Prospective Mathematics Teachers' Concept Images of Group Structure: From Binary Operations to Axiomatic Coordination

Abstract

Understanding group structure requires the coordination of a set, a binary operation, universal quantifiers, and the group axioms. Yet how these meanings are activated across connected tasks remains incompletely understood. This interpretive qualitative study examined prospective mathematics teachers' concept images as part of the prospective-analysis phase of Didactical Design Research (DDR). Thirty-six students enrolled in an abstract algebra course completed an eight-task diagnostic instrument, and eight purposively selected students participated in task-based interviews. Written responses, interview transcripts, and reflective notes were analyzed as meaning-making episodes and compared across tasks and representations. Three task-sensitive configurations were distinguished: operation-oriented reasoning, in which familiar calculations or visibly active operands dominated judgments; per-axiom reasoning, in which requirements were checked separately without consistent universal justification; and structurally coordinated reasoning, in which the set, operation, quantifiers, and relevant axioms jointly governed classification, proof, and refutation. Definition-governed reasoning sometimes weakened when the representation changed, particularly in operation-table tasks. These configurations describe evoked patterns of reasoning, not permanent learner types or developmental stages. The analysis connects binary-operation recognition to axiom coordination and suggests preliminary didactical needs involving non-prototypical operations, quantifiers, and representational flexibility. No didactical intervention was evaluated in this study.