Collateral learning and mathematical education of teachers

Abstract

This article explores the notion of collateral learning in the context of classic ideas about the summation of powers of the first n counting numbers. Proceeding from the well-known legend about young Gauss, this article demonstrates the value of reflection under the guidance of ‘the more knowledgeable other’ as a pedagogical method of making one's mathematics learning experience educative. This includes learning about the efficacy of generalizing by induction and the perils of overgeneralization. Recourse to geometry and computing as support system in understanding concepts of algebra is emphasized towards the end of promoting an experiential approach to mathematics. This article stems from the author's work with prospective teachers of secondary mathematics in a capstone course. Their reflections on the course are shared and analysed.

Keywords:

• teacher education
• reflection
• collateral learning
• geometrization
• induction
• overgeneralization
• computing

Notes

Notes

1. This geometric interpretation along with many similar geometrizations of basic algebraic identities was known to ancient Greeks 16.

2. For an alternative approach based on the method of recursion and thereby, serving a different didactic purpose, see 17, p. 63].