PIAGETIANS
2.4 The neo-Piagetians
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.
Leone and have attempted to incorporate both domain general and domain specific aspects of cognitive development. The theories of Bickhard (1978), Bruner (1964), Case (1985), Fischer (1980), Halford (1982), McLaughlin, (1963) and Pascual-Leone (1970) are efforts to re- conceptualise Piaget's progression in light of some of the criticisms levelled at the theory and to incorporate more recent data (all cited in Halford, 1989). Furthermore, many ofthe theories have attempted to combine the information-processing accounts of development with Piaget's structural approach (Case, 1987a).
Most ofthese theorists, according to Halford (1989), have left Piaget's developmental sequence relatively intact. All of the[:e theorists, and some others that do not fit into the category of the neo-Piagetian theories, propose that higher concepts are formed from the integration of lower ones, which is a fundamental Piagetian principle. However, Piaget's equilibration process is abandoned by all of the neo-Piagetian theorists in favour their own developmental mechanisms.
Miller (1956) argued; in a classical article, that there are memory imposed limits on our capacity to process information. He suggested that adults are able to process around seven units of information. Development involves 'chunking' increasing amounts of information into this fixed number of processing units. Therefore, it is not the amount of information that is limited, but rather the number of units into which the information is compressed. Some of the neo-Piagetian theorists appear to have incorporated a similar notion in their theories by emphasising children's developing processing capacities. They propose that development is linked to the child's short term storage space (STSS) and that the child's processing ability is constrained by their memory limitations.
Case (198 7b), for example, is one of the theorists who argues that children's mental processing
is constrained by the STSS. For Case (1987b), the total processing space comprises the operating space and the STSS. The operating space declines as the processing becomes more efficient and, since the total processing space remains constant, the STSS capacity increases. For Case (1987b) and others, the STSS is the workspace of higher cognitive processes and, therefore, the increase in storage results in the ability to process more complex information.
Halford (1987), like Case (1987b), argues that the information processing capacity increases with age, but unlike Case, he does not believe that the increases occur in a discontinuous fashion. For Halford (1987), development occurs as a result of increases in children's structure-mapping ability. Structure-mapping refers to the process where elements from an external structure are mapped to the internal representation of the structure, an essential aspect of reasoning. (He describes four levels of increasing structure-mapping ability.) Therefore, with age, children are able to map more complex relations between the elements ofthe structure. These more complex structure-mappings make greater information-processing demands on the child, but the trade-off is that children are increasmgly able to manage more complex concepts. Thus, the structure- mapping level constrains children's conceptual understanding, which, in turn, constrains the strategies that they are able to generate. Halford, Maybery, O'Hare and Grant (1994) state that development involves the process of representing increasing complexity in parallel. Strategy development, therefore, may involve a gradual shift from serial to parallel processing.
Importantly, Halford (1987) does not believe that the STSS is the workspace ofhigher cognitive processes, and, accordingly, does not believe that Case's (198 7b) trade-offoccurs. Processing and STSS, for him, are at least partly distinct. In other words, increasing operational efficiency does not necessarily facilitate storage. Halford, Maybery, O'Hare and Grant (1994) report numerous studies that indicate that information can be held in passive short term memory without interfering
with cognitive processes. They argue that the various memory studies support a multi-component view ofworking memory. Therefore, a more complete description ofthe various working memory components is required to better understand how memory and processing capacities are involved in development.
Two neo-Piagetian ideas make valuable contributions to our understanding of arithmetic development and are important for the purposes of this thesis. The first is the proposal that efficient problem solving methods free up working space (Case, 1987b). While the second is the suggestion that working space maturational development facilitate the use of more complex problem solving methods (Case, 1987b and Halford, 1987). Also important is the notion that this working space (whether this includes the STSS or not) is limited, which implies that the strategies children employ is, at least, partly dictated by their structural limits.
2.5 Summary
For Piaget, children's numerical development is a gradual process intertwined with their logical development. New mathematical structures are constructed from existing ones and it is believed that before the age of six or seven children are not ready for maths. Piagetian educators believe that children should be allowed to develop their concept of logic before they are taught mathematics (Dehaene, 1997). For this reason most pre-school activities involve playing with blocks of various colours and sizes. Also, Piaget describes a specific pathway in the acquisition of number concept where variability occurs only in the transition from the use of perceptually based reasoning to logical-mathematical reasoning. Piaget's sequence is invariant; children enter the world with no numerical ability and acquire a number concept in the first years of their lives.
However, it appears that others factors affect the developmental pathway. Language plays a