Gin McCollum, Portland State University
Mathematical cognition and the sensorimotor brain: Experiment on mental imagery Not all mathematicians use mental imagery, but many do: 9 out of 11, in this study. Participants were asked, with eyes closed, to visualize graphs of quadratic equations and to wrap spirals around cylinders and cones. There were seven such mental imagery tasks performed in five different body positions, each of which set the head in a different direction with respect to gravity.
The mind's eye obeys much the same geometry as the physical eye, preferring usually to look at imagery in front of the face and right side up. But which way is right side up, when body vertical differs from gravitational vertical? One mathematician commented: "If you'd asked me in advance what I thought would be the outcome, I wouldn't have been anywhere close to the reality."
Under these conditions, participants related to imagery with a mixture of directiveness and curiosity. For example: "Plane starts on wall, rotates to floor, I 'bring it back up.'… Graph emerges, sort of breathes a bit, but doesn't oscillate." For another mathematician, the imagery was also semi-autonomous: "I felt as though my brain wanted to move up on the graph where the graph was wider so that my brain could grab onto the side of the graph in order to stabilize it and keep it from wobbling."
Examining participants' descriptions has led me to the conclusion that imagery is typically embedded in a three-dimensional, body-centered, multisensory space, probably not Euclidean. Such an interior space is called a "peripersonal space". A large body of research suggests that we all carry peripersonal spaces around with us for tool use and personal safety. Although placing imagery in an internal peripersonal space resembles physically placing a picture on a wall, the sensorimotor process in both cases is not Euclidean. Besides, imagery moves, seemingly autonomously.
Imagery is a creative tool used by many mathematicians, but not all. What does it suggest for education? What does it tell us about the nature of mathematical cognition and creativity? It connects to my sensorimotor research. There are many stories of solutions appearing autonomously in well-prepared minds. Is the semi-autonomy of imagery the humble kin of deep creativity?
