As children learn to count, their thinking changes in a way that shapes their concept of number. Counting is not simply reciting the number word sequence. There must be items to count; and there must be a procedure to make each utterance of a number word correspond with one of the items to be counted.⁵ At first, these items are perceptual; they might be, for example, beads, marbles, fingers, taps, steps, or drumbeats. The child must not only be able to perceive the items but also to conceive of them as individual things to be counted. Later, children become able to count sets of things (e.g., “how many different colors of buttons are there?”) as well as items that may not be readily perceivable.⁶ The counter must always create a mental representation of the items that are counted. This process of creation is clearly demonstrated when a child appears to count specific items in a situation where no such items are visible, audible, or tangible. Counting in the absence of perceivable objects is the culmination of a rather intricate developmental process. The process includes the progressive development of an ability to create unit items to be counted, first on the basis of conscious perception of external objects and then on the basis of internal representations.⁷

Early research on children’s understanding of the mathematical basis for counting focused on five principles their thinking must follow if their counting is to be mathematically useful:⁸

1. One-to-one: there must be a one-to-one relation between counting words and objects;

2. Stable order (of the counting words): these counting words must be recited in a consistent, reproducible order;

3. Cardinal: the last counting word spoken indicates how many objects are in the set as a whole (rather than being a property of a particular object in the set);

4. Abstraction: any kinds of objects can be collected together for purposes of a count; and

5. Order irrelevance (for the objects counted): objects can be counted in any sequence without altering the outcome.

The first three principles define rules for how one ought to go about counting; the last two define circumstances under which such counting procedures should apply.