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.

Résumé

Nous présentons une approche théorique au problème de la transition entre Calculus et Analyse située au sein des programmes du licence. D'abord, nous précisons ce problème à l'aide de la théorie anthropologique du didactique, en particulier la notion de praxéologie, ainsi qu'une solution possible, liée au "Plan B" due à Klein: ici, relier la théorie de l'Analyse avec les savoirs pratiques du Calculus. Nous examinons deux cas fondés sur cette approche: (1) l'apport de l'Analyse Vectorielle aux fondations des fonctions trigonométriques, et (2) l'articulation entre la démonstration d'un théorème fondamental en Analyse de Fourier et le calcul de séries infinies élémentaires. Ces deux cas, avec des expériences à petite échelle, illustrent la nécessité, l'importance et la possibilité de nouvelles approches visant à aider l'étudiant à intégrer des théories et pratiques mathématiques qui sont autrement séparées dans l'enseignement.

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Correspondence to Margo Kondratieva.

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This paper draws on our contributions for the conferences INDRUM2016 (Montpellier, 2016) and CERME10 (Dublin, 2017), listed in the references.

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Kondratieva, M., Winsløw, C. Klein’s Plan B in the Early Teaching of Analysis: Two Theoretical Cases of Exploring Mathematical Links. Int. J. Res. Undergrad. Math. Ed. 4, 119–138 (2018). https://doi.org/10.1007/s40753-017-0065-2

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Keywords

  • Calculus
  • Analysis
  • Trigonometry
  • Transition
  • Praxeology