Klein’s Plan B in the Early Teaching of Analysis: Two Theoretical Cases of Exploring Mathematical Links

Abstract

We present a theoretical approach to the problem of the transition from Calculus to Analysis within the undergraduate mathematics curriculum. First, we formulate this problem using the anthropological theory of the didactic, in particular the notion of praxeology, along with a possible solution related to Klein’s “Plan B”: here, re-linking the theory of Analysis with practical knowledge from Calculus. We explore two cases based on this approach: (1) the contribution of Vector Analysis to the foundations of trigonometric functions, and (2) establishing the ties between the proof of a basic theorem in Fourier Analysis and the computation of elementary infinite series. These two cases, including small-scale experiences, illustrate the necessity, importance and possibilities of new didactical approaches aiming to help student to integrate mathematical theories and practices which are otherwise taught separately.