On Choice, Collaboration and Closeness in Problem Solving: Aesthetic Experiences of Pre-service Teachers

Abstract

The role of the aesthetic in the work of professional mathematicians has been well documented, even though relatively few mathematicians write about their problem solving processes. Based on this, several mathematics educators have suggested that mathematics education should promote the role of aesthetics in student problem solving. While some see the aesthetic as being relevant only to more sophisticated or advanced mathematics learners, others have argued that the aesthetic can be relevant at all ages—though perhaps operating in different ways than for professional mathematicians. In this chapter, we are interested in the role of the aesthetic in the problem solving experiences of pre-service teachers. Our goal is to investigate the way they experience the aesthetic, if at all, when they are given the opportunity to engage in mathematical problem solving. Given the distinctly different context in which pre-service teachers engage in mathematics, as compared to mathematicians, we expect some differences in their experiences of the aesthetic, differences that might be helpful in better understanding how to promote aesthetic responses in the mathematics classroom, both for teachers and for learners of mathematics. We begin by providing a brief overview of the relevant research on the role of the aesthetic in mathematics thinking and learning. We will discuss how contemporary views of the aesthetic frame it in a contextual manner, rather than being concerned about the objective beauty (or ugliness) of particular proofs or solutions. In so doing, we will be highlighting the sensory nature of the aesthetic, as its etymology suggests, so that aesthetic responses can be seen as fully embodied, rather than strictly detached from emotions or feelings. We will thus also discuss ways in which the aesthetic and the affective are closely intertwined and, furthermore, intertwined with the cognitive. Although we do not want to categorically and artificially distinguish these different dimensions of problem solving, which are enmeshed in the actual experience of problem solving, we temporarily isolate the aesthetic as an under-researched and potentially quite significant dimension of mathematics. In particular, we attend to the question of what, why and how a group of 66 pre-service teachers—who were enrolled in an elementary methods course taught by the second author—value as they engage in mathematical problem solving.