CONCEPTUAL CHANGE IN MATHEMATICS - FROM RATIONAL TO (UN)REAL NUMBERS

From an educational point of view, mathematics is supposed to have a c ompletely hierarchical structure in which all new concepts logically f ollow from prior ones. In this article we try to show that there are a lso concepts in mathematics which are difficult to learn because of pr oblematic continuity from prior knowledge to new concepts. We focus on the problems of conceptual change connected with the learning of calc ulus and the shift from rational to real numbers. We demonstrate the d ifficulty of this conceptual change with the help of historical and ps ychological evidence. In the empirical study 65 students of higher sec ondary school were tested after a 40 hour calculus course. In addition , 11 students participated in individual interview. According to the r esults the conceptual change from a discrete to a continuous idea of n umbers seems to be difficult for students. None of the subjects had de veloped an adequate understanding of real numbers although they had le arned to carry out algorithmic procedures belonging to calculus. We di scuss how appropriate recent theoretical ideas on conceptual change ar e for explaining learning problems in this domain. Also some education al implications are presented.