PROCEEDINGS OF THE 20TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2017
San Diego, California
Mathematicians’ interplay of the three worlds of the derivative and integral of complex-valued functions
Page: 1436
We engaged five research mathematicians in describing their images of differentiation and integration for functions of complex variables. Analyzing the data in terms of Tall’s three worlds, we explore the connections between the physical embodiments of the mathematicians’ reasoning, their descriptions for students, and their formalizations of these interpretations. For differentiation, the mathematicians relied heavily on direct application of concepts and analogies from differentiation of real-valued functions and employed rotation and stretching as a local linear description of the action of the function, corresponding to repeated mental imagery and physical gestures. For integrals, the mathematicians employed reasoning about line real- valued integrals, but acknowledged that they struggled to conceptually interpret what was being accumulated in the complex case. Instead, they all developed more personal meanings through a process of reconciling various aspects across their own conceptual-embodiment, operational- symbolic, and axiomatic-formal reasoning.