The premise of the article by Rips et al. – that characterizations of initial and mature knowledge must be congruent enough that there is a viable developmental pathway linking the two – underscores the contribution a developmental perspective can make to cognitive science. The fact that there must be a developmental pathway between initial and mature knowledge, however, does not mean that initial knowledge must have the same basic structure as mature knowledge. The determination of what is the relevant precursor knowledge for a given body of mature knowledge is therefore itself an important theoretical issue.
Rips et al. begin their inquiry with a consideration of the evidence of numerical discriminations that has emerged from research with infants. They question, however, whether generalizations derived from experience with physical collections provide an adequate basis for mature natural number knowledge. Instead, they propose that infants have schematic numerical knowledge that incorporates key structural properties of a mature concept of natural number. But the chief merit of this account appears to be that it simplifies the problem of explaining how children get from initial knowledge to mature knowledge by positing a close correspondence between the two. No evidence is cited to support the attribution to infants of the schematic knowledge posited by Rips et al.
An analysis of relations between number and other mathematical concepts that may be present early in life suggests an alternative way of thinking about the developmental foundations of natural number knowledge that better fits available evidence and that does not require the close correspondence between initial and mature knowledge that Rips et al. seem to take for granted. Davydov (Reference Davydov and Steffe1975) argued that, mathematically, the concept of number presupposes concepts of equivalence, and of less-than and greater-than relations, which are more fundamental than number in that they do not depend upon a concept of number (because they apply to unenumerated continuous quantities, such as lengths, as well as to numerical quantities), whereas a concept of number does entail concepts of equivalence and less-than and greater-than relations. This conceptual asymmetry in turn suggests that, developmentally, knowledge about number might build on ideas about equivalence and inequivalence relations that have their origins in comparisons among non-numerical quantities. Sophian (Reference Sophian2007) reviewed a wide range of empirical evidence consistent with this idea; for example, evidence that infants are sensitive to less-than and greater-than relations among non-numerical quantities in reasoning about the support relation between an object and the surface on which it rests (e.g., Baillargeon et al. Reference Baillargeon, Kotovsky, Needham, Sperber, Premack and Premack1995), and evidence that 3-year-old children pay more attention to correspondence relations between sets than to specific numerical values in selecting pictures to match stories that contains both kinds of information (Sophian et al. Reference Sophian, Harley and Manos Martin1995, Experiment 1).
What distinguishes numerical cognition from thinking about non-numerical magnitudes is the use of a unit. When we state the numerical value of a quantity, what we are actually doing is describing the relation between that quantity and some unit. The unit is often an individual object, however, conceptually what is important is not its objecthood but the fact that we can measure the quantity we are trying to enumerate against it. Whether we characterize a collection of shoes as six individual shoes or as three pairs of shoes, the numerical value we give essentially represents the multiplicative relation between our chosen unit and the total quantity (cf. Sophian Reference Sophian2007).
Although the notion of a discrete object appears to be an important part of how infants make sense of the world from a very early age (Spelke et al. Reference Spelke, Breinlinger, Macomber and Jacobson1992), it seems unlikely that infants start out with a general mathematical concept of unit or an understanding of the relation between units and number. Unfortunately, we know next to nothing about how children do acquire the concept of unit, beyond the fact that important aspects of the concept of unit are quite challenging throughout the preschool period and beyond (Gal'perin & Georgiev Reference Gal'perin, Georgiev, Kilpatrick and Wirszup1969; Shipley & Shepperson Reference Shipley and Shepperson1990; Sophian Reference Sophian2002; Reference Sophian2007). Instructionally, it is possible to introduce units as an extension of ideas about less-than and greater-than relations (e.g., as a way of characterizing how much greater one quantity is than another; cf., Dougherty et al. Reference Dougherty, Okazaki, Zenigami and Venenciano2005). Although it is not clear how closely this corresponds to the way children's knowledge develops in the absence of such instruction, the viability of instruction in units that builds on ideas about less-than and greater-than relations between unenumerated quantities demonstrates that a developmental pathway from one to the other is at least possible.
An important argument that Rips et al. give for rejecting the idea that the concept of natural number grows out of experience with physical quantities is that physical quantities are always finite, whereas the sequence of natural numbers is infinite. This objection becomes less compelling, however, if we construe children's numerical knowledge as building on their understanding of actions that are potentially infinitely repeatable (such as iterating a unit), rather than on their observations of the quantities generated by those actions, which are of course always finite. Although physical action sequences, like physical collections, can never actually be infinite, the repeatability of actions does correspond to a key insight that children express about the infinity of the number sequence – that you can go on and on (cf. Monaghan Reference Monaghan2001).
In sum, while there must always be a developmental pathway from initial knowledge to mature knowledge, there is no guarantee that that pathway is a direct one. Consideration of possible starting points for natural number knowledge other than specifically numerical abilities led to the suggestion that numerical knowledge may develop through the bringing together of initial knowledge about equivalence relations and less-than and greater-than relations with somewhat later-emerging knowledge about units. This account is consistent with several kinds of empirical evidence, and suggests potentially fruitful directions for further research as well.
