Representational Difficulty Across Cognitive Stages in Quadratic Functions: An APOS–Newman Analysis

Abstract

This study aims to identify the types of students’ mathematical representation errors using Newman’s error classification and to analyze their relationships with the stages of cognitive development in the Action, Process, Object, and Schema theory. The study employed a qualitative descriptive approach involving 42 tenth-grade students from a senior high school. Data were collected through a mathematical representation ability test and in-depth interviews, and were analyzed qualitatively by mapping the types of errors and students’ cognitive development stages. The results indicate that students’ mathematical representation ability remains low, with symbolic representation the most challenging aspect across all ability levels. Comprehension and transformation errors were dominant among low-ability students, most of whom were at the Action stage. Moderate-ability students showed an emerging uneven pattern of cognitive attainment, with Schema-level attainment in the visual representation and, to a limited extent, in the verbal representation, whereas no Schema attainment was observed in the symbolic representation. Meanwhile, high-ability students demonstrated Schema-level attainment in the visual and verbal representations, whereas their symbolic performance remained at the Object stage because of persistent process-skill difficulties. These findings indicate that cognitive development is specific to the type of representation and does not occur uniformly across representations. This pattern of uneven cognitive attainment across representational forms is conceptualized in this study as asymmetrical representational development. The implications of this study suggest that the teaching of quadratic functions should emphasize coordination among visual, symbolic, and contextual representations through multi-representation approaches and gradual scaffolding to consistently support students’ conceptual understanding.