ABSTRACT. The research reported in this paper explores the nature of student knowledge about group theory, and how an individual may develop an understanding of certain topics in this domain. As part of a long-term research and development project in learning and teaching undergraduate mathematics, this report is one of a series of papers on the abstract algebra component of that project.The observations discussed here were collected during a six-week summer workshop where 24 high school teachers took a course in Abstract Algebra as part of their work. By comparing written samples, and student interviews with our own theoretical analysis, we attempt to describe ways in which these individuals seemed to be approaching the concepts of group, subgroup, coset, normality, and quotient group. The general pattern of learning that we infer here illustrates an action-process-object-schema framework for addressing these specific group theory issues. We make here only some quite general observations about learning these specific topics, the complex nature of "understanding", and the role of errors and misconceptions in light of an action-process-schema framework. Seen as research questions for further exploration, we expect these observations to inform our continuing investigations and those of other researchers.We end the paper with a brief discussion of some pedagogical suggestions arising out of our considerations. We defer, however, a full consideration of instructional strategies and their effects on learning these topics to some future time when more extensive research can provide a more solid foundation for the design of specific pedagogies.
Research in mathematics education usually attempts to look into students' learning and other mental processes. It could therefore be expected to build on knowledge acquired within the academic discipline of cognitive psychology. Our aim in this paper is to show how some recent developments in cognitive psychology can help interpret empirical results from mathematics education. In particular, we will be looking into the heuristics-andbiases research by Kahneman and Tversky, the alternative views by Gigerenzer et al., and the more recent dual-process theory that has come to play a central role in interpreting this research. We first introduce the relevant background from cognitive psychology and survey its connections to previous work in mathematics education; then we apply this theoretical framework for re-interpreting previously-published empirical data from mathematics education research. We conclude with a discussion of potential theoretical and practical benefits of such synthesis.
This article is an attempt to place mathematical thinking in the context of more general theories of human cognition. We describe and compare four perspectivesmathematics, mathematics education, cognitive psychology, and evolutionary psychologyeach offering a different view on mathematical thinking and learning and, in particular, on the source of mathematical errors and on ways of dealing with them in the classroom. The four perspectives represent four levels of explanation, and we see them not as competing but as complementing each other. In the classroom or in research data, all four perspectives may be observed. They may differentially account for the behavior of different students on the same task, the same student in different stages of development, or even the same student in different stages of working on a complex task. We first introduce each of the perspectives by reviewing its basic ideas and research base. We then show each perspective at work, by applying it to the analysis of typical mathematical misconceptions. Our illustrations are based on two tasks: one from statistics (taken from the psychological research literature) and one from abstract algebra (based on our own research).
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