Defining and measuring conceptual knowledge in mathematics

A long tradition of research on mathematical thinking has focused on procedural knowledge, or knowledge of how to solve problems and enact procedures. In recent years, however, there has been a shift toward focusing, not only on solving problems, but also on conceptual knowledge. In the current work, we reviewed (1) how conceptual knowledge is defined in the mathematical thinking literature, and (2) how conceptual knowledge is defined, operationalized, and measured in three mathematical domains: equivalence, cardinality, and inversion. We uncovered three general issues. First, few investigators provide explicit definitions of conceptual knowledge. Second, the definitions that are provided are often vague or poorly operationalized. Finally, the tasks used to measure conceptual knowledge do not always align with theoretical claims about mathematical understanding. Together, these three issues make it challenging to understand the development of conceptual knowledge, its relationship to procedural knowledge, and how it can best be taught to students. In light of these issues, we propose a general framework that divides conceptual knowledge into two facets: knowledge of general principles and knowledge of the principles underlying procedures.