The relation between children’s conceptual understanding of counting and their mastery of conventional counting remains controversial. According to one viewpoint,9 children’s emerging understanding of these counting principles organizes and motivates their acquisition of conventional counting procedures. Other studies indicate that much of children’s conceptual understanding of counting follows (and may be based on) an initial mastery of conventional counting procedures.10 An intermediate view is that conceptual and procedural knowledge of counting develop interactively, with small changes in one contributing to small changes in the other.11
One reason it has been hard to resolve contrasting claims about how children come to understand the conceptual basis for counting is that preschoolers’ performance when they count is often quite variable, as it is with most other tasks.12 The many errors preschoolers make when counting could indicate that they fail to understand the importance of the counting principles. The variability of their performance makes fundamentally ambiguous the task of inferring their knowledge of principles from their behavior. A child’s difficulty in managing the complex processes involved in counting could mask a real understanding of its conceptual basis.
One way of circumventing the ambiguity of children’s counting behavior involves asking them to judge the adequacy someone else’s counting rather than perform the activity themselves. For example, asked to judge the accuracy of counting by a puppet who counted either correctly, incorrectly, or unconventionally (e.g., starting from an unusual starting point but counting all of a set of items), 3- to 5-year-olds demonstrated very good performance. Three-year-olds showed perfect acceptance of correct counting, 96% acceptance of unconventional but correct counting, and 67% rejection of real errors. Four-year-olds were better than 3-year-olds at rejecting true errors.13
Presented with a larger set of counting strategies to judge, children in a later study did not perform quite as well.14 In fact, 3-year-olds’ acceptance of unconventional correct counting was actually higher than that of 4-year-olds, suggesting that some of the acceptance of unconventional correct counting came from a blanket acceptance of the puppet’s performance. Finally, and most relevant to the relation between counting skill and judgment of another’s counting, the only children who failed to meet a criterion of 75% correct in rejecting the puppet’s counting errors also failed to meet the same criterion in their own counting. Thus, children’s own counting activity might form the basis for their judgments of what constitutes successful counting.