This chapter holds that curriculum appropriate to children in the elementary grades cannot be determined through mathematical analysis alone, but must be developed in conjunction with systematic work exploring how children's mathematical thinking develops. The chapter suggests that as children learn about addition, subtraction, multiplication, and division– developing an understanding of the kinds of situations that can be modeled by the operations, sorting out various representations for them, and figuring out how to compute–they observe and comment on regularities in the number system. The chapter examines the three fundamental properties from both points of view. It shows that the regularities children are prepared to notice are not usually of the same order of abstraction as the mathematicians' axioms. It steps into the elementary classroom to see how ideas related to the commutative, associative, and distributive properties can be engaged.