Date of Award

Spring 2007

Project Type

Dissertation

Program or Major

Mathematics Education

Degree Name

Doctor of Philosophy

First Advisor

Sonia Hristovitch

Abstract

This study discusses various theoretical perspectives on abstract concept formation. Students' reasoning about abstract objects is described based on theoretical proposition that abstraction is a shift from abstract to concrete. Existing literature suggested a theoretical framework for the study. The framework describes process of abstraction through its elements: assembling, theoretical generalization into abstract entity, and articulation. The elements of the theoretical framework are identified from students' interpretations of and manipulations with elementary abstract algebra concepts including the concepts of binary operation, identity and inverse element, group, subgroup, cyclic group. To accomplish this, students participating in the abstract algebra class were observed during one semester. Analysis of interviews conducted with seven students and written artifacts collected from seventeen participants revealed different aspects of students' reasoning about abstract objects. Discussion of the analysis allowed formulating characteristics of processes of abstraction and generalization.

The data showed that the students often find it difficult to reason about abstract algebra concepts. They prefer to deal with "concrete" objects and often are confused if the problem is stated in more general terms. Moreover, number of students based their arguments about a certain object on their understanding of a concrete structure. For example, some students said that if integer 1 does not belong to a given structure then this structure cannot be a group. Also, since abstract algebra concepts are complex structures, participating students repeatedly missed some elements of these structures during problem solving. One of the frequently missed elements was quantification of objects. Students often were confused how to use quantifiers.

The study elaborates on these problems and offers theoretical explanations of the difficulties. The explanations, therefore, provide implications for instructions and future research.

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