Are conceptual knowledge and procedural knowledge empirically separable? The case of fractions.

Abstract

Concerning students' difficulties with fractions, many explanatory approaches are based on the distinction between conceptual knowledge and procedural knowledge. For further research in this field, it is thus crucial to make these constructs accessible to valid measurement. In this study, we aim at developing a test instrument that affords valid measurement of students' conceptual and procedural fraction knowledge, including in particular empirical validation of this distinction. The data used in this study were from 8th- and 9th-grade students (N=235) in Germany. Facilitated by expert discussions, items from previous studies were developed further and assigned to either a conceptual scale or procedural scale. Confirmatory factor analysis was used to investigate the underlying structure of the data including model comparisons (1-dimensional; conceptual-procedural, verbal-non-verbal). Further analyses in terms of validation focused on reliability and on correlations of the knowledge types with general cognitive abilities. It was found that the theoretically assumed 2-dimensional model fitted the data best. Correlations of the two knowledge types with general cognitive abilities differed significantly. Furthermore, the latent constructs could be reliably estimated from its indicators. Our findings indicate that the empirical separation of conceptual and procedural fraction knowledge is possible: A theoretically grounded test instrument was developed that allows measuring the knowledge types with a sufficient degree of validity. These findings address a research gap that was pointed out repeatedly and gives rise to further research into reasons and remedies for students' difficulties in dealing with fractions.