Adding diversity to mathematical connections to counter Klein’s second discontinuity

For instructors that try to make university mathematics courses relevant to future secondary school teachers, doing so generally involves making connections between university mathematics content and school mathematics content–in attempts to counter what Felix Klein referred to as a “double discontinuity.” In this paper, I consider the nature of the mathematical connections that bridge these two domains, and common distinctions made in extant literature between them, such as directionality. Through this analysis, I point out another aspect of these connections that has been left implicit: university mathematics is primarily–and reasonably–framed as a superset of school mathematics content. In this paper I consider alternatives, in particular conceptualizing connections that invert this typical relational connection–i.e., a subset relational connection–and I exemplify these connections with concepts from university courses such as real analysis and abstract algebra. Then, I consider the rationale for doing so in terms of secondary teacher education, and the ways that diversifying our framework of connections in this way can be used to help counter Klein’s second discontinuity.