A study of methods of teaching and learning that promote concept formation of geometric figures

本研究旨在提供促进数学中几何图形概念形成的有效教与学方法，通过强调理论上指出的促进两个几何图形之间关系的理解状态转换的因素的课程，验证了因素实施的可能性。在此基础上提出了二、五、八年级几何图形的教与学方法。在二年级，教与学的要点是：通过比较矩形定义后除正方形以外的平行四边形和矩形，注意矩形的四个角都是直角；通过折纸做矩形、画矩形、用三角形确定矩形的直角等活动，了解矩形有四个直角。五年级的教学重点是注意矩形的四个直角。在五年级，通过折纸做矩形、画矩形和用三角形确定矩形的直角的活动，了解矩形有四个直角。五年级的教学重点是通过比较矩形定义后的正方形和平行四边形来发现矩形的四个角教与学的要点是考虑几何图形的性质来测量几何图形的面积，将平行四边形、菱形和正方形的面积测量方法与已学过的利用几何图形及其之间的关系测量几何图形的面积的方法联系起来并加以确认。八年级几何图形定义方法的要点是不使用负表达式来表示几何图形的定义，即边不同，角度不是直角等，并讨论了几何图形定义不包含负表达式的原因。

The purpose of this study is to offer the efficient methods of teaching and learning that promote concept formation of geometric figures in mathematics.Through the lessons with emphasizing the factors which were pointed out theoretically to promote the transitions of states of understanding of the relations between two geometric figures, the possibility of implementation of factors was verified. The methods of teaching and learning of geometric figures in second, fifth and eighth grade that proposed based on the results are as follows.In the second grade, points of the method of teaching and learning are to notice that four angles of a rectangle are right angles by comparing parallelograms except for rhombi and rectangles to rectangles except for squares after defining of a rectangle and to understand that a rectangle has four right angles through the activities which are making rectangles by folding papers, drawing rectangles and confirming right angles of a rectangle by using triangles.In the fifth grade, points of the method of teaching and learning are to measure area of geometric figures by considering their properties and to measure area of parallelograms, rhombi and squares by relating the ways of measuring area of each of them with ways of measuring areas of geometric figures that have already learned by using of the geometric figures and the relations between them and to confirm it.In the eighth grade, points of the method of defining of geometric figures are to represent definitions of geometric figures without using of negative expression, i.e. the sides are different, the angles are not right angles, etc. and to discuss about the reason why definitions of geometric figures do not contain negative expression.