Embodiment, Symbolism and Formalism in Undergraduate Mathematics Education

Plenary Speaker
David Tall
The University of Warwick

This presentation will relate to a range of contemporary theoretical perspectives by presenting a framework of three modes of thinking that operate so differently as to present essentially three distinct ‘worlds of mathematics’—conceptual embodiment, proceptual symbolism and axiomatic formalism—which will be abbreviated in the context of this theoretical framework to embodiment, symbolism and formalism. Human learning will be addressed in terms of compression of knowledge in which important aspects of a situation are named and built into rich thinkable concepts that enable thinking to be performed by making connections between the thinkable concepts to build successively more sophisticated conceptual structures. In this way, any new concept needs to be seen in the light of previous experience, building on aspects within the individual’s concept image that I call met-befores. Symbolism builds on earlier embodied and symbolic met-befores; formalism builds on earlier embodied, symbolic and formal met-befores. All three operate with ongoing interchange between them.

To illustrate the power of embodiment to underpin formal thinking, the embodied notion of local straightness will be contrasted with the symbolic notion of local linearity to show that while the latter has great computational power, it lacks the embodied meanings that lead naturally to significant formal meanings in mathematical analysis. On the other hand, proceptual computations in many areas such as groups, vector spaces, mathematical analysis will be shown to lead not only to formal definitions but also to structure theorems that provide deeply meaningful embodiments.

The presentation will consider a range of recent studies which show how embodied met-befores can both enhance and impede formal thinking and discuss how a combination of (conceptual) embodiment, (proceptual) symbolism and (axiomatic) formalism relates to the all-embracing Lakoffian concept of embodiment, and process-object encapsulation that is the main focus of APOS theory.

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