In contrast, children succeeding at the give-N task are usually referred to as “Cardinal Principle Knowers” (hereafter,
CP-Knowers). Becoming a CP-Knower has been thought to mark a crucial induction Wortmannin price where children construct a new concept of exact number (Carey, 2009; Piantadosi et al., 2012; although see Davidson, Eng, & Barner, 2012). Thus, to address the debate on the origins of exact numbers, in the rest of this paper we focus on the number concepts of children who have not yet mastered counting: subset-knowers. Do subset-knowers understand that number words refer to precise quantities, defined in terms of exact equality? In the small number range, by definition, subset-knowers apply their known number words to exact MAPK inhibitor quantities, as do adults. To be classified as a “two-knower”, for example, a child must systematically give exactly one and two objects when asked for one and two objects
respectively, and he/she must not give one or two objects when asked for other numbers. In line with this competence, for quantities within the range of their known number words, children’s interpretation of number words accords with the relation of exact numerical equality (Condry & Spelke, 2008): children choose a different number word after a transformation that affects one-to-one correspondence (such as addition), but not after a transformation that does not affect the set (such as rearrangement). Nevertheless, these abilities are open to the same three interpretations as is children’s performance in Gelman’s “winner” task (Gelman, 1972a, Gelman, 2006 and Gelman and Gallistel, 1986): Known number words may designate exact cardinal values; they may designate approximate numerosities (and yield exact responding
because of the large ratio differences between sets of 1, 2, and 3); or the meaning of these words may be defined Chloroambucil through representations constructed in terms of parallel object tracking, a mechanism that is not available for larger numerosities. Studies of subset-knowers’ application of larger number words are needed to determine whether subset-knowers interpret exact numerals in terms of exact numbers. In contrast to their performance with words for small numbers, subset-knowers do not consistently apply words for larger numbers to precise quantities, even for words that they use when they engage in the counting routine. Results are mixed across studies (Brooks et al., 2012, Condry and Spelke, 2008 and Sarnecka and Gelman, 2004), and different interpretations have been proposed for these discrepant results: children’s responses may either reflect limits to their conceptual competence, or variations of their strategic performance (Brooks et al., 2012). We will return to this debate in the General Discussion; at this point, it suffices to note that subset-knowers do not consistently generalize number words according to exact number.