Year of Award

2025

Document Type

Dissertation

Degree Type

Doctor of Philosophy (PhD)

Degree Name

Mathematics

Department or School/College

Department of Mathematical Sciences

Committee Co-chair

Frederick Peck, Matthew Roscoe

Commitee Members

Bharath Sriraman, Nikolaus Vonessen, Georgia Cobbs

Keywords

abstract algebra, group, isomorphism, realistic mathematics education, reinvention

Abstract

This study explores the reinvention of the concepts of group and isomorphism in abstract algebra through design-based research. It investigates how students can reinvent these concepts and how mathematical artifacts transform their mathematical reasoning during the reinvention process. The study identifies a reinvention ecology comprising three interrelated components: students' mathematical activity, social interaction, and the continual emergence of mathematical artifacts. Using the guided reinvention principle of realistic mathematics education, students engaged with contextual situations, such as a 12-hour analog clock and the symmetries of an equilateral triangle, to progressively develop structural observations, formalize group axioms, and assemble them into the group concept. Later, students compared isomorphic groups with the aid of Power Wheels and Cayley tables to develop the notion of group isomorphism. The study highlights the transformative role of artifacts in bridging abstraction levels, reconceptualizing mathematical ideas, and providing meaning to mathematical artifacts. It culminates in a refined local instructional theory for the reinvention of groups and isomorphisms, offering practical guidance for educators and contributing to the theoretical understanding of the interplay between mathematical activity, artifacts, and social interaction.

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