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. 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 bodily functions.
Mathematical cognition is observed as a doubly embodied ongoing action in an environment.
The environment is seen to provide the occasion and the space for the action. Thus both are co-implicated in any mathematical activity by a person. 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.