The principal aim of this chapter is to provide a structure for mathematical topics as an aid to ‘psychologizing the subject matter’, as Dewey (1933) put it. The secondary aim is to reveal just how complex a matter preparing to teach a topic effectively can be, beyond trying to make the definitions and theorems as clear as possible.

The chapter develops the notion of a concept image (Tall & Vinner 1981) into a description of a framework based on a threefold structure of the psyche. Two mathematical topics, quotient groups and L'Hôpital's rule, are used to illustrate how the framework can be used as a reminder to direct attention to structurally different aspects of any topic when preparing to teach it. The framework can be used both at a more abstract level (for example, treating groups or limits as the topic) or at an even more detailed level (for example, quotient groups of cyclic groups or the relation between L'Hôpital's rule and derivatives). When combined with awareness of learners' mathematical powers and with ubiquitous mathematical themes and heuristics, the framework can be used to inform the design of pedagogically effective tasks and interactions with learners.

Concepts and Concept Image

Concepts are not isolated entities floating about in our minds but rather familiar ‘lines of thought’ triggered by concept labels.