A Continuous Path from School Calculus to University Analysis

Abstract

It is common to describe university-level mathematics as virtually a different subject from school-level mathematics, even when their subject matter overlaps. The difference is particularly keenly felt in analysis, where there is a big contrast between a typical first course in calculus and the more rigorous epsilon-delta approach that one encounters at university. I shall argue that this appearance is misleading, and that the epsilon-delta definitions and proofs are more intuitive than they might at first appear. I shall focus in particular on the treatment of the real number system, the definition of continuity, and the proof of the intermediate value theorem. Mathematics Subject Classification (2000). 97-XX If I was asked to name the two most notable ways in which university-level mathematics differs from school-level mathematics, then I would say that they were abstraction and rigour. Early courses at university in subjects such as group theory and linear algebra will introduce students to the axiomatic way of thinking, while a first course in mathematical analysis introduces them to rigorous proofs of statements that they will hitherto have justified only informally, if at all. It is often claimed that mathematical analysis is difficult to learn because in order to understand it one must learn to think in a new way. In this short presentation I would like to suggest that there are many connections between the advanced, rigorous way of thinking and the more naive way of thinking that would come naturally to a schoolchild. How these observations should influence the way we teach analysis is far from clear, but it cannot do any harm to draw attention to them. I plan to discuss three aspects of basic real analysis: the axiomatic approach to the real number system, the definition of continuity, and the proof of the intermediate value theorem. In each case, I shall compare how they are treated ∗Royal Society 2010 Anniversary Research Professor, University of Cambridge, UK. E-mail: W.T.Gowers@dpmms.cam.ac.uk