-researched domains) in the solving of nonroutine mathematical problems. Especially, as in the case of the nonroutine problem-solving involved in the Problem@Web Project, when students are not under compulsion to solve the mathematics problems, a significant question is as follows: What is it that gives them the motivation to do so and to persist in the face of difficulty in order to reach a solution? After presenting a vignette of a high school teacher who was particularly aware of the aesthetic elements of mathematics, and who stressed these in his teaching, I address some theoretical formulations of “the aesthetic”, taken from both the arts and the sciences. Aesthetics and affect are intimately connected, but they are not the same because they function differently in problem-solving processes. Further, I provide research evidence, by analysing both negative and positive affective pathways, that both are also intimately connected to the cognitive domain. Visual aspects of aesthetic sense are also examined. The patterning aspects of aesthetic sense often result in metaphoric thinking that has the potential to be fruitful and to overcome some of the affective states, such as discouragement, which threaten to derail the problem-solving processes. Some examples of metaphors that were ultimately fruitful are presented. Connections, patterns, and metaphors are recurring themes in my research that related to affect and to aesthetics. I present a brief vignette from the case of Sam, whose initial compartmentalization prevented him from solving a trigonometric problem, and an elegant visual solution by Leonor, to one of the problems presented in the Sub14 mathematics competition in which she participated. Finally, I close with suggestions for further research that might prove fruitful.