Back to the Future: From the University Lecture Room to the School Classroom

Abstract

In his classic Elementarmathematik von hoheren Standpunkte aus Felix Klein pointed out a 'double discontinuity', namely, the transition from school mathematics to university mathematics a beginning undergraduate goes through and later the transition from university mathematics back to school mathematics if the graduate becomes a school teacher. In Klein`s own words: 'The young university student found himself, at the outset, confronted with problems which did not suggest, in any particular, the things with which he had been concerned at school. Naturally he forgot these quickly and thoroughly. ….. When, after finishing his course of study, he became a teacher, he suddenly found himself expected to teach the traditional elementary mathematics in the old pedantic way; and, since he was scarcely able, unaided, to discern any connection between this task and his university mathematics, he soon fell in with the time honoured way of teaching, and his university studies remained only a more or less pleasant memory which had no influence upon his teaching.' This leads one to ask the question, 'Will a tertiary graduate with academic distinction in advanced mathematics be a successful school teacher in mathematics?' The answer seems to be, 'Maybe yes, but not always.' Why not always? What kind of frame of mind and focal point of attention, and related to that what kind of subject knowledge, would a school teacher in mathematics do well to possess and to be nurtured in? We believe school teachers in mathematics should have research experience in mathematics, which, though similar in spirit as that of a researcher in mathematics, can be quite different in form and content, because a school teacher has to explain mathematics in a language and at a level of sophistication suitable to the mental development of school pupils. Mathematics learnt in the university provides the background and the general upbringing in the discipline, but it needs research experience of the kind mentioned above to enable a teacher to work on designing the teaching sequence in class and to better re spond to pupils` questions. We will illustrate with many examples gleaned from classroom teaching or historical sources. On the way the connection between school mathematics and university mathematics will be revealed, addressing the issue of Klein`s 'double discontinuity'.