Restoration of Geometry in Secondary School Mathematics (II)

本報においては,ベルギーのW.Servaisとその協力者たちによる"Mathématique"1,2,3の幾何について分析と検討を試みる。この教科書は,集合・関係・写像・群・構造の同型性などの現代的諸概念を基盤として,学校数学全体(幾何だけでなく)を構成しようと意図するものであって,幾何においても,これらの概念を基盤として,まずアフィン平面を構成し,ついで計量を導入して,2次元ユークリッドベクトル空間へ移行する。以下においては,次の諸点について考察を加える。(1)この教科書とA.H.Колмогоровの教科書との公理系の対比(2)公理系設定の教育的意義(3)このアプローチによって,どのような幾何学像を期待するか(4)実数系と数直線の同時構成Although what to teach in the geometry course has been, and is now, one of the most crucial issues in the teaching of secondary school mathematics, there have been recently developed several noticeable projects, in which it would be expected to avoid the existing obstacles and to improve the present situation in geometry, resulting in the "comprehensive" teaching of geometry. One of them is the USSR programs which the authors have discussed in their previous paper. Another pioneering project is the one designed and executed by the Belgian group led by W.Servais. The textbooks "Mathématique 1,2,3" written by W.Servais and his collaborators are under consideration in the present paper. There is obviously no doubt that their distinguished efforts give rise to the production of these highly organized and elaborated textbooks, and that no one who has interest in school geometry would be allowed to pass by silently without reference to their textbooks, even if they would be slightly open to the charge of too much richness in unimportant details. The most remarkable part in the textbooks can be found in the chapters devoted to the birth of the symbiosis of the real numbers system and the straight line with graduation. Contrary to the USSR programs which try to unify various approaches comprehensively, W.Servais and his collaborators do not intend to include various approaches in their textbooks, but they make an attempt to build up the comprehensive school mathematics based on the modern concepts, such as sets, relations, mappings, groups and isomorphic structures. The geometry course proceeds, by introducing progressively axioms involved, through the affine structure of plane to the metric structure, leading to the two-dimensional Euclidean vector space. In this paper the authors try to give an analysis on the following: 1) the comparison between A.N. Kolmogorov's system of axioms and the system of axioms revealed in those textbooks, 2) the pedagogical significance of the axiomatic setting in school mathematics. and 3) the comprehensive construction of the whole range in school mathematics