Connecting theories in mathematics education: challenges and possibilities

This paper is a commentary on the problem of networking theories. My commentary draws on the papers contained in this ZDM issue and is divided into three parts. In the first part, following semiotician Yuri Lotman, I suggest that a network of theories can be conceived of as a semiosphere, i.e., a space of encounter of various languages and intellectual traditions. I argue that such a networking space revolves around two different and complementary “themes”—integration and differentiation. In the second part, I advocate conceptualizing theories in mathematics education as triplets formed by a system of theoretical principles, a methodology, and templates of research questions, and attempt to show that this tripartite view of theories provides us with a morphology of theories for investigating differences and potential connections. In the third part of the article, I discuss some examples of networking theories. The investigation of limits of connectivity leads me to talk about the boundary of a theory, which I suggest defining as the “limit” of what a theory can legitimately predicate about its objects of discourse; beyond such an edge, the theory conflicts with its own principles. I conclude with some implications of networking theories for the advancement of mathematics education.