幾何問題解決における思考過程の研究 (I)

Abstract

Five experiments were conducted to study the thinking process in geometrical problem-solving using single problem. Exp. I was for the effect of problem conditions in thinking process, Exp. II for the effect of positive conditions which accelerate the problem-solving and negative conditions which disturb it, as the supplement of Exp. I, Exp. III for the relation between the first perception of figure and the after-thinking direction, Exp. IV for the relation between the increase of problem conditions and the degree of difficulty in its solving, Exp. V for the role of positive and negative previous learning in this problem-solving. The geometrical problem was that when the circle O had a radius of 5cm and its two diameters AB and CD were perpendicular to each other and PE⊥CD and PF⊥AB were constructed from any point P on the circle, subjects were asked to estimate the length of the line EF (cf. Fig. 1).The Ss were 140 junior high school students (3rd grade) who were divided into seven groups in Exp. I, 240 students into 5 groups in Exp. II, 12 students (2nd grade) and 65 college students in Exp. III, 25 students (3rd grade) and 30 elementary school children in Exp. IV, 45 students (3rd grade) into 3 groups in Exp. V. All groupings in each experiment were expected to be homogeneous as regards mathematical achievment. All Ss were used only once through the whole experiment. The Ss were put to group test in Exp. I, II and to individual test in others. The main findings were as follows:1) There are many great and small sections in the thinking process of geometrical problem-solving. The greatest section is the key of its solving. The solving comes out from inferring the meaning of the key-figure from an appropriate theorem. This is what is called “insight” in geometrical problem-solving (Exp. I).2) The positive conditions which accelerate the recall of the needful theorem in problem-solving…words, figures, previous learnings…make its solving easy, but the negative conditions which disturb its recall do not necessarily disturb the problem-solving (Exp. II, V).3) The first perception of geometrical figure exercises certain effects on theafterthinking fairly. The poor students in mathematics is more easily swayed by it than the superior students. The latter is free and behaves with flexible attitude (Exp. III).4) The more the conditions of problem are given, the deeplier the key of a problem is covered and its finding becomes more difficult. If it is solved, it will take longer time to solve (Exp. IV).