How working memory and cognitive modeling functions of the cerebellum contribute to discoveries in mathematics

A theory of how connections between working memory (Science 255 (1992) 556; in: G. Bower (Ed.), The Psychology of Learning and Motivation, Vol. 8, Academic Press, New York, p. 47) and cognitive functions of the cerebellum (Trends Neurosci 16(11) (1993) 448; Curr. Opinion Neurobiol. 9 (1999) 718; Behav. Neurosci. 100 (1986) 443, Behav. Neurosci. 103 (1989) 998) lead to mathematical discovery is presented. It is proposed that (a) patterns of repetitious working memory processing are formed in the cerebellum, and (b) when these cerebellar patterns are subsequently fed back to control processing in working memory, they may become cognized in visuospatial imagery and language as the concepts and axioms that underlie mathematical discovery. It is concluded that a neurophysiological explanation of the cognitive origins of mathematics (L. English (Ed.), Mathematical Reasoning: Analogies, Metaphors, and images, Lawrence Erlbaum, Mahwah, NJ, p. 21, where Mathematics comes from: How the embodied mind brings mathematics into being, Basic Books, New York) can be based upon how conceptual constructions arise from the collaborative interactions of working memory and the cognitive functions of the cerebellum.     

GENERALITY AND SIDE EFFECTS OF OVERCORRECTION1

The effects and side effects of overcorrection for self-stimulatory behaviors of two children in a specialized day-care program were evaluated. For one child, a “hand” overcorrection procedure involving arm and hand exercises was introduced contingent upon inappropriate hand movements and later contingent upon inappropriate foot movements. After “hand” overcorrection was withdrawn for inappropriate foot movements, a “foot” overcorrection procedure involving foot and leg exercises was introduced contingent upon inappropriate foot movements. For a second child, the “hand” overcorrection procedure was introduced contingent upon inappropriate hand movements during a free-play period, and later contingent upon inappropriate vocalizations at naptime. “Hand” overcorrection was withdrawn and then re-introduced sequentially for both behaviors. Several concurrent behaviors were measured to assess multiple effects of treatment. Results for both children indicated the “hand” overcorrection procedure suppressed inappropriate hand movements and inappropriate behaviors that were topographically dissimilar. In addition, inverse relationships were observed between the second child's inappropriate hand movements and appropriate toy usage during free play and between his inappropriate vocalizations and inappropriate foot movements during naptime. Results suggest that overcorrection procedures that are effective for one behavior can be used to reduce the frequency of topographically different behaviors. This finding is discussed in terms of its practical implications for therapists.