Conceptual and procedural knowledge in students’ fraction division thinking: Insights from a hermeneutic phenomenological study

Abstract

While both conceptual and procedural knowledge are central to mathematical competence, their interaction in students’ reasoning remains insufficiently understood, particularly in the context of fraction division—a domain widely recognized as conceptually demanding. This study addresses this gap by qualitatively examining how different configurations of conceptual and procedural knowledge shape students’ approaches to fraction-division tasks and the challenges they experience. The originality of this work lies in applying a hermeneutic phenomenological lens to investigate the interplay of conceptual and procedural knowledge beyond numerical performance measures, identifying three student profiles that reveal the fragility of rote-driven procedural mastery and the limitations of conceptual understanding in the absence of procedural fluency, and offering theoretical and pedagogical insights for integrating conceptual and procedural domains in mathematics instruction. Data were collected from 74 fifth-grade students across three Indonesian elementary schools through written assessments and follow-up interviews and analyzed using qualitative content analysis supported by NVivo. The analysis produced three distinct profiles: students with low levels of both conceptual and procedural knowledge; students with high procedural but low conceptual knowledge; and students with high conceptual but low procedural knowledge. Notably, none of the participants demonstrated strong proficiency in both domains. Students with relatively advanced procedural knowledge were able to carry out symbolic calculations but had difficulty solving contextualized problems. Conversely, students with stronger conceptual knowledge exhibited flexible reasoning in real-world contexts but lacked fluency with algorithms. These patterns provide evidence that conceptual and procedural knowledge may develop along independent trajectories, underscoring the importance of instructional designs that deliberately integrate the two. Finally, this study contributes to mathematics education research by highlighting the need for teaching approaches that intentionally bridge conceptual understanding and procedural fluency in fraction division and beyond.