Dydaktyczna rola skończonych i konkretnych modeli struktur matematycznych

Abstract

In the article a finite and concrete model of a mathematical structure is called a system of sets and relations defined on those sets, being a model for the given structure and such that cardinal numbers of those sets are presented to the student by means of figures and are practically "small", while the relations are defined in a way enabling him to solve problems connected with structural properties of the model by direct manipulations with elements of the sets or their subsets, which makes him step by step conscious of the structure. Modern didactics of mathematics emphasizes more and more such models as means of transition from the concrete to the abstract, natural for the student, (algebraic and order structures, combinatorics, combinatorial topology, metric spaces etc.). The role of such models, in opinion of didacticians advocating their use in teaching, consists in that the student can reason quite precisely, perform investigation and, in particular, prove theorems by concrete, manipulative exhaustion of finite and accessible for manipulations number of cases, without necessity to deal with abstract conceptual classes and general inferences. But it is the lack of this necessity that arouses reservation, for it limits forms of student's mathematical thinking and often keeps them on the stage of purely empirical investigation. On the grounds of some typical examples, forms of student's activities both those that are being developed in the course of his exploration of a finite and concrete model and those that are not, are discussed in the article. Also the heuristic role of such a model in finding solution of a more general problem is presented, as well as the role of models for transition from the concrete to the abstract, such that although finite are of such a great cardinal number that "manipulative" investigation is no longer possible for the student and naturally suggests the necessity of conceptual thinking using general inferencies. In particular the method of generalisation by variation of constants, natural for the student, is discussed. The article includes some didactic implications, in particular it emphasizes the general principle, often passed by in evaluation of a didactic experiment that consists in veryfying new means of transition to mathematical abstraction, namely the following principles all such means should be investigated not only in view of their advantages but also in view of what they cannot give or of what is lost by using them.