<![CDATA[Purpose: This study investigated the conceptions of proof of prospective mathematics teachers’ learning limits, with a particular focus on how these conceptions were manifested in representational activity. Method: A qualitative teaching-research design was used to collect data from 82 first-year prospective mathematics teachers enrolled in a Calculus I course at a public university in Indonesia. The main data were prospective mathematics teachers’ written responses to an ε–δ limit proof task, classroom discussions, and reflective teaching notes. Data was analysed using iterative, thematic coding and constant comparative analysis, following a representational-epistemic framework. Findings: Four qualitatively different conceptions of proof were identified: proof as computation, proof as algebraic procedure, proof as formal display, and proof as logical argument. Most participants used computational or procedural approaches and a small proportion of participants constructed logically coherent deductive arguments. The results indicate that prospective mathematics teachers’ difficulties are not only related to algebraic manipulation but also to understanding the epistemic function of proof in university mathematics. Specifically, ε–δ notation was often taken as a formality of notation rather than as a relational structure for reasoning. Significance: The study adds to research on the secondary–tertiary transition in mathematics education by demonstrating the relationship between prospective mathematics teachers' representational practices and different conceptions of proof. The findings are also relevant to designing introductory calculus instruction to support prospective mathematics teachers’ transition from verifying procedures to deductive justification.]]>
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