Three Approaches to the Problem of Double Discontinuity

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''.