Beyond procedural mastery: a semiotic analysis of students’ understanding of the first derivative definition

Abstract

In calculus learning, students often rely on procedural methods when studying the first derivative, which limits their ability to grasp its deeper conceptual meaning. This procedural focus represents a significant learning challenge that needs to be addressed. This study aims to analyze students’ semiotic processes in interpreting or expressing the definition of the first derivative using Peirce’s triadic framework, which emphasizes the interrelation among the representamen, the object, and the interpretant. A qualitative case study approach was employed with 21 mathematics education students, using written tests and in-depth interviews with three students who demonstrated the most complete semiotic competence. Thematic analysis identified three categories of understanding: comprehensive, partial, and pseudo. Results indicate that students with comprehensive understanding can coordinate symbolic, visual, and verbal representations, forming stable interpretants that conceive the derivative as the slope of a tangent line or an instantaneous rate of change. In contrast, students with partial or pseudo understanding exhibit fragmented semiotic processes and a dominance of procedural approaches. These findings provide important educational value by offering diagnostic insight into students’ conceptual difficulties, informing instructional design that integrates multiple representations, and supporting the development of more valid assessment strategies that capture students’ conceptual understanding beyond procedural performance.