Affect, behavioural schemas and the proving process

Abstract

In this largely theoretical article, we discuss the relation between a kind of affect, behavioural schemas and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our focus to a particular kind of affect–nonemotional cognitive feelings. We then mention the position of feelings in consciousness because that bears on the kind of data about feelings that students can be expected to be able to report. Next we introduce the idea of behavioural schemas as enduring mental structures that link situations to actions, in other words, habits of mind, that appear to drive many mental actions in the proving process. This leads to a discussion of the way feelings can both help cause mental actions and also arise from them. Then we briefly describe a design experiment–a course intended to help advanced undergraduate and beginning graduate mathematics students improve their proving abilities. Finally, drawing on data from the course, along with several interviews, we illustrate how these perspectives on affect and on behavioural schemas appear to explain, and are consistent with, our students’ actions.

Keywords:

• tertiary level
• design experiment
• proof
• proving
• affect
• nonemotional cognitive feelings
• behavioural schemas
• habits of mind

Acknowledgements

We would like to thank Educational Advancement Foundation for its support in helping us develop and study the Modified Moore Method course mentioned above. Ideas expressed in this article are those of the authors and not of the Foundation.

Notes

Notes

1. This is consistent with Libet and his collaborators’ psychological/neurological work suggesting that the immediate cause of (simple physical) actions, such as raising a finger, is an unconscious mental state–rather than the conscious intention to act, as one might expect [18, pp. 329–333].

2. This is similar to Selden et al.'s analysis of students’ problem-solving processes. In that analysis, ‘problem situations’ were treated as similar to concepts and were seen as sometimes linked to ‘tentative solution starts’ in one's knowledge base [20, p. 146]. Bringing a tentative solution start to mind is a mental action, and consistently pairing a problem situation with the action of bringing to mind a corresponding tentative solution start, can be viewed as a behavioural schema.