Negative Numbers: Obstacles in their Evolution from Intuitive to Intellectual Constructs*

Abstract

Negative numbers: mathematically simple but didactically demanding Today the negative numbers present no mathematical difficulties. They can be elegantly constructed using the set of natural numbers. The method involved is that of embedding a semigroup in a group and is one of the fundamental techniques of mdern algebra. The practical computational mastery of the enlarged number system and its relevant use in applications have become obvous and routine on the university level and need hardly be discussed. The inclusion of the negative numbers in the domain of matters obvious to the professional can easily lead to underestimation of the difficulties associated with the teaching of this topic at the school level. Hence this topic can be used very as proof that the defensive thesis that He who has understood mathematics can teach it effectively oversimplifies the interplay between professional knowledge and teaching, and describes understanding itself in too onesided a manner. This thesis obscures the many possible facets of understanding as well as the fact that for teaching to generate enthusiasm and insights it must disclose the richness of the phenomenology of the mathematical subject matter.