Learning fractions is difficult for children in general and especially difficult for children with mathematics difficulties (MD). Recent research on developmental and individual differences in fraction knowledge of children with MD and typically achieving (TA) children has demonstrated that U.S. children with MD start middle school behind their TA…

Learning of fractions is difficult for children in general and especially difficult for children with mathematics difficulties (MD). Recent research on developmental and individual differences in fraction knowledge of MD and typically achieving (TA) children has demonstrated that U.S. children with MD start middle school behind TA peers in…

Children's failure to reason often leads to their mathematical performance being shaped by spurious associations from problem input and overgeneralization of inapplicable procedures rather than by whether answers and procedures make sense. In particular, imbalanced distributions of problems, particularly in textbooks, lead children to create…

We tested the hypothesis that encoding the numerical-spatial relations in a number board game is a key process in promoting learning from playing such games. Experiment 1 used a microgenetic design to examine the effects on learning of the type of counting procedure that children use. As predicted, having kindergartners count-on from their current…

Recent research on fractions has broadened and deepened theories of numerical development. Learning about fractions requires children to recognize that many properties of whole numbers are not true of numbers in general and also to recognize that the one property that unites all real numbers is that they possess magnitudes that can be ordered on…

This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds…

Children's thinking is highly variable at every level of analysis, from neural and associative levels to the level of strategies, theories, and other aspects of high-level cognition. This variability exists within people as well as between them; individual children often rely on different strategies or representations on closely related problems…

Longitudinal associations of domain-general and numerical competencies with individual differences in children's understanding of fractions were investigated. Children (n = 163) were assessed at 6 years of age on domain-general (nonverbal reasoning, language, attentive behavior, executive control, visual-spatial memory) and numerical (number…

This special issue is dedicated to research that adopts the microgenetic method in order to investigate change as it is happening. In this commentary we reflect on the diversity of the articles included in this special issue, and examine how the findings from these articles relate to five critical features of change: path, rate, breadth,…

We applied overlapping waves theory and microgenetic methods to examine how children improve their estimation proficiency, and in particular how they shift from reliance on immature to mature representations of numerical magnitude. We also tested the theoretical prediction that feedback on problems on which the discrepancy between two…

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…

Identifying the types of mathematics content knowledge that are most predictive of students' long-term learning is essential for improving both theories of mathematical development and mathematics education. To identify these types of knowledge, we examined long-term predictors of high school students' knowledge of algebra and overall mathematics…

This paper argues in favor of using interactional strategies in the study of formal operations reasoning. Interactional designs allow a convergent approach to specifying processes underlying the interaction of variables. In contrast, current methodologies contain two inherent disadvantages: they have limited utility in specifying the processes…

This paper questions evidence for the thesis that causal reasoning of older children is more logical than that of younger ones, and describes two experiments which attempted to determine (1) whether there are true developmental differences in causal reasoning, and (2) what explanations for developmental differences can be supported. In the first…

Describes and discusses the rule-assessment approach, a new research strategy for studying developmental sequences in children's acquisition of knowledge. Four experiments were conducted to illustrate the utility of this approach across a variety of concepts and a wide range of ages (three-year-olds to college students). (Author/CM)