One of the most striking characteristics of human cognition is its variability. Both children and adults often possess multiple strategies, rules, concepts, and theories that they use to think about a given phenomenon or solve a given type of problem. For example, in such diverse domains as arithmetic, spelling, serial recall, and moral reasoning, children know and use multiple strategies. Recent trial-by-trial analyses have shown that the variability is present even in domains that have given rise to classic stage theories. Thus, when 5-year-olds are presented number conservation problems, they not only judge on the basis of the relative lengths of the rows, as stated in Piaget's theory and virtually all developmental psychology textbooks, but also sometimes rely on the type of transformation and other times rely on the results of counting (Siegler, 1995).
Abstract Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first grade predicted knowledge of fraction arithmetic in middle school, controlling for whole number magnitude knowledge in first grade and the other control variables. In contrast, neither type of early whole number knowledge uniquely predicted middle school reading achievement. We discuss the implications of these findings for theories of numerical development and for improving mathematics learning.
Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge-rather than understanding of mathematical concepts and procedures-to guide choices of solution strategies. They further proposed that this associative knowledge reflects distributional characteristics of the fraction arithmetic problems children encounter. To test these hypotheses, we examined textbooks and middle school children in the United States (Experiments 1 and 2) and China (Experiment 3). We asked the children to predict which arithmetic operation would accompany a specified pair of operands, to generate operands to accompany a specified arithmetic operation, and to match operands and operations. In both countries, children's responses indicated that they associated operand pairs having equal denominators with addition and subtraction, and operand pairs having a whole number and a fraction with multiplication and division. The children's associations paralleled the textbook input in both countries, which was consistent with the hypothesis that children learned the associations from the practice problems. Differences in the effects of such associative knowledge on U.S. and Chinese children's fraction arithmetic performance are discussed, as are implications of these differences for educational practice. (PsycINFO Database Record (c) 2018 APA, all rights reserved).
A new field of children's learning is emerging. This new field differs from the old in recognizing that children's learning includes active as well as passive mechanisms and qualitative as well as quantitative changes. Children's learning involves substantial variability of representations and strategies within individual children as well as across different children. The path of learning involves the introduction of new approaches as well as changes in the frequency of prior ones. The rate and the breadth of learning tend to occur at a human scale, intermediate between the extremes depicted by symbolic and connectionist models. Learning has many sources; one that is particularly promising for educational purposes is self-explanations. Overall, contemporary analyses show that learning and development have a great deal in common.
