RUME 14 (Volume 2)
2011
Portland, Oregon
DYNAMIC VISUALIZATION OF COMPLEX VARIABLES: THE CASE OF RICARDO
Page: 488
Quantitative reasoning combined with gestures, visual representations, or mental images has been at the center of much research in the field of mathematics education. In this research we extend these studies to include the arithmetic of complex numbers and the analysis of complex valued functions. Our data consists of videotaped interviews with experts, including mathematicians, physicists, and graduate students. Microethnography and phenomenological methods were used to analyze and interpret the data. In this case study we synthesize how one mathematician, Ricardo, employs geometric representations, gestures, metaphor, diagrams, and models to describe his understanding of complex variables topics. Linear transformations were fundamental in his connections between analytical and geometrical representations. He routinely characterized complex linear functions as linear transformations on R2 with added symmetries. These findings may serve as a foundation for creating teaching experiments to help students develop geometrical representations of the mathematics behind complex variables.