Математичко мишљење: општа или посебна правила закључивања?

Abstract

Vygotsky proposes that the highest so called structural level of knowledge comprises formal structure of knowledge (built-in intelligence) along with patterns of reasoning establishing a foundation for this structure. There is no doubt that structural level of knowledge differs from area to area (compare, for example, mathematics with history). On the other hand, some patterns of reasoning such as modus ponens are general, applicable to many (perhaps all) areas of school/scientific knowledge. Being concerned with typical patterns of reasoning applied in (school) mathematics, we examined in more detail the following question: Are patterns of reasoning general or nevertheless particular, distinctive to school/scientific area? Our analysis evidences that some patterns of reasoning are general (applicable to knowledge in general or to knowledge of several school/scientific domains), whereas others are primarily reserved for mathematical thinking (compare, for example, reasoning by induction with reasoning by mathematical induction). Along with that, patters of reasoning may be context dependent even within a single domain. In case of specialization, we can examine different mathematical objects (particular numbers, factorable polynomials, regular polygons, etc.). Having in mind research in artificial intelligence (from general problem solver to expert systems), it is clear that, beside general patterns of reasoning, competent problem solving also requires those patterns that are specific to a particular domain and probably sensitive to context of application.