On the Teaching Situation of Irrational Numbers : Epistemological Considerations as the Basis of Designing for Conceptual Change

Abstract

Generally we can point out two different ways in introducing new kinds of numbers as follows. The first is to represent a result of measurement The second is to solve algebraic equations. Although the relation between these two ways might have been overlooked in any teaching situations, this can be didactically explicit in the teaching situation of irrational numbers from the conceptual change perspective. The purpose of this paper is to derive some didactical implications for a conceptual change situation by focusing on a knowing of "incommensurability" that can be an essential aspect of irrationals. For attaining this purpose, the epistemological considerations take place in three contexts: curricular contents, history and teaching experiment. In conclusion, three points as didactical implications are shown: 1) problematic situation regarding the representation of number; 2) eliminating the tendency to cling to the "concrete"; 3) shifting attitudes toward the mathematical knowledge.