Unveiling Students’ Cognitive Obstacles in Constructing Quadratic Function Representations: An APOS Theory-Based Analysis

Abstract

Quadratic functions are a fundamental topic in the high school curriculum; however, many students struggle to construct and connect various forms of mathematical representation. These difficulties are not merely procedural but manifest as cognitive obstacles, namely, when prior knowledge that is usually sufficient becomes irrelevant in a new context. This study employs the APOS theory (Action, Process, Object, Schema) to systematically map students’ cognitive obstacles. The research used a descriptive qualitative approach involving 35 tenth-grade students at a public high school in Pekanbaru. This study employed two research instruments: a mathematical representation test, encompassing visual, symbolic, and verbal indicators, and semi-structured interviews. Data were analysed using Miles and Huberman’s model through reduction, display, and conclusion drawing. The findings indicate that students’ cognitive obstacles emerged in almost all stages of APOS, with a predominance at the early stages, resulting in most students not reaching the Object or Schema stages. Visual representations were generally stronger than symbolic and verbal ones, but remained procedural rather than conceptual. Students also struggled to recognise their misconceptions, resulting in repeated errors. These findings underscore the need for instructional strategies that facilitate transitions across APOS stages, reinforced by reflective practices and the integration of multiple representations. The study recommends the development of quadratic function instruction that emphasises meaningful connections among symbolic, visual, and verbal representations, as well as further research on the effectiveness of APOS-based strategies in other mathematical topics.