Why the verbal counting principles are constructed out of representations of small sets of individuals: a reply to Gallistel.

The seminal work by Gelman & Gallistel (1978) overturned the (then) Piagetian orthodoxy concerning the ontogenesis of the capacity to represent the natural numbers. Piaget (1952/1941) offered his famous studies on number conservation as evidence that children do not have a concept of number until age 6 or so. He explained this putative late emergence in terms of the absence of logical abilities required to support number representations, which he argued are not achieved until the stage of concrete operations. Gelman and Gallistel (1978) replied that any child who counted could thereby represent natural number, so long as the child followed what they called the “counting principles” (stable order, 1-1 correspondence and the cardinal principle that the last numeral reached in a count represents the cardinal value of the enumerated set). Indeed, these three counting principles guarantee that verbal numerals represent quantities that satisfy the successor function. That is, for any set whose cardinality n is represented by a given numeral, the next numeral in the list will represent cardinality n + 1. Since the late 1970’s, Gelman and Gallistel have systematically studied the acquisition of verbal counting in childhood as a window onto the ontogenetic sources of knowledge of the natural numbers (Gelman & Gallistel, 1978; Gallistel & Gelman, 1992; Gelman & Lucariello, 2002). They have argued that the way children acquire verbal counting shows that knowledge of the natural numbers is innate and is embodied in a system of preverbal counting. The more recent formulations of the hypothesis have taken preverbal knowledge of the counting principles to be implemented in the mechanism that generates analog representations of number (e.g. Gallistel & Gelman, 1992; Gelman & Lucariello, 2002) This is the mechanism Gallistel has in mind when he refers to the “preverbal counting system.” Like virtually all researchers in this field, we agree with Gallistel and Gelman that the verbal numeral list deployed in a count routine is the first explicit representation of positive integers mastered by children growing up in numerate societies. Indeed, our project derives from the work Gelman & Gallistel initiated almost thirty years ago: we have studied the acquisition of verbal numerals and of verbal counting as a means of understanding the ontogenesis of knowledge of the natural numbers. However, we disagree with Gallistel (and Gelman) on two major points. First, we believe that knowledge of the counting principles is not innate, but rather is constructed as a result of children’s attempt to make sense of the verbal count list. We believe the evidence shows that the count list is first mastered much as children learn to recite the alphabet, that is, without attributing any significance to the order. Second, although we fully agree that analog magnitudes are part of our innate cognitive resources and that they eventually provide an important part of the meaning of verbal numerals, we take our data and that of others (e.g., Condry & Spelke, in press) to convincingly show that knowledge of the verbal counting principles is not constructed out of analog magnitudes but out of representations provided by a system we call “enriched parallel individuation.” In rejecting a role of the analog magnitude system in the early development of knowledge of the meaning of numerals, we stand in stark opposition to Gallistel’s (and Gelman’s) theory of the acquisition of verbal counting. In what follows, we explain why we disagree with Gallistel, addressing his criticisms along the way. But first we clarify the logic of our project, for Gallistel’s comments suggest it may not have been clear.