<![CDATA[Proving is an important ability for pre-service mathematics teachers. However, previous studies have rarely examined proving in Real Analysis using the APOS framework, which provides insight into students’ cognitive development in constructing mathematical proofs. Although Real Analysis plays an important role in developing proving abilities, students’ proving abilities from an APOS perspective remain unclear. Therefore, this study aims to analyze pre-service mathematics teachers’ proving abilities in Real Analysis based on the APOS framework. This qualitative study involved 59 university students. Data were collected through a test consisting of four proof problems representing different levels of reasoning in Real Analysis. The data were analyzed using content and thematic analysis guided by APOS indicators: action, process, object, and schema. The results indicate that students constructed proofs without deep reflection and had difficulty identifying appropriate proof methods. Some students used examples to justify general statements, indicating limited understanding of mathematical proof. Most students remained at the action conception, only a few reached the process conception, and none demonstrated the object or schema conception. These findings indicate that students mainly relied on procedural manipulation rather than conceptual understanding. Therefore, instruction should strengthen prerequisite abilities and provide opportunities to construct various Real Analysis proofs to facilitate students’ transition from action to higher APOS levels.]]>
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