An investigation of the effectiveness of the use of computer animation on understanding transformations of the complex plane

There is a trend in mathematics instruction at the college level toward using computers for numerical processing and graphing. This is evident at all levels from remedial and developmental courses to graduate courses with a concentration in elementary courses. By the very nature of the field, computer graphics would seem to be an effective instructional tool in complex analysis. The purpose of this study is to investigate the idea that viewing an animation of a homotopy starting with a set of curves in the complex plane and moving to their images under functions of the complex plane will aid students in understanding those functions. A sample of students was drawn from a course in complex analysis, and separated into two groups, an animation group comprising three subjects and a non-animation group comprising two subjects. In individual sessions the members of the animation group viewed animated graphs as part of instructional exercises, and the members of the other group viewed static pictures as part of the same exercises. There were three sessions for each group, planned to correspond to three sets of complex functions: (1) Linear and quadratic functions; (2) Conformal mappings; and, (3) Roots and poles. In addition to the responses of students to the questions in the exercises, the researcher asked questions about each student's perceptions of their understanding, both in written form and in interviews. Because of the extremely small sample size and the fact that two subjects did not complete the study, the planned quantitative statistical tests were not completed. The basic statistics are included and are presented as qualitative statistics. A qualitative analysis of the raw data from the sessions with the subjects led to two major findings: (1) The effectiveness of the animations is greatest in functions which are not linear. The differences between the groups were very slight for the questions dealing with the linear functions, and were the greatest in the questions dealing with the power functions and functions with multiple poles. (2) All the subjects in the animation group found the animations useful in understanding functions presented to them in class and in assignments. No subjects in the non-animation group found the diagrams helpful in similar circumstances.