| dc.description.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. | en_EN |
