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system. The top section of Box 2-1 provides an overview of the number core from the perspective of children’s learning; this is discussed in more detail in Chapter 5. Here we discuss the number core from a mathematical perspective, as a foundation for the discussion of children’s learning.

###### Numbers Quantify: They Describe Cardinality

Numbers tell “how many” or “how much.” In other words, numbers communicate how many things there are or how much of something there is. One can use numbers to give specific, detailed information about collections of things and about quantities of stuff. Initially, some toy bears in a basket may just look like “some bears,” but if one knows there are seven bears in the basket, one has more detailed, precise information about the collection of bears.

Numbers themselves are an abstraction of the notion of quantity because any given number quantifies an endless variety of situations. We use the number 3 to describe the quantity of three ducks, three toy dinosaurs, three people, three beats of a drum, and so on. We can think of the number 3 as an abstract, common aspect that all these limitless examples of sets of three things share.

How can one grasp this common aspect that all sets of three things share? At the heart of this commonality is the notion of 1-to-1 correspondence. Any two collections of three things can be put into 1-to-1 correspondence with each other. This means that the members of the first collection can be paired with the members of the second collection in such a way that each member of the first collection is paired with exactly one member of the second collection, and each member of the second collection is paired with exactly one member of the first collection. For example, each duck in a set of three ducks can be paired with a single egg from a set of three eggs so that no two ducks are paired with the same egg, no two eggs are paired with the same duck, and no ducks or eggs remain unpaired.

###### The Number List

The counting numbers can be viewed as an infinitely long and ordered list of distinct numbers. The list of counting numbers starts with 1, and every number in the list has a unique successor. This creates a specific order to the counting numbers, namely 1, 2, 3, 4, 5, 6, …. It would not be correct to leave a number out of the list, nor would it be correct to switch the order in which the list occurs. Also, every number in the list of counting numbers appears only once, so it would be wrong to repeat any of the numbers in the list.

The number list is useful because it can be used as part of 1-to-1 ob-

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