Date of Award

Winter 2001

Project Type

Dissertation

Program or Major

Mathematics Education

Degree Name

Doctor of Philosophy

First Advisor

Directors: Joan Ferrini-Mundy

Second Advisor

Karen Graham

Abstract

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 of all cosets. This kind of understanding was characterized as being immersed in the process of generating all of the cosets of a subgroup, a characterization that described and explained several instances of the phenomenon of failing to distinguish between a set and its elements.

The students managed their relationships with abstract ideas through metaphor, process and object conceptions, and proficiency with concepts, examples, and representations. For example, some students understood a particular group by relying upon its operation table, which they sometimes took to be the group itself rather than a representation. The operation table supported an object conception even when a student had a fragile understanding of the processes used in forming the group.

Making distinctions and managing abstraction are elaborated as fundamental characteristics of mathematical activity. Mathematics thereby becomes a dialectic between precision and abstraction, between logic and intuition, which has important implications for teaching, teacher education, and research.

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