de Villiers M. (1996)
The Future of Secondary School Geometry.

La lettre de la preuve, Novembre-Décembre 1999.

Editorial comment:
This is the text of a Plenary lecture given by Michael de Villiers at the "Geometry Imperfect" conference which was held in October 1996 at the University of South Africa (UNISA), Pretoria, South Africa. It deals with the "future of secondary school geometry". A refined version of this paper has been published in the December 1997 issue of PYTHAGORAS, the official journal of AMESA (Assoc of Math Ed of SA).
© Michael de Villiers

 

Introduction

Recently in a Mathematical Digest (Jul '96, no. 104:26) published by the Mathematics Department at UCT someone wrote the following:

"... South Africa is the habitat of an endangered species, for Euclidean Geometry has disappeared from the syllabuses of most other countries ... "

Such a statement is rather common amongst mathematicians and mathematics educators in South Africa, but speaks of great ignorance, as nothing could be further from the truth. In fact, geometry is alive and well and experiencing an exciting rebirth in many countries; not only at school level, but at university level as well. There is great danger if policy makers in mathematics education in South Africa are unaware of these dramatic new developments.

 

Some developments in contemporary geometry

   Some developments in geometry education  

The Van Hiele theory
Russian research on geometry
The primary & middle school geometry curriculum

Process versus product teaching in geometry
The USEME experiment

Dynamic Geometry Software

Concluding comments

So what are some of the crucial changes necessary in secondary school geometry as we approach the year 2000? Basically the changes can be summed up as changes in content, process and teacher education. In terms of content there is a need to contemporize by including possible content such as fractals, graph theory, transformations, non-Euclidean geometry, etc. at various grades and at various levels of formality. In particular, the study of transformations could form a valuable golden thread through the entire curriculum, and in the high school show the powerful integration of algebra and geometry (see De Villiers, 1993). However, even before any changes in the high school, many changes are necessary to our primary school geometry curriculum. Apart from content such as tessellations, vision- and 3D-geometry as described by Van Niekerk (1995, 1996) and Witterholt & Heinneman (1995) is absolutely essential for developing visualisation and spatial orientation skills, not only for formal geometry later on, but also for further study in woodwork, metalwork, architecture, art, computer graphics, engineering design, etc. More use could also be made of accurate scale drawings to solve complicated real world problems, and to develop an intuitive understanding of the process of modelling. These changes also have to be contextualised meaningfully in different contexts geographically, culturally, linguistically, etc.
   However, perhaps even more important than changes in geometrical content, we need to focus far more on teaching and developing the process aspects of mathematics. It needs to be acknowledged that geometry content should not be presented in a ready-made form to pupils, but that they should actively (re)construct it in the class. In order to realize such a radical change in objectives, it is also necessary to change our evaluation procedures. Joubert (1980) and De Vries (1980) have for example developed several examples of how one could evaluate pupils' abilities to conjecture, define, axiomatize, classify, read critically, refute, etc. (For example see Joubert, 1988 & 1989).
   Lastly, it is important to point out that none of the above would be realizable unless radical changes are made to teacher education programs around the country; both in pre-service and in-service. In particular, most high school teachers, even those with good qualifications, know hardly any more geometry than the pupils they have to teach. The reason is simple: most tertiary institutions (with the exception of UPE) do not teach any further geometry in their undergraduate courses. It is therefore important to seriously consider the (re)introduction of geometry in tertiary courses for secondary teachers, not only Euclidean, but different kinds of geometry (compare with Baart, 1992). However, the geometry education of primary school teachers also needs urgent attention. Burger (1992) for example has proposed an interesting geometry curriculum for primary mathematics teachers based on the Van Hiele model that could provide the basis for the development of a new college geometry curriculum.

References