Reconsidering conceptual knowledge: Heterogeneity of its components.

Cognitive arithmetic classically distinguishes procedural and conceptual knowledge as two determinants of the acquisition of flexible expertise. Whereas procedural knowledge relates to algorithmic routines, conceptual knowledge is defined as the knowledge of core principles, referred to as fundamental structures of arithmetic. To date, there is no consensus regarding their number, list, or even their definition, partly because they are difficult to measure. Recent findings suggest that among the most complex of these principles, some might not be "fundamental structures" but rather may articulate several components of conceptual knowledge, each specific to the arithmetic operation involved. Here, we argue that most of the arithmetic principles similarly may rather articulate several core concepts specific to the operation involved. Data were collected during a national mathematics contest based on an arithmetic game involving a large sample of 9- to 11-year-old students (N = 11,243; 53.1% boys) over several weeks. The purpose of the game was to solve complex arithmetic problems using five numbers and the four operations. A principal component analysis (PCA) was performed. The results show that both conceptual and procedural knowledge were used by children. Moreover, the PCA sorted conceptual and procedural knowledge together, with dimensions being defined by the operation rather than by the concept. This implies that "fundamental structures" rather regroup different concepts that are learned separately. This opens the way to reconsider the very nature of conceptual knowledge and has direct pedagogical implications.