The specific aim of this project is to determine what cognitive capacities infants and children possess when reasoning about arithmetic, with the broader long-term objective of pinpointing inventive ways to tap into intuitive mathematical skills. There is an emphasis on the "number sense", an evolutionarily-ancient ability to roughly estimate the number of objects in a scene without the benefit of symbols such as Arabic numerals. The project explores areas of continuity or discontinuity in the development of reasoning about the mathematical operations of addition, subtraction, multiplication, and division. Thus, these studies contribute to the NIH mission to pursue knowledge about the behavior of living systems, as well as the NICHD Mathematics and Science Cognition Learning program mission to encourage basic research on the normal development of mathematical proficiency. The proposed participation of undergraduate research assistants in every aspect of this project aims to encourage early interest in basic research, and to foster mentoring in advanced developmental psychology methods despite the college's status as a non-graduate degree granting institution. As such, this set of discrete research projects is suitable for the AREA program, which is geared towards institutions which provide undergraduate degrees for a significant number of future scientists, but have not been significantly supported by NIH. Three main sets of studies are proposed. In all experiments, the participants will be engaged via animated videos depicting various types of non-symbolic arithmetic operations, and their comprehension of the outcomes to these operations will be determined by speech, pointing, or mouse-clicking (children), or looking time (infants). One set of studies examines whether infants and children can determine a divisive relationship between two visual arrays of objects, and generalize this proportional constant to a new array of objects. A second set of studies probes whether children exhibit an unschooled understanding of the principle of inversion, in which a x b / b must necessarily equal a. Finally, a third set of studies explores the new phenomenon of operational momentum, in which outcomes to non-symbolic addition and subtraction problems are systematically over- or under-estimated, respectively. Operational momentum also captures a unique spatial-numerical interaction, with addition problems enhancing attention to the right side of space, and subtraction problems to the left side of space. Studies are proposed which look at the developmental time-course of this phenomenon and the cultural determinants of this phenomenon, such as increased left-to-right reading fluency. PUBLIC HEALTH RELEVANCE: The research proposed here is relevant to public health in two main ways. First, mapping the development of this unschooled 'intuitive math'leads to better methods for conveying difficult math topics, when children eventually enter into the educational system. Second, research on non-symbolic arithmetic provides an invaluable baseline from which one can detect and begin intervention on abnormal mathematical development and mathematical disabilities such as dyscalculia.