Mapping Undergraduate Students' Misconceptions and Their Developmental Trajectory in Fundamental Concepts of Complex Analysis

Abstract

Purpose: Conceptual understanding is fundamental in Complex Analysis because students must coordinate algebraic, geometric, and polar representations to construct advanced mathematical concepts. However, existing studies have primarily focused on identifying students' errors rather than explaining how misconceptions develop across interconnected concepts. This study aimed to map undergraduate students' misconception profiles and identify the developmental trajectory of misconceptions in fundamental Complex Analysis concepts. Methodology: A descriptive qualitative approach was employed involving nine mathematics education students who had completed a Complex Analysis course. Data were collected through an open-ended diagnostic test and analyzed using open, axial, and selective coding. Findings: The findings revealed that the most frequent misconception occurred in the application of De Moivre's Theorem (66.7%), followed by the polar form (55.6%), argument and argument in polar form (44.4%), analyticity of complex functions (44.4%), and complex number representation and the Cauchy–Riemann equations (33.3%). Significance: The analysis suggested a hierarchical pattern in students' misconceptions, beginning with complex number representation and extending to argument, polar form, De Moivre's Theorem, and analytic functions. This pattern indicates a possible developmental trajectory of misconceptions across interconnected concepts. These findings suggest a possible developmental trajectory of misconceptions and provide implications for designing instruction that emphasizes conceptual connections and the coordination of multiple mathematical representations.