Semiotics in Mathematics Education: Topological Foundations and Diagrammatic Methods

The question of mathematical pedagogy depends on the perceptual and intellectual capacities of teachers and students on the one hand and on the intrinsic demands for abstract understanding and rigorous formal proof on the other. The chapter sketches a semiotic sequence from metaphysics through category theory to topology to applied topology; and revisits the philosophies of Plato, Deleuze and others to elucidate the relevant mathematical problematics. While mathematics is intrinsically caught up in the dialectic of sense and idea, edusemiotics takes this distinctive feature of conceptual knowledge and learning into account. The use of diagrams as a semiotic tool is shown to be an essential component of any mathematics teaching and learning. An edusemiotic approach to processes of teaching and learning mathematics demonstrates that topological concepts of continuity and free variation support a diagrammatic framework for experimenting with and appropriating mathematical knowledge. This framework, consistent with the intuitive approach and formal notation of category theory, helps cultivate both ‘upward’ and ‘downward’ transits between abstract and concrete domains.