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ABSTRACT
The use of technology is now widespread at the school as well as college levels. Instructors, however, continuously debate the usefulness of hardware/software in enhancing the learners' knowledge and their understanding of mathematical concepts. This research explores whether technology can contribute to a better student understanding of geometrical concepts and theories. In particular, we explore whether a learning environment that uses Dynamic Geometry Software (DGS) can help students determine the truthfulness of certain geometrical conjectures or whether such an environment can mislead students. Results show that the continuous variations of a geometric figure may sometimes be misleading and that a strategy guiding the investigation process and exploiting the potential of the new technologies are in many cases necessary to properly investigate a given geometrical problem. Results also show that the role of the teacher is, in many cases, critical for the development of the solution strategy.
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INTRODUCTION
The advancement of dynamical computer software has contributed to fundamental changes in the way mathematics is being taught at various levels. Not only has a renewed emphasis on visualizing mathematical concepts been observed in teaching various topics, but also such a dynamic environment has added the potential for more elaborate student investigations. G. Hanna writes [8, pp. 12-13]: "The availability in the classroom of software with dynamic graphic capabilities has given a new impetus to mathematical exploration"; in particular Hanna adds that "dynamic software has the potential to encourage both exploration and proof, because it makes it so easy to pose and test conjectures." This is particularly important in geometry where research results have shown that dynamic geometry software programs help students understand geometric propositions because such programs allow them to more easily perform geometric constructions with high level of accuracy. According to Gawlick [6, p. 300], "DGS expands the scope of activity", by making the ruler and compass constructions more dynamical. Thus, "students can easily test conjectures by exploring given properties of the constructions they have produced" [8, p. 12]. In line with these claims, Michael de Villiers [3] had asserted that explorers are now able to more easily investigate whether conjectures are true or false through continuous variations of geometric configurations.
The focus of this paper is two-fold: First,...