This book developed as an outgrowth of a two-year study of 300 children discovered during preschool or kindergarten to be advanced in their mathematical thinking. Among other findings, the study revealed that the children tended to remain advanced in math over the two-year period, that their spatial reasoning related more closely to their math reasoning than did their verbal reasoning (although they were advanced in all three domains), and that the math scores of the boys started and remained somewhat higher than those of the girls. The biweekly Saturday Clubs to which half the group were assigned also proved effective in enhancing mathematical reasoning. This book presents the ideas developed in the Saturday Club program. The book describes characteristics of math-advanced young children; ways to "tune into" children's ideas and questions through informal play; and the power of "big ideas" like infinity and zero. A wide assortment of real-life contexts for mathematical explorations, such as gardening, cooking, going out to eat, and making collections are described. A final chapter presents a variety of alternatives by which schools and parents, working in partnership, can create optimal ways to support the development of highly capable children. (Contains 96 references.) (Author/DB)