PIAGETIANS
2.3 Criticisms of Piaget's Theory of Numerical Development
As mentioned, Dehaene (1997) argues that children are able to conserve at an age far younger than Piaget would have believed. Dehaene claims that when children are asked to compare two sets they fail to understand the question rather than fail to conserve. These children witness the experimenter move the objects before asking the same question and therefore reason that the experimenter wants a different answer. MeWer and Bever (1967) replaced the objects in this type of experiment with a type of sweet. Children were allowed to pick up one ofthe rows and eat the sweets. Under these conditions, young children were usually able to choose the row with the greater number ofelements regardless ofthe perceptual appearance. However, Dehaene concedes that Piaget' s criteria for the concept ofconservation are likely to be far more stringent. Being able to demonstrate conservation in one particular task does not necessarily mean that children have acquired the concept.
Siegler (1995) cites a number of other studies that expand Piaget's original findings on the classical number conservation task which support Dehaene's (1997) conclusion. These studies suggest that young children often perform better if the rows include fewer objectsl~if the rows are transformed by adding or subtracting objects~if the wording of the questions is facilitative;
if the transformation was 'accidental' rather than deliberate; and if the children are trained in various ways. This, however, as noted earlier, was only one of the many tasks to which Piaget's
IChapter Six presents some evidence that suggests that the addition operation is performed more efficiently when small numbers are involved.
subjects were required to respond. Also, even if children are able to respond to the tasks in advanced ways under very specific conditions, one could not necessarily conclude that children acquire the number concept earlier than at the age proposed by Piaget or that they acquire this concept in a notably different way. Halford (1989) points out that despite the proliferation of studies designed to refute Piaget's findings, his account of the course of development remains relatively intact.
Piaget describes how young children, at the first level ofnumerical development, treat continuous and discontinuous quantities in the same way (and fail to conserve either). Piaget, however, does not demonstrate the role that language plays in this distinction between these two types of quantities. In The Child's Conception of Number (1952), the role that language plays in the acquisition of a number concept is not acknowledged. For example, the English language distinguishes betweenmassandcountnouns. Mass nouns are superordinate terms such as butter, furniture and money (McShane, 1991). The discontinuous quantities in Piaget's (1952) experiment included beads (count nouns) while the continuous ones included water (mass noun).
Markman (1985, 1989 as cited in McShane, 1991) argues the mass nouns assist children to develop their understanding of class inclusion relations. While on this point, there is other evidence, cited by Devlin (2000b), that indicates that Chinese and Japanese children outperform their English speaking counterparts in school mathematics. This disparity appears to be largely the result of language differences. A related finding, cited by Deheane (1997), indicates that children's digit spans differ according to their language, with Chinese children having an advantage over most of their western peers. Their advantage comes from speaking a language with short number words. Miller, Smith, Zhu,
&
Zhang (1995) report that Chinese children are better able to recite the counting sequence when compared to their American counterparts. The difference, once again, is to do with the short length ofChinese number words and the reality thatChinese number grammar directly parallels the structure ofthe arabic system. This provides these children with a numerical head-start. Thus, language, in a number of different ways, plays an important role in the development of the number concept.
Russell (1978) offers a critique of the role that conservation plays in Piaget's theory of number development. Inat least one conservation task involving the ability to judge the area bound by a loop of string, which is then elongated, adults believe that the changes in dimension are compensated while children are able to recognise the change in area (Russell, 1976 cited in Russel, 1978). Children appear to outperform adults on this particular task. Russell believes that children's success has to do with their greater familiarity with these sorts oftasks. But, as Russell (1978) suggests, if conservation indicates a structural equilibrium then it should really be insusceptible to the effects of familiarity.
Possibly the most damaging of the criticisms levelled against Piaget's theory has to do with his rejection ofany innate numerical ability. A fundamental premise ofPiaget's theory is that children enter the world a tabula rasa, whose contents are constructed with experience. The recent works by Dehaene (1997), Butterworth (1999), and Devlin2(2000b) all argue that children are born with an inherent number module or number sense. Their evidence comes from a collection of studies from a variety of domains. There is evidence to suggest that many animals are able to quantify small collections ofobjects, suggesting that this number sense is a feature ofmany different animal species. Also, Karen Wynn's (1992) famous experiment indicates that even infants as young as five months old will stare for longer at events that violate numerical concepts compared to those events that don't. This is a finding that strongly supports the 'innate' argument. Furthermore,
2Devlin (2000b) uses tlle termnumber gene,but does so in a metaphorical sense. He seems to prefer the term used by Dehaene (1997), hlenumber sense.
people who suffer specific brain injuries are rendered number blind, evidence supporting the modular approach to understanding the working of the mind.
This, according to the thr~e authors, means that we are born (if not born then very soon afterwards) with the ability to see the world in numbers just as we perceive it in colour or shapes.
The notion is similar to Noam Chomsky's concept of a language acquisition device, a hypothesised innate mechanism facilitating the learning ofgrammatical rules. Therefore,itappears that infants are not the blank slates that Piaget suggests, but rather that they enter the world with some core competencies. The concepts ofthe language and number modules are further supported by the proliferation of domain specific theories which have emerged as a consequence of the failure of the various domain general theories to adequately explain all aspects of cognitive development. If this view is correct, thenitposes a serious challenge to Piaget's entire theory since he postulates that knowledge is constructed with higher concepts being built on the foundation of lower ones. However, some of the higher concepts appear to be present without the foundation of lower ones. Piaget vigorously dismissed any claims ofa priori abilities.
Nevertheless, there is growing evidence that in this respect he may have been wrong, although, perhaps not entirely wrong, since the innate numerical abilities may be very limited when compared with the final abilities3.