This essay clarifies what it means to know mathematics by examining ways of knowing multiplication and explores what those ways of knowing imply for the teaching and learning of mathematics in schools. It reviews the perennial argument about whether computational skill or conceptual understanding should guide the school curriculum. A mathematical analysis of the process of multiplication, a conceptual analysis of mathematical cognition, and speculative research on classroom teaching and learning are presented to support this argument. Included are descriptions of several lessons in which children are being taught about multiplying large numbers. The descriptions focus on the connections that can be made in teaching among students' naive, concrete, computational, and conceptual knowledge. (Author)