Relational understanding of the derivative concept through mathematical modeling: A case study

Abstract

The purpose of this study was to investigate three second-year graduate students' awareness and understanding of the relationships among the "big ideas" that underlie the concept of derivative through modeling tasks and Skemp's distinction between relational and instrumental understanding. The modeling tasks consisting of warm-up, model-eliciting, and model-exploration activities were used to stimulate participants to reflect and construct ideas about the concept of change. The data indicated that the participants' understanding of derivative was rather instrumental. Their explanations couldn't reveal the role of the big ideas regarding the concept of derivative with respect to what they mean, why and how they are related to the derivative and to each other. The results highlighted the fact that even if one of big ideas is ignored, the concept of derivative may not be fully understood relationally due to the compartmentalization of these big ideas in students' conceptual systems. If this happens to be the case, even though students can solve differentiation tasks/problems correctly, which implies procedural understanding, they may not be actually making sense of what the concept of derivative conceptually means.