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This book is an outgrowth of a 2-year study of 284 children discovered during preschool or kindergarten to be advanced in mathematics. In addition to psychometric and cognitive testing conducted at the beginning, middle, and end of the study, half of the children attended biweekly interventions designed to enrich their experience with mathematics. Results found the children remained advanced in math over the 2-year period, their spatial reasoning related more closely to their math reasoning than did their verbal reasoning, and the math scores of the boys started and remained higher than those of the girls. The intervention proved effective in enhancing mathematical reasoning. The book discusses ways of identifying very young math-advanced children as well as a variety of educational strategies to meet their needs. Its primary emphasis is on creating an open-ended approach to teaching mathematics that provides an opportunity for children at different levels of advancement and different personal styles to engage with mathematical challenges in a playful way, to conceptualize math broadly, to pose problems, and to make sense of the mathematical system. Appendices include a questionnaire for parents, jobcards and other activities, annotated bibliography, and recommendations for the classroom. (Contains 84 references.) (CR)