<![CDATA[Despite its foundational role, pre-service mathematics teachers frequently struggle with the high level of mathematical abstraction and the deductive arguments required to construct abstract algebra concepts, particularly Ring Theory. This study maps and analyzes the cognitive variations and mental structures of pre-service mathematics teachers facing these challenges within the APOS framework. Addressing single-case design limitations, this qualitative study employs a multi-case comparative design involving three high-ability pre-service teachers selected through strict criterion sampling. Data were gathered via a written diagnostic test and task-based clinical interviews for the primary subject, complemented by textual content analysis of written artifacts for the corroborative subjects. The results reveal that in the Action stage, subjects relied heavily on external axiomatic checklists. In the Process stage, they successfully internalized algebraic properties as mental operations. Transitioning to the Object stage, they encapsulated dynamic processes into static mathematical entities by treating rings as unified structures, ultimately synthesizing these into a coherent Schema. Crucially, cross-case analysis establishes that advanced algebraic schema development relies on bidirectional flexibility—the fluid capability to rapidly encapsulate processes into objects and de-encapsulate objects back into processes.]]>
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