Gifted Mathematicians Constructing Their Own Geometries — Changes in Knowledge and Attitudes

Abstract

A previous paper (1) looked at how gifted mathematicians in primary school respond to the study of Non-Euclidean geometries. In this study children in the secondary school are asked to construct their own Non-Euclidean Geometries. Using Taxi-Cab Geometry (2) as the starting point, children are asked to focus on the non-Euclidean elements of this geometry, and consider the differences between Euclidean and Non-Euclidean geometries. They are then asked to construct their own geometry, and consider the non-Euclidean elements within it. Young children find this task quite complex, but slightly older children find it highly challenging. Working with secondary age pupils has proved particularly rewarding, as they gain a totally new perspective of geometry, and of mathematics as a discipline. This type of work can only be done in very small groups or on a one-to-one basis. It requires a specialist, who would not be readily available within the school. It is, in effect, type 3 enrichment, using the Renzulli and Reis model (3). The new knowledge the children acquire has an immediate effect on their attitude towards geometry in particular and mathematics in general. The change in attitude depends on a number of factors, not least of them, the attitude of the school involved.