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 8th and 9th graders' (n=122) knowledge of fraction magnitudes and its cross-grade gains. Performance on the fraction magnitude measures predicted 9th grade algebra achievement. Understanding and fluently identifying the numerator-denominator relation in 7th grade emerged as the key predictor of later fraction magnitudes knowledge in both 8th and 9th grades. Competence at using fraction procedures, knowledge of whole number magnitudes, and the central executive contributed to 9th but not 8th 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.

Rethinking speed theories of cognitive development. Increasing the rate of recall without affecting accuracy.

Researchers have suggested that developmental improvements in immediate recall stem from increases in the speed of mental processes. However, that inference has depended on evidence from correlation, regression, and structural equation modeling. We provide counter-examples in two experiments in which the speed of spoken recall was manipulated. In one experiment, second-grade children and adults recalled lists of digits more quickly than usual when the lists were presented at a rapid rate of two items per second. In a second experiment, children received lists at a rate of one item per second; half the children were trained (successfully) to speak their responses more quickly than usual, at a rate similar to adults' usual rate. Recall accuracy was completely unaffected by either of these response-speed manipulations. Thus, although response rate is a strong marker of an individual's maturational level, it does not appear to determine the accuracy of immediate recall. These results have important methodological and theoretical implications for human development.