Three Approaches to the Problem of Double Discontinuity

TL;DR

This paper examines three distinct approaches to addressing the 'double discontinuity' in mathematics education, analyzing their conceptual differences and implications for bridging the gap between high school and university mathematics.

Contribution

It introduces and compares three innovative methods from Klein, Sally, and Stanley to tackle the persistent educational gap in mathematics.

Findings

First two approaches follow a vertical trajectory in problem-solving.

Third approach proceeds horizontally, offering a different perspective.

Reframes the debate on the 'double discontinuity' in mathematics education.

Abstract

The gap between high school and university level mathematics has long been deemed problematic. Felix Klein referred to this gap as the ''double discontinuity'' meaning that students come to university unprepared for university courses and university level courses do not adequately prepare students to teach. In this paper we look at three possible solutions to the problem. The first, from Klein himself, involving logarithms, the second from Paul and Judith Sally involving lattice geometry, and the third from Dick Stanley involving an investigation of the box problem from high school calculus. We see that the first two examples proceed upon a vertical trajectory while the third example proceeds horizontally. This difference is key to reframing the century long debate about the ''double discontinuity''.