| Research Abstract |
1.We conducted some experiments for pupils of elementary, lower and upper secondary schools. One of main aims was to clarify characteristics of pupils' spatial thinking in solving problems having to construct and to operate mental image, for example, rotation, composition or division, projection, development, section of solids, decision of points in 3-D space, changes and its' integration of points of view, visualization(drawing9 of 3-D fugures.
On the base of pupils' writings/drawings and interview with some of them, we summarize characteristics on pupils' spatial images and their mental operations as follows.
【encircled1】Spatial images are 3-D, 2-D, 1-D or 0-D figural.
【encircled2】Spatial images change between 3-D figurals or between 3-D and lower-D figurals, and inversely, in process of mental operations.
【encircled3】Dimensional changes of images show flexibility of spatial thinking.
【encircled4】Changes of images and process of reasoning are represented in writings/drawings.
【encircled5】Mathematical knowledge, methods, and visualization(drawing) lie between dimensional changes of images.
【encircled6】Construction of spatial images and their mental operations, mathematical knowledge and methods, and visualization(drawings) are closely connected and their synergy is cognitively for proficiency in spatial thinking.
2.We propose the followings :
【encircled1】Construction of spatial images and their mental operations, mathematical knowledge and methods, and visualization(drawings) are basic for learning geometry in school mathematics.
【encircled2】Developing of pupils' spatial thinking, we need to develop teaching materials and teaching-learning methods.
【encircled3】We show teaching-learning practices for elementary, lower and upper secondary schools.
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