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 After “hand” overcorrection was withdrawn for inappropriate foot movements, a “foot” overcorrection procedure involving foot and leg exercises was introduced contingent upon inappropriate foot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 “Hand” overcorrection was withdrawn and then re-introduced sequentially for both behaviors.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.     

Physical features vs meaning: A difference in decay

The encoding of either physical or semantic features of words was biased in an intentional learning situation. A modified recognition test was then employed to assess the effectiveness of this study manipulation and its consequences for retention. The Ss were required to select test items that were either physically similar, semantically similar, or identical to a study word. Results revealed that Ss biased toward physical encoding were more successful in selecting physically similar than semantically similar test items, while the opposite was true of Ss biased toward semantic encoding. The Ss in the two study conditions did not differ in their ability to select test items that were identical to a study word. This pattern of results was interpreted as evidence that semantic and physical information can be equally well retained over the long term. Limits on the generality of prior findings of rapid decay for physical information are discussed.     

On the form of stimulus generalization curves for visual intensity

Twelve pigeons were given successive discrimination training involving variable-interval reinforcement for key pecking in the presence of one intensity of monochromatic light randomly alternated with extinction for pecking during another intensity. All of the pigeons were then tested in extinction for generalization along the intensity dimension, and all showed a displacement of maximal responding from the positive stimulus in the direction opposite the negative stimulus. For six of the pigeons, for which the test included only three values beyond the positive stimulus, four showed peaked gradients but two did not, showing monotonic gradients with maximal responding to the most extreme test value. For another six pigeons tested over a wider range, all showed peaked gradients. Thus, when a sufficiently wide range of test values is employed, generalization gradients for visual intensity have the same peaked form as do gradients for qualitative visual dimensions such as wavelength or line angle.