The mistakes children make in mathematics are usually not just NmistakesO - they are often intelligent generalizations from previous learning. Following several decades of academic study of such mistakes, the phrase Nerrors and misconceptionsO has recently entered the vocabulary of mathematics teacher education and has become prominent in the curriculum for initial teacher education.The popular view of childrenOs errors and misconceptions is that they should be corrected as soon as possible. The authors contest this, perceiving them as potential windows into childrenOs mathematics. Errors may diagnose significant ways of thinking and stages in learning that highlight important opportunities for new learning.This book uses extensive, original data from the authorsO own research on childrenOs performance, errors and misconceptions across the mathematics curriculum. It progressively develops concepts for teachers to use in organizing their understanding and knowledge of childrenOs mathematics, offers practical guidance for classroom teaching and concludes with theoretical accounts of learning and teaching.ChildrenOs Mathematics 4-15 is a groundbreaking book, which transforms research on diagnostic errors into knowledge for teaching, teacher education and research on teaching. It is essential reading for teachers, students on undergraduate teacher training courses and graduate and PGCE mathematics teacher trainees, as well as teacher educators and researchers.