Rips et al. bring a fresh perspective to the study of numerical development, and their points are well taken. In particular, they make a thorough and compelling case against the possibility of a direct transition from infant quantification to natural number concepts, proposing instead that these concepts are induced from verbal counting. This is a worthy proposal, indeed, and one that merits empirical study. However, in and of itself, it is unlikely to provide a complete account of the way number concepts are constructed. In short, achieving a genuinely developmental account is going to require greater sensitivity to developmental issues.
One of these issues is timing, and, more specifically, how inputs, contexts, and responses interact in developmental time. It can be tempting to oversimplify these relations to make the researcher's task more tractable. For example, we might ask, as many – including the present authors – have, whether language comes before or after concepts. The problem is that elements of both are in the mix, right from the beginning, and there is no clear guideline for determining when a child “has” either one (see Mix et al. Reference Mix, Sandhofer, Baroody and Kail2005). It is true that modern theorists have given short shrift to the contribution of verbal counting. However, shifting the explanatory weight from nonverbal processes to words is just as unbalanced. Taking such a view obscures the subtle layering of experiences and insights that occur as development unfolds over real time (see also Mix Reference Mix2002).
Second, developmentalists have found that cognitive change rarely reduces to a single cause. Instead, it emerges from within a complex, ever-shifting, multifactorial system (e.g., Smith & Thelen Reference Smith and Thelen2003). This means there is no single basis for number concepts. Advanced counting is no more the basis of number concepts than is experience matching object sets, object tracking, or estimates of continuous amount. The empirical fact is that children have access to all of this information and more. Moreover, by adulthood, all these inputs have been coordinated into an interrelated conceptual structure, suggesting that none of them winds up cognitively vestigial. Thus, although there may be a key role for advanced counting, it is not the only relevant input. The challenge is to explain how these various streams of information coalesce into stable behaviors. On a related note, it is not clear why researchers need to adopt one definition of number over another. Though it is true that recent research has emphasized number as category, developmentalists have traditionally defined number as the combination of class and seriation. Piaget, for example, studied both aspects extensively and focused on how children integrate the two rather than assigning precedence to one or the other.
The present argument is similarly overdrawn with respect to the competence-performance distinction. Its crux seems to be whether quantification in infancy constitutes “true number concepts,” and if it does not, what does. The only way to evaluate this question is to assume we have some way of defining true number concepts; that there is a clear division between the advent of veritable number concepts and their illegitimate precursors. However, development is not like turning on a light: Children do not lack concepts one day, and then experience them full-blown the next. Instead, they acquire new insights in fits and starts, enjoying moments of competence in one context and then losing it in others, until finally their performance is stable across a range of situations (see Mix Reference Mix2002; Sophian Reference Sophian1997; Thelen & Smith Reference Thelen and Smith1993). They may grasp various pieces of the concept at different times, such that partial understandings interact for years before children reliably exhibit what might be considered mature understanding (Mix Reference Mix2002; Mix et al. Reference Mix, Sandhofer, Baroody and Kail2005). How, then, can we say whether infants have true number concepts? And what does it profit us to do so? We can admit a role for what infants and young children seem to understand, along with a role for advanced counting, and lose nothing.
An analogy from motor development might be helpful. Children crawl before they can walk. Crawling is not walking, and it is unclear how a child could suddenly start walking with only a background in crawling. However, the experience of crawling is not developmentally irrelevant. It tells babies about navigation through space, the properties of various surfaces, the feeling of balance, and so forth – information they will need when they start to walk. Furthermore, there is a temporary bridge that helps many babies transition from crawling to walking, called cruising – moving while upright but using furniture for support. Again, cruising is not “true” walking, but it binds the previous experience with navigation (crawling) with the new skill to be learned (walking), via a self-generated scaffold. What if number development is the same? Maybe young children have a way to track objects and estimate continuous amounts. This is not the same as natural number, and, as Rips et al. contend, they may need advanced counting to get to there. But these precursors give children important information about one-one correspondence, ordinality, and equivalence – ideas they will need to make sense of conventional counting. What might bridge the two (i.e., nonverbal and verbal enumeration) is labeling the cardinality of small sets; abundant research tells us children can do so (see Mix et al. Reference Mix, Sandhofer, Baroody and Kail2005). And researchers long ago proposed that the juxtaposition of labeling with counting was the mechanism for coordinating class and seriation, or cardinality and counting, in the verbal realm (see Klahr & Wallace Reference Klahr and Wallace1976; Schaeffer et al. Reference Schaeffer, Eggleston and Scott1974). This critical insight, that a set of three is counted 1-2-3, seems necessary for making sense of advanced counting, but it can only be achieved via these precursor experiences.
In summary, Rips et al. raise important questions that require answers. However, these answers are likely to be more complex than the present analysis suggests.
