Experts’ Construction of Mathematical Meaning for Derivatives and Integrals of Complex-Valued Functions

Abstract

We engaged five mathematicians who conduct research in the domain of complex analysis or use significant tools from complex analysis in their research in interviews about basic concepts of differentiation and integration of complex functions. We placed a variety of constructivist, social-constructivist, and embodied theories in mathematics education in conversation with one another to explore the development of the expert participants’ construction of mathematical meanings while moving between varying levels of abstraction from embodied concepts and real-world contexts to symbolic manipulation and formal theories. The mathematicians relied heavily on direct application of concepts and analogies from differentiation of real-valued functions and employed rotation and dilation as a local linear description of the action of a complex differentiable function with attendant repeated mental imagery and physical gestures. They also employed reasoning about real-valued line integrals to interpret contour integrals but acknowledged significant struggle to conceptually interpret what was analogously accumulated in the complex case. Instead, they all developed more personal meanings through a process of reconciling various aspects across their concrete to formal domains of reasoning. Much of the observed construction of meaning was manifested through contextualizing well-understood aspects of formal mathematical theory. We consequently explore implications for characterizing mathematical conceptual development as an interplay between concrete and formal reasoning rather than a development from one to another.