Assessing Cognitive Obstacles in Learning Number Concepts: Insights from Preservice Mathematics Teachers

Abstract

Persistent difficulties in learning abstract algebraic concepts--particularly among preservice mathematics teachers--continue to hinder students' mathematical development. While prior studies have documented general misconceptions, few have grounded their analysis in comprehensive learning theories. Addressing this gap, the present study adopts the APOS (Action, Process, Object, Schema) theoretical framework to examine the cognitive obstacles encountered in understanding logarithmic, matrix, and quadratic function concepts. This qualitative study employed a descriptive case study design involving six preservice mathematics teachers with varying levels of mathematical ability (high, moderate, and low). Data were collected through written responses, semi-structured interviews, classroom observations, and cognitive mapping. The findings revealed that most participants were at the action stage, relying on procedural steps without deep conceptual understanding. Key cognitive obstacles included errors in applying logarithmic properties, difficulties integrating logarithms with matrices, and an inability to perceive systems of equations as unified entities. Group discussions proved effective in helping participants transition through the learning stages. Collaborative interactions enabled participants to identify errors, correct misconceptions, and strengthen conceptual understanding through reflection and validation. Furthermore, the use of visual tools, graphical representations, and real-world contexts supported deeper conceptual integration. This study underscores the importance of implementing APOS-based instructional strategies, including group discussions, exploratory exercises, and problem-based learning, to facilitate transitions between stages. The implications of these findings highlight the need for developing APOS-based diagnostic tools and innovative instructional designs to address cognitive obstacles effectively.