The Role of Continuous Processes in Cognitive Development  

Many scientists observing the cognitive development in children have noted distinct phases in the way they learn. One phase appears to be a gradual accumulation of experience. Another phase appears to be a reorganization of those experiences to make them more useful. In this paper we show how mathematical closure concepts can be used to abstractly model these cognitive processes. Closed sets, which we will call knowledge units, represent tight collections of experience, facts, or skills, etc. Associated with each knowledge unit is the notion of its generators consisting of those attributes which characterize it. Finally, we provide a rigorous mathematical model of these di erent kinds of learning in terms of continuous and discontinuous transformations. There are illustrations of both kinds of transformation, together with necessary and su cient criteria for certain kinds of transformation to be continuous. By using a rigorous de nition, one can derive necessary alternative properties which may be more easily observed in experimental situations. The formal mathematics is illustrated with reference to Lev Vygotsky's view of cognitive psychology, but it is not a veri cation of his model. We believe that this concept of \continuity" can be re ned to test, and possibly verify, his and other models of cognitive behavior