The reciprocal relationship between conceptual and procedural knowledge—a case study of two calculus problems

Abstract

The literature describes different stances concerning the focus on how mathematics should be taught, with some preferring a conceptual knowledge approach and others a procedural knowledge approach. The current study investigated the relationship between students’ conceptual and procedural knowledge in a calculus context. To better understand the relationship between students’ conceptual and procedural knowledge, we conducted a content analysis of the solution approaches of three subject specialists, as well as student responses to two mathematical problems. The students (n =192) were enrolled for a first-year mathematics module which forms part of an extended engineering degree in South Africa. The solutions to the two problems were analysed based on the number and nature (conceptual or procedural) of experts’ and students’ steps to solve each problem. Each step in the solution was categorised based on the approaches used to solve the problem. The study found that solutions are not unique and could follow more than one approach. More importantly, the study found that the relationship between conceptual and procedural knowledge is complex and integrated as solutions require both procedural and conceptual knowledge. The findings reveal that calculus problems cannot be uniquely described as mainly conceptual or procedural. Both procedural and conceptual thinking is required to solve calculus problems and is often iterative. Student techniques to solve the calculus problems included algebraic, graphical and unexpected approaches. The analyses of student solutions suggest that lecturers and teachers should compare and discuss multiple solution strategies with their students to enhance mathematical proficiency and understanding.