This definitive fourth edition synthesizes the research and wisdom of expert teachers on how young children learn mathematics and how effective teachers support their development with learning trajectories. The learning trajectories approach builds on what children know and can do, using activities tailored to develop the next level of thinking. By focusing on the inherent delight and curiosity behind young children's mathematical reasoning, learning trajectories make teaching more joyful. Through helping teachers understand the varying levels of children's thinking and learning, this approach allows teachers to better meet the learning needs of all children. This thoroughly revised edition builds on the comprehensive research base of past volumes while drawing on numerous new research studies, expanded international examples, and updated illustrations. Chapters are also more closely linked with Learning and Teaching with Learning Trajectories [LT]²-an open-access, web-based tool that includes videos, interactive games, and practical classroom resources to help early childhood educators understand how children think and learn about mathematics. Ideal as a curriculum planning resource for mathematics administrators and leaders, as a professional development resource for early math teachers, and as a resource for math methods courses, this book brings the authors' nationally recognized learning trajectories approach directly into your classroom.