Elsevier

Cognition

Volume 98, Issue 3, January 2006, Pages B57-B66
Cognition

Brief article
Preschool children master the logic of number word meanings

https://doi.org/10.1016/j.cognition.2004.09.013Get rights and content

Abstract

Although children take over a year to learn the meanings of the first three number words, they eventually master the logic of counting and the meanings of all the words in their count list. Here, we ask whether children's knowledge applies to number words beyond those they have mastered: Does a child who can only count to 20 infer that number words above ‘twenty’ refer to exact cardinal values? Three experiments provide evidence for this understanding in preschool children. Before beginning formal education or gaining counting skill, children possess a productive symbolic system for representing number.

Introduction

What do children understand about large-number words before they can count to large numbers or master the base-ten counting system? Do preschool children who cannot count beyond 40 understand that words such as ‘eighty-six’ refer to exact cardinal values, changing their application when items are added to or removed from a set?

Although most 3-year-old children can recite the ordered list of number words at least to 6, children take many months to learn the meanings of these words (Wynn, 1990, Wynn, 1992). Prior to this learning, children show inconsistent understanding of the logic of number words: They judge that each word picks out a specific, unique and exact cardinal value in some tasks (Sarnecka & Gelman, 2004) but not others (Condry et al., 2002, Sarnecka and Gelman, 2004). Young children either fail to understand that their counting words denote unique, specific numerosities, or their understanding is fragile and task-dependent.

By the end of the fourth year, most children have mastered the meanings of the smallest counting words (Wynn, 1990, Wynn, 1992), but their count list typically is limited to 20 or fewer items. Over the next year, children learn to count to higher numbers, and they map number words in their expanded count list to non-symbolic, approximate numerosities (Siegler & Opfer, 2003). What do such children understand about number words beyond those to which they can count?

In one series of experiments (Lipton & Spelke, in press), we have shown that preschool children map the number words within their counting range onto nonsymbolic numerosities, but that they show no such mapping for number words beyond that range. Children first were given an estimation task, in which they were shown arrays of 20–120 dots and gave verbal estimates of the number of dots in each array. Verbal estimates were linearly related to the presented numerosities for adults and for children who could count to 100. In contrast, preschool children who could not count beyond 60 showed this linear relation only for numbers within their counting range and produced number words at random for larger numerosities. Children next were given a comprehension task, in which they were shown two dot arrays, one twice as large as the other, and were asked to pick the array with a given verbally presented number of elements. Performance was above chance within children's counting range but at chance beyond that range. Finally, children were shown two dot arrays (one with twice as many elements as the other), were told how many dots were in one array, and were asked how many dots were in the other array. When tested with numbers within their counting range, children nearly always produced a number word in the correct direction: a smaller word when the to-be-estimated set was smaller than the named set, and a larger word when it was larger. In contrast, children produced larger and smaller words randomly when tested with numbers outside their counting range.

These findings provide evidence that children begin to map number words onto nonsymbolic representations of numerosity only when they become able to count to those words reliably. It does not follow, however, that unskilled counters know nothing about the meanings of words beyond their counting range. Here we ask whether these children understand the logic of such number words.

Adults understand that any counting number, such as ‘1,814,’ refers to an exact cardinal value: its application changes as items are added or removed. This knowledge is productive: It applies not only to number words whose meanings have been verified by experience (e.g. by counting a set, removing one number, and counting it again), but also to number words whose meanings have never been directly verified. Adults, however, have mastered the recursive, place-value system that would allow them to count reliably to any large number in principle, even if they have not done so in practice. Adults also can use addition and subtraction to solve large-number problems. Children, in contrast, learn to count small sets of objects well before they master recursive counting rules or symbolic arithmetic. As children become proficient at counting small sets of objects, do they possess the same, productive understanding of the logic of number words?

The present experiments test for this understanding against two alternatives. First, because infants' preverbal representations of large numerosities are imprecise (Lipton and Spelke, 2003, Xu and Spelke, 2000), children may infer that words such as ‘eighty-six’ refer to sets with approximately 86 members. We test, therefore, whether children who are told that a large set contains 86 items will continue to apply the term ‘eighty-six’ to the set if a single item is removed. Second, some linguists have proposed that number words place a lower bound on cardinal values: When a speaker claims that a set of six items contains ‘five items,’ her statement is strictly true, though in many contexts it is pragmatically inappropriate (Grice, 1957). Although this proposal has been disputed (Koenig, 1991), it is difficult to evaluate in adults, who possess a complex mix of semantic and pragmatic knowledge. Studies of children provide evidence against Grice's proposal (Huang et al., 2004, Musolino, 2004, Papafragou and Musolino, 2003), but their findings are mixed and may reflect effects of experience with number words, since children were tested with words for small numbers. We investigate whether children who cannot count to 86 have a lower-bound interpretation of the corresponding number word by asking whether they continue to apply ‘eighty-six’ to a labeled set after one object is added to the set.

Section snippets

Experiment 1

Experiment 1 investigates whether children understand that round number words beyond their counting range (e.g. ‘eighty’) change their application when one or more elements are removed from a set to which the words are applied.

Experiment 2

Experiment 2 extended the findings of Experiment 1 in two ways. First, whereas Experiment 1 used only round number words, Experiment 2 tested children's understanding of number words that are not multiples of 10. Second, Experiment 2 used an altered reasoning task in which a number word is applied to a set, one individual is removed, and a different individual is restored to the set. If children take large number words to refer to all members of a designated set, they should fail to apply a

Experiment 3

Experiments 1 and 2 provide evidence that children understand that a large number word ceases to apply to a set after a single item is removed. Do children also infer that a large number word ceases to apply to a set when a single item is added to the set? If number words define only a lower bound of numerosity, as Grice (1957) has suggested, then children might judge that a number word continues to apply to a set when its cardinal value increases. Experiment 3 tested this possibility. After

General discussion

Three experiments provide evidence that five-year-old children take large number words to apply to specific, unique cardinal values. Children understand that the number word that correctly labels a large set of items changes when items are removed from or added to the set, but remains the same when items are rearranged. Moreover, children understand that if an item is removed from a set and a different item is added to the set, the original number word labeling the set will apply. Importantly,

Acknowledgements

This research was supported by a Harvard University Fellowship to JSL and NSF grant REC-0087721 to ESS.

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