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.