Construction of Teaching Method of Geometry to help students' cognitive development for geometrical figures

本研究的目的是构建一种有助于学生几何图形认知发展的教学方法。为此，我们进行了为期三年的两项研究。第一项研究的目的是基于纵向法阐明学生对几何图形认知发展的特征。我们计划在A、B两所初中进行调查。研究对象为A学校的90名学生和B学校的200名学生。第二项研究的目的是构建一种有利于学生几何图形认知发展的教学方法。以日内瓦学派的学习实验方法和布鲁索的教学情境理论为基础，构建“认知冲突法”和“辩证教学法”等教学情境教学方法框架。我们以一年级学生的等腰三角形问题、二年级学生的平行四边形问题和三年级学生的相似三角形问题为A学校制定了教学实验计划。我们可以得出以下结论：在第一项研究中，学生对作文和图形解析的认知发展随着年级的进步而提高，而学生对假设演绎解释的认知发展从一年级到三年级一直处于较低水平。第二项研究中，在几何图形的定理与证明应用中，“辩证教学法”课程（实验组）和“认知冲突法”课程（实验组）的成绩依次高于一般问题解决教学法课程（对照组）。我们可以接受这种教学方法的有效性。

The purpose of this study is to construct a teaching method to help students' cognitive development for geometrical figures. For the purpose, we carried out two kinds of studies for three-year period. The aim of the first study is to clarify characteristics of students' cognitive development for geometrical figures based on longitudinal method. We planed investigations in two junior high schools A and B. The Subjects were 90 students in the school A and 200 students in the school B. The aim of the second study is to construct a teaching method to help students' cognitive development for geometrical figures. Based on based the method of learning experiments of Genevan School and G. Brousseau's theory of didactical situation, we could construct a framework of teaching method by the didactical situations including “method of cognitive conflict" and “dialectic-didactical method". We made plans of didactical experiments in the school A by using problems of isosceles triangle for the first grader students, parallelogram for the second grader students and similitude of triangles for the third grader students.We could draw conclusions as follows: In the first study, students' cognitive development of compositions and resolutions of figure improved with progress of their grades whereas students' cognitive development of hypothetical-deductive explanations were retained in low level from the first from to the third grade. In the second study, in applications of theorems and proof for geometrical figures, subjects of course of “dialectic-didactical method"(experimental group) and subjects of course of “method of cognitive conflict"(experimental group) could gain higher performance in the order as compared with subjects of course of teaching method of problem solving in general (control group). We could accept the effectiveness of the teaching method.