Making Sense of Calculus: Unpacking the Conceptual Knowledge of Instantaneous Rate of Change among Malaysian Pre-Service Mathematics Teachers

Abstract

While derivatives are often introduced through symbolic manipulation and graphical interpretations, their function as tools for modeling real-world change is rarely emphasized in secondary mathematics. A significant body of research has identified a persistent disconnect between students’ procedural fluency and their conceptual knowledge of rate of change. This study investigates how Malaysian pre-service mathematics teachers conceptualize the derivative beyond its procedural definition, focusing specifically on how they interpret the concept of instantaneous rate of change across different forms: equations, tables, graphs, and symbols. Using a qualitative case study approach, three mathematics majors participated in task-based interviews that included interpretive questions. Data from interviews and written responses were analyzed through directed content analysis. Findings reveal that although participants could define derivatives as a "rate of change" or "slope," none could explain the underlying principles, such as the relationship between and or interpret how derivatives describe change in various representations. Their limited knowledge was closely tied to static procedures, lacking the flexibility needed for pedagogical application. This highlights an urgent need to shift teacher education from symbolic fluency towards conceptual adaptability, enabling future educators to teach derivatives as powerful tools for reasoning about change in diverse real-world settings.