Actualizing the virtuality of the cauchy-riemann equations

Abstract

In this case study we explored how a mathematician’s teaching of the Cauchy-Riemann (CR) equations actualized the virtual aspects of the equations. Using videotaped classroom data, we found that in a three-day period, this mathematician used embodiment to animate and bind formal aspects of the CR equations (including conformality), metaphors, himself, and his students. We found that the mathematician’s creative introduction of matrices led to a discussion of transformations which made the CR equations mobile and hence gave him a space to virtualize the CR equations. In our results we summarize how the mathematician's assemblage of unusual, unexpected, and unscripted, and without given content creative acts with materials embedded in algebraic and geometric inscriptions and metaphors introduced or catalyzed the new—the virtuality of the CR equations. In our discussion, we highlight how the mathematician bridged the virtual with the abstract via his conceptualizations of the CR equations. Didactic implications include adopting the mathematician’s conceptualizations and asking students to bind them. This could stress the mobility of conformal maps which are generally not taught in an undergraduate class. We propose offering professional development for educators focused on learning how to engineer didactic practices that showcase mobility, support binding, and exhibit animation of mathematical concepts.