To be or to become: how dynamic geometry changes discourse

Abstract

In this article, we investigate the impact of the introduction of a dynamic geometry environment on mathematical thinking by identifying changes in discourse engendered by its introduction in a high school geometry class. Our focus is on the teacher, and we find significant differences between static and dynamic geometry in terms of the ways in which the teacher talks about geometric objects, makes use of visual artifacts and models geometric reasoning. Even though these changes have major implications for the geometry being studied, they are made only very implicitly in the classroom.

Keywords:

• dynamic geometry
• mathematical discourse
• geometry

Notes

1. This internship (school-based placement) was a required component of her two-year pre-service education programme. It involved her taking on increasing responsibility for teaching two or three high school mathematics courses.

2. It seems that Thomas meant to ask whether it is possible for a parallelogram to be a rhombus, given the fact that the sketch is showing a parallelogram and his follow-up statement that Sarah would try to drag the parallelogram to look like a rhombus.

3. We say implicit because the actual turn of phrase “be made into” is in the passive form, indicating perhaps a move toward generality (that anyone would do it, not just Sarah).

4. Although this principle was never accepted as sufficiently rigorous by modern geometers, Poncelet's goal was to provide a way for geometry to continue the tradition of Ancient Greek geometry, in which visual reasoning is central, but provide the level of generality - greater than that of the Greeks - that his contemporaries were achieving with their algebraic methods of analytic geometry. The particular relationship between objects of one class is not evident in the geometry of static figures: both Poncelet and dynamic geometry have their figures in motion.

5. Although Thomas did not go down this road, we find it useful to consider the way in which one could answer the new question in relation to the old one. In fact, the new question does not fully address the original form of the question (“Is <this shape> a <that shape > ?”), because it is just concerned with showing that one can make at least one rhombus out of a particular parallelogram. In order to address the original question, one must show that any parallelogram can be dragged to make any rhombus.

6. This struck us as an interesting move, perhaps indicating his wish to return to a more comfortable or familiar discourse where objects are static and continuous transformations are impossible.