幾何問題解決における思考過程の研究 (II)

Abstract

This report contains four experiments on thinking process in the geometrical problem……“When two points, A and B, are on one side of a straight line XY, find a point P on XY so that AP+BP is shortest.” These experiments were designed to examine the role of a point of symmetry in the original problem-solving, chiefly in view of the direction of line and points, the effect of which T. Obonai found in the Figugre Dividing Test. Exp. I attempted to clarify how much the superior students in mathematics use “symmetry” in this problem-solving, when they have not learned about it at school. Ss were 10 junior high school pupils. The result was very unfavourable. Exp. II was designed to clarify how much the normal children who have not learned about “symmetry” response symmetrically to simple geometrical figures, when they were allowed to response freely to them. Each figure was given twice in different directions, vertical and horizontal. Ss were 88 elementary school children and 38 junior high school pupils. As a result, the symmetric responses were found more than expected and the difference between the number of the symmetric responses in the vertical direction and that in the horizontal was significant. Exp. III was planed to verify, in the original problem, the effect of different directions of figures which was clarified in Exp. II. Ss were 96 junior high school pupils. The result showed that the effect of direction was found as expected. Exp. IV was for the verification, in two other geometrical problem, of the fact which was found in Exp. III. Ss were 196 junior high school pupils. In Exp. III, IV, Ss were divided into four homogenious groups. The main findings were as follows:(i) Some degree of effect of the direction of lines or points in the thinking process was generally found in geometrical problem-solving. Especially the vertical direction of line was more favourable to Ss than the horizontal in the case of the original problem.(ii) Generally, when the key of the geometrical problem was exposed perceptually by the direction of its figure, the problem was solved easily, but when not, it was difficult to solve.(iii) When the structure of a problem, as in the original problem, called the subject's attention to the part where the key for solving was not given even the necessary response for its solution, such as symmetric response which appears easily every day, was suppressed and the problem was hard to solve.