This book describes children's mathematical thinking to guide and support teachers in their reflection upon student thinking and teaching and learning. It is organized into five parts: (1) reasoning, student thinking, and invented strategies independent of the mathematics content domain which addresses the role of discourse in helping students clarify their thinking, communicate their thinking to other students, and communicate their thinking to the teacher, thus permitting access to their thinking; (2) children's thinking in the specific content domains of numbers, number sense, number operations, place value, fractions, geometry, geometric reasoning, spatial sense, algebra and algebraic reasoning, and statistics and statistical reasoning; (3) research on tools for representing mathematical ideas including concrete materials, electronic tools, and mathematical symbols as tools; (4) equity issues and research that can help teachers reduce inequity in their classrooms; and (5) additional resources and ideas for using research in the classroom. (KHR)