Modeling the dynamic interplay between general and specific change in children's conceptual understanding.

In introducing this chapter, I pointed out that traditional theories of learning and of cognitive development were in conflict with regard to the effects of specific learning. Developmental theorists saw general structures as influencing specific learning but not being affected by it, whereas learning theorists took the opposite view - that general structures (if they existed) were affected only by specific experiences. In the formulation of neo-Piagetian theory, both general and specific effects were acknowledged; however, general effects were assigned to mental capacity and specific ones to the child's schematic repertoire. Thus, the possibility of reciprocal influence did not emerge (or at least was not explored). In the present chapter, I have proposed the existence of such a reciprocal influence and explored its consequences. At a general level, the two consequences that follow are (1) that the overall pace of development is accelerated and (2) that the profile of development is evened out because benefits obtained from high-frequency learning experiences are passed on, via the mediation of the central conceptual structure, to low-frequency ones. These two effects were then advanced as one possible explanation for the difference in the data obtained between different cultures and different social classes. In the former case, the explanation utilized the notion that the benefits of high-frequency learning could be passed on to low-frequency situations via the mediation of general structures; in the latter case, the explanation drew on the notion that experiential loops can accelerate or decelerate development by magnifying experiential differences that are relatively small but that prevail across most of the tasks that a child encounters. The last half of the present chapter was devoted to specifying the dynamics of this sort of interaction in mathematical terms. The data that were obtained in Chapters III and V were extremely regular and showed an even pattern of development across different tasks; hence, they could conceivably be modeled with single curves or even straight lines. The mathematical model chosen to fit these findings was much more complex, however. Each growth curve was generated by an expression that contained a dynamic tension between two opposing categories of effect: those whose tendency is to make different developmental pathways disperse (different growth rates and the effect of compounding) and those whose tendency is to hold development to a single course (the constraints imposed by a growing carrying capacity and the "binding together" or "squeezing" effect generated by the reciprocal feedback loop). The disadvantage of this sort of modeling is clearly its complexity. An equally clear advantage, however, is that it allows one to provide a unified explanation for a set of data that might otherwise seem quite disparate and to express relations in quantitative rather than merely qualitative terms. This, in turn, permits one to check the entire set of proposed relations for their consistency, and to explore the dynamic pattern of their interaction, by conducting "intellectual experiments" and checking them against common sense and/or existing data sets. In the present chapter, this approach has been used for the effects of social class and of culture. In principle, however, it could potentially be used equally to explore the effects of other variables, such as those that underlie intellectual retardation and/or "giftedness". At least for the moment, then, the mathematical modeling approach looks promising.