Conceptual change in mathematics: From rational to (un)real numbers

Abstract

From an educational point of view, mathematics is supposed to have a completely hierarchical structure in which all new concepts logically follow from prior ones. In this article we try to show that there are also concepts in mathematics which are difficult to learn because of problematic continuity from prior knowledge to new concepts. We focus on the problems of conceptual change connected with the learning of calculus and the shift from rational to real numbers. We demonstrate the difficulty of this conceptual change with the help of historical and psychological evidence. In the empirical study 65 students of higher secondary school were tested after a 40 hour calculus course. In addition, 11 students participated in individual interview. According to the results the conceptual change from a discrete to a continuous idea of numbers seems to be difficult for students. None of the subjects had developed an adequate understanding of real numbers although they had learned to carry out algorithmic procedures belonging to calculus. We discuss how appropriate recent theoretical ideas on conceptual change are for explaining learning problems in this domain. Also some educational implications are presented.