Sources of Mathematical Thinking

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To understand mathematical cognition both as it develops in the young child and as it is taught in school, one must understand the cognitive systems from which it is constructed and the processes by which those systems are coordinated to produce new concepts and skills. Based on previous research, we hypothesize that elementary school mathematics builds on three representational systems: a system for representing exact small numerosities, a system for representing approximate large numerosities, and natural language with its system of number words and other quantifiers. The proposed research investigates each of these building block systems and their interactions through experiments on human infants, non-human primates, preschool children learning counting, elementary school children learning arithmetic and fractions, and adults. To study the building block cognitive systems directly, experiments investigate spontaneous number representations in human infants and in untrained adult monkeys, using in each population the same three converging behavioral measures: looking time to arrays of different numerosities and to addition or subtraction events (building on the finding that both infants and monkeys look longer at novel arrays or unexpected events), manual search (building on the finding that the number of times that an infant or monkey will search in a container depends on the number of objects it represents within the container), and locomotor approach to containers with different numbers of attractive objects (building on the finding that infants and monkeys will approach the container with the greater number of objects). Further experiments investigate how preschool children assemble these components in learning number words and the counting routine, by using verbal and pointing tasks to assess developmental changes in children's understanding of number words and counting procedures. To uncover the neural substrates underlying mathematical cognition, both behavioral and neuroimaging experiments investigate whether and how human adults use each of the three representational systems in performing numerical comparisons and elementary arithmetic. Finally, experiments investigate number concepts and arithmetic learning in elementary school children. Training studies in which children are taught new facts or concepts and then are tested on a range of related problems will serve to investigate the subsystems involved in this learning, to probe the processes by which those subsystems are assembled to meet new educational challenges, and to explore ways of enhancing mathematics learning in elementary school. This research promises to shed light on the teaching and learning of mathematics through coordinated, laboratory-based studies in which monkeys, infants, children and adults are given the same stimuli and often the same tasks. This coordinated effort should provide a broad portrait of the sources of mathematical thinking, from its phylogenetic and ontogenetic origins to its culmination in educated adults.