## Effects of Geometrical Construction in Computer-based Learning Environments.– The Case of Cabri-Geometry –

*Hiroko Tsuji*

`htsuji@human.tsukuba.ac.jp`

Doctoral program

**University of Tsukuba**

1-1-1 Tennodai

Japan

### Abstract

The purpose of this paper is to consider the meaning and the effects of the drawing figure in computer-based learning environments. Then, we will investigate its effects on geometrical learning in Junior high schools in Japan. For this study, I refer to G. Brousseau’ Ideas, the “milieu” in particular.
Brousseau explains that the “milieu” is a system opposing the taught system (students) (Brousseau 1997). This system is related to mathematical knowledge of teaching. Brousseau says that students adjust to the milieu, which is one of representations of mathematical knowledge that can be controlled by students, although the conditions that the milieu holds can not be. Computer is considered a tool setting the milieu as the learning softwares are very much developed in resent years (Balacheff & Kaput 1996, etc.).
The drawing figures are a means of constructing elements in the geometrical object. This activity is essential to leaning Geometry. we defined drawing figures in computer-based environments as follows, based in the definition of construction and the four-steps of problem-solving with construction in elementary geometry; analysis, drawing, demonstration and test (Hayashi 1927). :
The construction in computer-based environments is to create drawings as examples that satisfy the conditions. In other words, it is to create drawings that keep the conditions even when being moved with drag-mode on the software such as Cabri-Geometry.
From the definition, we can be said that the construction in computer-based environments is a part of the milieu. The drawing has conditions including processes of the construction. In the steps of demonstration and test of the construction, students can confirm the validity. Moreover it gives students opportunities of confirming their conception of the figures.
We think that this is related to interactions between students and the milieu proposed by Brousseau.
: Construction, Milieu, Computer-based leaning environment
Brousseau, G. (1997). Theory of didactical situation in mathematics, (Edited & translated by N. Balacheff, et al.) Kluwer.
Balacheff, N and Kaput, J. (1996). Computer-based learning environments in mathematics, In A. J. Bishop et al.(eds.), International Handbook of Mathematics Education, pp. 469-501, Kluwer.
Hayashi, T. (1927). The Form of Elementary geometry, Kohdoh-Kan.
Pratt, D. and Ainley, J. (1997). The construction of meanings for geometric construction: Two contrasting cases, International Journal of Computers for Mathematical Learning, 1(3), 293-322, Kluwer.

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