alization of counting involves taking as a unit each object they are counting so that each object can receive one count word. For example, when they are counting toy animals, each animal is a unit regardless of how big it is, what color it is, or what kind of animal it is. Later, 2- and 3-year-olds continue to generalize the range of objects they can count. Children with little experience with print may have more difficulty counting pictures of objects rather than objects themselves, and so they may especially need practice counting pictures of objects (Murphy and Wood, 1981).

The next crucial coordination of components is connecting counting and cardinality (Fuson, 1988; Gelman and Gallistel, 1978). When counting things (objects or pictures), the counting action matches each count word to one thing (see discussion above and in Chapter 2). But a cardinal number word refers to how many things there are in the whole set of things. So when anyone counts, they must at the end of the counting action make a mental shift from thinking of the last counted word as referring to the last counted thing to thinking of that word as referring to all of the things (the number of things in the whole set, i.e., the cardinality of the set). For example, when counting 7 toy animals 1, 2, 3, 4, 5, 6, 7, the 7 refers to the one last animal you count when you say 7. But then you must shift to thinking of all of the animals and think of the 7 as meaning all of them: There are 7 animals. This is a major conceptual milestone for young children.

When children discover this relationship, they tend to apply it to all counts no matter the size of the set of objects (Fuson, 1988). Therefore, this is a type of rule/principle of learning that children immediately generalize and apply fairly consistently. It is relatively easy to teach children that the last word said in counting tells how many there are (see Fuson, 1988). For example, a statement of this principle followed by three demonstrations followed by another statement of the principle was sufficient to move 20 of 22 children ages 2 years 8 months to 3 years 11 months who did not use the principle to using it (Fuson, 1988).

However, not all children really understand cardinality, even when they understand the importance of the last counted word (Fuson, 1988). Some children initially understand only that the last word answers the “How many?” question. They do not fully grasp the more abstract idea of cardinality. Thus, they give their last counted word when asked how many there are, but they do not point to all of the objects when asked the cardinality question “Show me the seven animals.” Instead, they point at the last animal again. It is important to note that responding with the last word is progress. Earlier when asked “How many are there?” children may have recounted or given a number other than the last counted word. Children who recount are understanding the question “How many are there?” as a request to count, not as a cardinal request. Such children may recount several if the question is repeated and may protest But I already did it or I already said it because they don’t understand the reason for the repeated



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