It is common sense to think about mathematical knowing and understanding
as entailing a very specific response to a pre-given problem from the environment.
Further it is easy to think of the product of that mathematical activity
as an "answer" which matches pre-given conditions. While such a view is
popular with those who develop mathematics tests and some cognitive scientists,
it is problematic on several counts. As Wittgenstein and Lakatos have pointed
out many years ago, this view disembodies mathematical thinking. It divorces
mathematical reasoning from the broader personal histories of the person
doing the reasoning and ignores or reduces the complexities of the situations
in which mathematical thinking by humans actually occurs. The research
described here asks us to think otherwise about cognition in general and
mathematical cognition in particular.
Some Tenets Of An Enactive View Of Mathematical Cognition
* Mathematical cognition is viewed as an embodied interactive process
with the environment in which the person acts. It is not a reactive representation
of the environment which attempts to match the environment. Nor is it observed
simply to be an emergent phenomenon arising from more primary brain and
* Mathematical cognition is observed as a doubly embodied ongoing action
in an environment. A person's structure
determines the action which the person takes (in-person embodiment). The
environment is seen to provide the occasion and the space for the action.
(person-in embodiment). Thus both are co-implicated in any mathematical
activity by a person.
* Mathematical cognition and understanding are seen as a non-linear,
processes through which one builds and acts in a mathematical world.
* The teacher or the teacher-researcher is seen to be in the middle of the students' mathematical actions and is observed as being a key part of that environment which provides the occasions for the cognitive actions observed.
If mathematical cognition is viewed this way, then research on
it cannot simply consider disembodied tests and test results. Such research
must consider and trace the patterns of mathematical activity and understanding
as it occurs, must look at the mechanisms and beliefs by which persons
act mathematically, must attempt to account for the ways in which the environment
occasions or creates space for personal mathematical activities, and must
account for the interactions and conversations through which mathematical
activity occurs and by which it is bounded. The Enactivism Research Group
seeks to observe mathematical cognition as such a recursive and self-organizing
process co-emerging with a community and in an environment. It is hoped
our research will provide basic insights into how mathematics as an effective
way of acting in one's world might be taught and how better spaces for
its learning might be provided.
For references, see the Enactivism reading list.
This page maintained by David
A. Reid. email: firstname.lastname@example.org