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Abstract. In this chapter some special features of mathematical knowledge are considered in order to better understand the nature of conceptual change in this domain. In learning mathematics, every extension to the number concept demands, not only accepting new concepts, but new logic as well. This new logic more or less contradicts the prior fundamental logic of natural numbers. Therefore, misconceptions and learning difficulties are possible at every enlargement. To understand the problems students have in the conceptual change pertaining to the enlargement of the number concept a test was administered to 564 students (mean age 17.3) from randomly selected Finnish upper secondary schools. The test included identification, classification and construction problems in the domain of rational and real numbers. We found that changes of number conceptions, which was measured through questions in the domain of rational and real numbers, was not adequately carried out by the majority of the students who had just finished their first calculus class. While working on the tasks on the more advanced numbers they spontaneously used the logic and general presumptions of natural numbers or based their answers on their everyday intuition. The number concept of the majority of these students seemed to be based on the spontaneous logic of natural numbers but had also fragmented pieces of more advanced numbers. The students tended to overestimate the certainty of their answers when they used the logic of natural numbers even if it was erroneous.

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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 difJicult to learn because of problematic continuity from prior knowledge to new concepts. Wefocus on the problems of conceptual change connected with the learning of calculus and the shift from rational to real numbers. We demonstrate the difficulty oj 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. fVe discuss how appropriate recent theoretical ideas on conceptual change are for explaining learning problems in this domain. Also some educational implications are presented.

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