the following: “Children at different stages cannot learn the same content. They cannot learn about number, for example, until they reach the concrete operational stage [roughly ages 7 to 11, according to Piaget].”^{2}

Preschool children in fact know quite a bit about number before they enter school.

Research over the last 25 years, however, suggests that preschool children in fact know quite a bit about number before they enter school. Much of that knowledge is tied up with their understanding of counting. Even for preschoolers, the act of counting a set of objects is not entirely a rote activity but is guided by their mathematical understanding.

Babies show numerical competence almost from the day they are born,^{3} and some infants younger than six months have shown they can perform a rudimentary kind of addition and subtraction.^{4} These abilities suggest that number is a fundamental component of the world children know. Whether and how this early sensitivity to number affects later mathematical development remains to be shown, but children enter the world prepared to notice number as a feature of their environment.

Much of what preschool children know about number is bound up in their developing understanding and mastery of counting. Counting a set of objects is a complex task involving thinking, perception, and movement, with much of its complexity obscured by familiarity. Consider what you need to do to count a set of objects: The items to be counted must be identified and distinguished from items not to be counted, as well as from those that have already been counted. Items are counted by pairing each one with some sort of verbal representation (typically a number name). An indicating act is needed that pairs each object in space with a word said in time. Finally, you need to understand that counting results in a number that represents how many things are in the set that was counted.

Competent counting requires mastery of a symbolic system, facility with a complicated set of procedures that require pointing at objects and designating them with symbols, and understanding that some aspects of counting are merely conventional, while others lie at the heart of its mathematical usefulness. We discuss issues related to competent counting, including the learning of number names, in the section on procedural fluency below. In this section, we discuss children’s understanding of the conceptual aspects of counting. This separation is somewhat artificial because counting is a good example of the way in which the different strands of mathematical proficiency are interwoven.