Relational Reasoning about Numbers and Operations - Foundation for Calculation Strategy Use in Multi-Digit Multiplication and Division

Theoretical analysis of whole number-based calculation strategies and digit-based algorithms for multi-digit multiplication and division reveals that strategy use includes two kinds of reasoning: reasoning about the relations between numbers and reasoning about the relations between operations. In contrast, algorithms aim to reduce the necessary reasoning processes. In a sample of 221 German fourth graders, both kinds of relational reasoning were operationalized, as well as the use of strategies and algorithms in multiplication and division. The multi-dimensionality of the constructs and their discriminant validity were confirmed by a confirmatory factor analysis. The theoretically proposed, unidirectional relations between the constructs were investigated using a structural equation model: Abilities in reasoning about relations between numbers had a significant positive impact on strategy use in multiplication and division. Abilities in reasoning about relations between operations influenced strategy use in multiplication only. The use of algorithms in multiplication and division was exclusively affected by abilities in reasoning about relations between numbers, and not by abilities about relations between operations. Moreover, a negative effect of the use of digit-based algorithms on the use of whole number-based strategies was identified. Finally, the results of the theoretical and empirical analysis were integrated into a synthesis of existing models about calculation strategy use and development.