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.

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The growing demand for higher education worldwide, along with global expansion of telecommunication technologies, give online distance education a potential world-wide reach for institutions in many countries. Given the persistent international digital divide and the potential for the host institutions and languages to be those of wealthy, industrialized countries, international online distance higher education (IODHE) has great potential for educational and cultural imperialism. Therefore institutions contemplating expansion into international distance education must accommodate a number of changes that would enable building the safe learning/teaching environment needed for the development of a successful course. Drawing on literature of domestic and international online teaching and learning, multicultural studies, as well as personal international teaching and learning experience, the authors examine four domains where these changes are of critical importance: 1) host institution, 2) technology, 3) learning models of students, and 4) teaching models of faculty. The paper discusses issues and practices in each of these domains and offers general recommendations for institutions participating or planning to participate in cross-border/cross-culture online education.

This study contributes to the ongoing discussion of visualization and analysis in mathematical thinking. On the basis of data gathered from clinical interviews with 32 students in their first abstract algebra course, we consider the tasks of listing the elements of the dihedral group D4 and finding the product of two such elements. These problems can be solved either using a “visual” approach of transforming a square or an “analytic” approach of multiplying permutations. Rather than clearly preferring either a visual or analytic strategy, most students in our study used some combination of these approaches. Our results suggest that the conventional analyzer/visualizer dichotomy may not be an appropriate classification scheme for describing learning processes or for designing instruction. We propose an alternative model, the Visualizer/Analyzer or VA model, that assumes visualization and analysis to be mutually dependent in mathematical problem solving, rather than unrelated opposites. Our model provides one description of how this mutual dependence might function. We end by considering how pedagogical approaches might be designed in consonance with this model to help students coordinate visual and analytic thinking.

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