Although human infants readily develop some capacities for representing number and geometry, children's thinking extends far beyond these capacities when they learn school mathematics. The basic concepts and skills of arithmetic and geometry that children learn in elementary school are critical for all subsequent learning and practicing of mathematics and science, but their acquisition, in relation to younger children's preexisting numerical and geometrical concepts, is not well understood. The proposed experiments are guided by the hypothesis that children learn school mathematics by using symbols- both language and visual-spatial symbols such as pictures and maps- to combine productively the early-developing numerical and spatial representations that emerge in infancy. To test this hypothesis, the proposed experiments probe the relationship between children's performance on tests of core numerical and geometrical abilities and their performance on tests of symbolic arithmetic, map reading, and abstract geometrical reasoning. Some experiments use individual difference methods to test whether variability in children's sensitivity to core numerical or geometric information is associated with variability in the mastery of language or symbols or of more abstract mathematical concepts. Further experiments use training methods to investigate whether tasks that exercise children's core abilities or use of symbols enhance children's numerical or geometrical reasoning in tests of school mathematics. The training experiments probe both the existence of relationships between core and constructed mathematical abilities and the underlying cognitive and motivational processes that may account for these relationships. Through these investigations, the experiments aim both to contribute to basic understanding of mathematical cognition and to inform efforts to enhance children's mathematical learning and reasoning.
The proposed research has two complementary goals. First, it aims to shed light on the fundamental cognitive capacities that allow humans to develop knowledge of elementary school mathematics, especially the abstract concepts and rules of arithmetic and Euclidean geometry. Second, the research aims to develop better ways to aid children's learning of mathematics, by building on basic research findings concerning the nature and early development of reasoning in the core cognitive domains on which elementary school mathematics may be founded. To accomplish these goals, tasks that previously were developed as tools to probe the fundamental numerical and geometrical abilities of infants, children, and adults in diverse cultures are deployed both to assess patterns of variability in children's mathematical abilities and to serve as training interventions with the potential to enhance school math abilities. The experiments investigate not only the direct relations between early and later developing numerical and geometrical concepts, but also the ways in which children's developing mastery of symbols, cognitive control, and attitudes toward learning may modulate those relations and impact on children's learning and performance of school mathematics. Through these laboratory experiments, the research aims to achieve a better understanding of mature mathematical reasoning and to discover new ways to aid children's mastery of this critical domain of knowledge.
By investigating the sources of variability in children's mathematical reasoning and the experiences that enhance children's reasoning, the proposed research promises to contribute to efforts both to foster all children's learning of mathematics and to aid children who experience difficulty with learning in this domain. Because children's learning of mathematics is affected by motivational patterns as well as cognitive abilities, the experimental interventions aim to enhance not only the cognitive processes underlying mathematical reasoning but also children's attitudes towards mathematics, belief in the malleability of mathematical skills with practice, and sense of their own competence as mathematical learners. Because children often learn best in social contexts, the interventions aim to create a set of training materials that children can use to play with others, and experiments will compare these interventions to those instantiated in individualized interactive computer-training programs. The findings of this research should inform efforts to enhance children's learning of mathematics both in and outside of school, for 4- to 10-year-old children at all ability levels.
