Mental construction of completeness property of real numbersbased on APOS Theory

In this study, the research was conducted in the context of student conceptions of the completeness property of real numbers, which is the most basic structure of analysis, a field of modern mathematics. In this context, the historical development of the concept was examined based on the APOS theory, the learning difficulties in the literature related to the concept were revealed, students' conceptions of the completeness property were determined, and a genetic decomposition of the completeness property was created by utilizing the experiences of the researchers, and a study was conducted to verify and refine this genetic decomposition. For this purpose, a conceptual understanding test based on the hypothetical genetic decomposition was created and applied to students studying in the mathematics education program as a pilot intervention. Mathematical tasks were created in line with the refined genetic decomposition, a problem situation was posed based on these mathematical tasks, and an instructional application was carried out based on the ACE instructional cycle. As a result of the implementation of the instruction, students' mental constructions related to the property of completeness were examined and the results were revealed. Accordingly, it was concluded that the role of group discussions as external stimuli was important in the development of students' mental structures, the dense structure of rational numbers played a fundamental role in the schemas related to the completeness property of real numbers, and mental structures and mechanisms were important for the consolidation of mathematical justifications.