The Development of Cardinal Extension: From Counting to Exact Equality

Numerate adults know that when two sets are equal, they should be labeled by the same number word. We explored the development of this principle-sometimes called "cardinal extension"-and how it relates to children's other numerical abilities. Experiment 1 revealed that 2- to 5-year-old children who could accurately count large sets often inferred that two equal sets should be labeled with the same number word, unlike children who could not accurately count large sets. However, not all counters made this inference, suggesting that learning to construct and label large sets may be a necessary but not sufficient step in learning how numbers represent exact quantities. Experiment 2 found that children who extended labels to equal sets were not actually sensitive to exact equality and that they often assigned two sets the same label when they were approximately equal, but differed by just one item (violating one-to-one correspondence). These results suggest a gradual, stagelike, process in which children learn to accurately count, learn to extend labels to perceptually similar sets, and then eventually restrict cardinal extension to sets that are exactly equal.