ject counting to tell how many objects are in a collection. Although the number of objects in small collections (up to 3 or 4) can be recognized immediately—this is called *subitizing*—in general, one uses the number list to determine the number of objects in a set by counting. Counting allows one to quantify exactly collections that are larger than can be immediately recognized. To count means to list the counting numbers in order, usually starting at 1, but sometimes starting at another number, as in 5, 6, 7, …. (Other forms of counting include “skip counting,” in which one counts every second, or third, or fourth, etc., number, such as 2, 4, 6, …, and counting backward, as in 10, 9, 8, 7, ….)

Although adults take it for granted because it is so familiar, the connection between the list of counting numbers and the number of items in a set is deep and subtle. It is a key connection that children must make. There are also subtleties and deep ideas involved in saying and writing the number list, which adults also take for granted because their use is so common. Because of the depth and subtlety of ideas involved in the number list and its connection to cardinality, and because these ideas are central to all of mathematics, it is essential that children become fluent with the number list (see Box 2-2).

* Connecting the number list with cardinality.* In essence, counting is a way to make a 1-to-1 correspondence between each object (in which the

All of the work on the relations/operation core in kindergarten serves a double purpose. It helps children solve larger problems and become more fluent in their Level 1 solution methods. It also helps them reach fluency with the number word list in addition and subtraction situations, so that the number word list can become a representational tool for use in the Level 2 counting of solution methods. To get some sense of this process, try to add or subtract using the alphabet list instead of the number word sequence. For counting on, you must start counting with the first addend and then keep track of how many words are counted on. Many adults cannot start counting within the alphabet from D or from J because they are not fluent with this list. Nor do they know their fingers as letters (How many fingers make F?), so they cannot solve D + F by saying D and then raising a finger for each letter said after D until they have raised F fingers. It is these prerequisites for counting on that kindergarten children are learning as they count, add, and subtract many, many times. Of course as they do this, they will also begin to remember certain sums and differences as composed/decomposed triads (as |