操作的 數學敎育 프로그램

Abstract

Motivated by his educational philosophy of 'activism', H. Freudenthal pronounced the use of mathematizing as one of the major problems of mathematics education. This, in the plenary address delivered at the Fourth International Congress on Mathematical Education, held at Berkeley, U.S.A., in August, 1980. The present study was undertaken to discuss the major problem of mathematics education indicated by Freudenthal and propose "the Operational Program of Mathematics Education" as an answer to the problem from the viewpoint of "operationalism". "Teaching to think mathematically" through the process of mathematization, interpreting and analysing mathematics as an activity, is a means to embodying academicism, realism and humanism in mathematics education. The essence of mathematical activities are the "operational schemes" reconstructed by reflective abstraction which start from the coordination of the subjects activities, and which take into account the historical and psychological development of mathematics, together with its present trend. In this case, the operations as the means of organization of the lower level become the subject matter on the higher level. Adopting this interpretation of the development of mathematical thinking and its operational nature gives orientation to the construction of curriculum and to teaching methods in mathematics instruction, and suggests a way of humanising mathematical education. In this vein H. Aebli and A. Fricke developed the operational learning principle, which aims to construct operational schemes from the subjects actions which are isomorphic to those through internalization. According to H.B. Griffiths & A. G. Howson, if we interperete mathematical activities in terms of categories of various levels the operational learning principle of Aebli and Fricke may be thought of as concerning the learning phase of moving the student from the category of concrete objects and its operations to the category of natural numbers. Thus it should be capable of being generalized to higher lovels. From this point of view, we may think that teaching to think mathematically should take the form of moving the student from operational schème Cn-1 to Cn, through reflective abstraction, in such a way that at the higher level, operational schemes of the lower level become objects of analysis, and the awareness of subject increases. Here the learning phases proposed by the van Hieles, G. Polya, E. Wittmann, and Z.P. Dienes could be metaphorically interpreted to form a functor from category Cn-1 to Cn. This operational program of mathematics education is supported by the operational constructivism of Jean Piaget and the spirits of the Erlanger Program of geometry by F. Klein and the Erlanger Programm of algebra, that is, category by S. Eilenberg and S. MacLane and could make a contribution to the huge task of humanising mathematical education.