Topology in the high school

There has long been a debate as to how geometry should be taught in the High School, and many papers contributed at this conference will doubtless be concerned with this important question. My own attitude is based on certain principles which should, I believe, inform one’s approach to the choice of content in a High School mathematics curriculum and to the choice of methods of presentation. A topic is only deserving of inclusion if it enriches the student’s experience by illuminating past or current interests and concerns, or if it is capable of future — but not too distant future — application. The experience referred to may be mathematical or non-mathematical; likewise, the application may be outside mathematics, or it may be the application of a general mathematical procedure to a more specialized mathematical situation. On this criterion, geometry deserves its place as an application of algebra and as a model of the world of experience. It is not, however, obvious that, in an inevitably crowded curriculum, one can justify the inclusion of geometrical material (for example, problems of coordinatization or Hilbertian axiomatics) which, while elegant and deep parts of mathematics, are somewhat sui generis and do not make a strong impact on the rest of mathematics or on the world of scientific knowledge served by mathematics.