Developmental Foundations of Children's Fraction Magnitude Knowledge.

The conceptual insight that fractions represent magnitudes is a critical yet daunting step in children's mathematical development, and the knowledge of fraction magnitudes influences children's later mathematics learning including algebra. In this study, longitudinal data were analyzed to identify the mathematical knowledge and domain-general competencies that predicted 8(th) and 9(th) graders' (n=122) knowledge of fraction magnitudes and its cross-grade gains. Performance on the fraction magnitude measures predicted 9(th) grade algebra achievement. Understanding and fluently identifying the numerator-denominator relation in 7(th) grade emerged as the key predictor of later fraction magnitudes knowledge in both 8(th) and 9(th) grades. Competence at using fraction procedures, knowledge of whole number magnitudes, and the central executive contributed to 9(th) but not 8(th) graders' fraction magnitude knowledge, and knowledge of whole number magnitude contributed to cross-grade gains. The key results suggest fluent processing of numerator-denominator relations presages students' understanding of fractions as magnitudes and that the integration of whole number and fraction magnitudes occurs gradually.