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Learning and understanding in abstract algebra

LEARNING AND UNDERSTANDING IN ABSTRACT ALGEBRA by Bradford R. Findell University of New Hampshire, December, 2001 Students’ learning and understanding in an undergraduate abstract algebra class were described using Tall and Vinner’s notion of a concept image, which is the entire cognitive structure associated with a concept, including examples, nonexamples, definitions, representations, and results. Prominent features and components of students’ concept images were identified for concepts of elementary group theory, including group, subgroup, isomorphism, coset, and quotient group. Analysis of interviews and written work from five students provided insight into their concept images, revealing ways they understood the concepts. Because many issues were related to students’ uses of language and notation, the analysis was essentially semiotic, using the linguistic, notational, and representational distinctions that the students made to infer their conceptual understandings and the distinctions they were and were not making among concepts. Attempting to explain and synthesize the results of the analysis became a process of theory generation, from which two themes emerged: making distinctions and managing abstraction. The students often made nonstandard linguistic and notational distinctions. For example, some students used the term coset to describe not only individual cosets but also the set