This book is a guide to the development of mathematical talent in students in grades K through 8. The first chapter is on developing mathematical promise and considers characteristics of students who are mathematically promising, the goals of mathematics instruction, how to find and/or create good problems, models for increasing the numbers and levels of mathematically promising students, and assessment strategies. The remaining chapters offer suggested investigations into the following subject areas: number and operations, algebra, geometry and measurement, and data analysis and probability. Investigations offer activities at three levels of difficulty and are based on an open problem solving heuristic. Each investigation is developed to follow the following steps: (1) relate (connects to prior learning and National Council of Teachers of Mathematics principles and standards); (2) investigate (poses the initial problem to start students thinking); (3) evaluate and communicate (provides solutions, probing assessment questions, and suggestions for teacher responses); (4) create (offers ideas for extending and deepening the investigation); and (5) discussion (gives teachers hints on what to look for in student solutions and ideas for encouraging students). Individual chapters also include reproducible forms. A list of selected resources is provided. (Contains 12 references.) (DB)