Using dynamic geometry to expand mathematics teachers’ understanding of proof

Abstract

This paper gives a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers’ understanding of other functions of proof than just the traditional function of ‘verification’. These other functions of proof illustrated here are those of explanation, discovery and systematization (in the context of defining and classifying some quadrilaterals). A solid theoretical rationale is provided for dealing with these other functions in teaching by analysing actual mathematical practice where verification is not always the most important function. The activities are designed according to the so-called ‘reconstructive’ approach, and are structured more or less in accordance with the Van Hiele theory of learning geometry.

Notes

This is an extended and refined version of a paper originally presented at the Working Group on The Use of Technology in Mathematics Education, ICME-9, July 2000, Tsukuba, Japan.

Rasmussen and Zandieh [39] give an example of where these two defining processes appear to be intertwined. Here students adapted/generalized their definitions for straight lines and triangles of the plane to the sphere after an initial, informal experiential exploration of these concepts on the sphere.

Many reasons for this can be given. Due to the shortage of mathematics teachers generally, many black teachers currently teaching mathematics are under-qualified or unqualified to teach the subject. Such teachers tend to feel that high school algebra and calculus are much easier topics to teach as they believe it can be taught algorithmically, whereas the solution of typical riders in our high school geometry often requires far more creative thinking, and is not only more difficult for themselves to accomplish, but also to teach. However, even well-qualified teachers (in all population groups) often have difficulty with geometry. The reason is that in most teacher education institutions in South Africa, such as universities and colleges, there is a heavy focus on calculus and algebra in the mathematics courses, with hardly any geometry being done. So many return to teach only with high school geometry as their highest qualification in geometry, and so perpetuate the cycle.

Although most Van Hiele literature places hierarchical class inclusion at Level 3, Van Hiele [17, p. 93) argues as follows that it could occur at Level 2 (analysis of properties):

The development of a network of relations results in a rhombus becoming a symbol for a large set of properties. The relationship of the rhombus to other figures is now determined by this collection of properties. Students who have progressed to this level, will answer the question of what a rhombus is by saying: ‘A rhombus is a quadrilateral with four equal sides, with opposite angles equal and with perpendicular bisecting diagonals which also bisect the angles.’ On the grounds of this, a square now becomes a rhombus. (freely translated from the Dutch)

In a reconstructive teaching approach in 1977/78 with Grade 10 high school pupils it was similarly found in a classic control/experimental group comparison that the experimental group had developed a higher ability to correctly, and economically, define an unknown object [18].