Teaching and learning complex analysis at two British Columbia universities.

摘要

The objective of the present study was to investigate teaching and learning in typical university level complex analysis classes in British Columbia. Three classes were studied at two universities. Presentations of material in class by instructors was carefully recorded. A total of 20 subjects were studied during 54 audiotaped interviews of one hour duration. During interviews students worked on 6 to 10 questions, recording their work on worksheets. Transcripts were made of the interviews and these together with the worksheets were analyzed to study student understanding of the many concepts covered in a beginning complex analysis course.;In this thesis we have confined our analysis to the data we collected on just one theme: multirepresentations of complex numbers. We found that students have many misconceptions and difficulties with the basic representations of complex numbers such as, z = x + iy, z = (x, y), z = reitheta , and the symbolic representation, in which z is used directly. In addition, we studied how well students were able to judge when to shift from one representation to another. Finally, we have examined the data we collected that shows how insufficient understanding of basic material affects student ability to do problems from material covered later in the course.;Where possible we have attempted to identify different levels of understanding of the various representations. Although we found very little Mathematical Education literature on complex analysis, our analysis of our data supports results reported in the literature on representations of fractions and more general representations.;We found that students were competent with the z = x + iy and z = re itheta representations, and most made reasonable shifting decisions between these two forms, but there is little evidence that students understood the symbolic representation of complex numbers. In any case, from our data we have identified four characteristics of understanding a given representation that are consistent with results reported in the literature.;We have also studied how well students were able to understand the z = (x, y) representation simply as a mathematical form with certain rules. Four stages of understanding of this question were identified.