Throughout the present study, two approaches to cognitive development have been considered.
The Piagetian tradition emphasises structural development and how this determines the way in which children approach simple addition and other types of problems. Halford (1987), in particular, regards cognitive development as a function ofchildren's increasing structure-mapping abilities. Structure-mapping ability constrains children's conceptual understanding, which, in turn, dictates the strategies that they are able to develop. Siegler has positioned himself in direct opposition to the Piagetian tradition. His theory emphasises cognitive variability and the role it plays in development.
There is much evidence to suggest that children employ a host of different problem solving strategies, and not only the strategy associated with their current developmental level defined by their structural ability. In this respect, at least, Siegler appears to be correct. The selection of strategies from the available possibilities has been the empirical focus ofthis work. On the whole, the findings indicate that strategy choice is largely determined by the problem presented, as well as children's conceptual and procedural knowledge, but apparently not necessarily on the situation. The situation largely determines how the strategy will be executed.
The first research question, suggesting that a principle of least effort applies to the selection of strategies, was supported by the prediction analysis. The result indicates that children will attempt to match the problem presented with the most efficient strategy that they are able to execute successfully. Children are likely to retrieve the answer to small problems and easy problems such
as tie-problems. With greater experience, children will be able to make more and more accurate associations between addition problems and their answers, thereby increasing the range of their retrieval strategy. The actual decision to retrieve or to resort to a backup strategy appears to be determined by problem familiarity. First grade children will tend to count on from the larger addend by the amount indicated by the smaller addend when the answer is not retrieved. If the smaller addend comes first they will tend to reverse the order ofthe addends, and are most likely to do this when the difference between the two addends is great or when the first addend is small and the second large.
However, children may never reach the level of perfect strategy selections, since selection- optimality is compromised by variability. Variability offers a short-term cost for a long-term benefit, the benefit being greater learning potential. Variability means that children are less likely to stagnate and also that they are better equipped to adapt to new situations.
The second research question considered whether, under conditions of cognitive stress, children execute their strategies in an overt manner in order to extend their working memory capacity, or resort to faster strategies, specifically retrieval, that minimise working memory decay or use less of the valuable workspace. The results do not support the prediction that children will use retrieval with greater frequency and, therefore, the context does not appear to influence the actual strategy selected. This conclusion is supported by Siegler's (1990) finding that emphasising speed over accuracy, or vice versa, does not influence the strategy selected but rather influences how the strategy is executed. The results do suggest that, under cognitively loaded conditions, children will execute their strategies in an overt manner. Itis proposed that the overt execution ofbackup strategies aids the child's memory limitations. This finding supports the limited-processing-space hypothesis promoted by some of the neo-Piagetians. When children's cognitive processing
resources are exceeded, they resort to external aids much in the same way as adults will resort to pen and paper or to a calculator. Given that the evidence supports the notion of structural constraints, then it seems likely that these structural constraints dictate the sequence of strategy discoveries.
Itis further argued that the backup strategies are discovered in a sequence corresponding with increasing information-processing demands and decreasing completion times. This type of sequence, even if there is debate around the exact description of the sequence, fits the neo- Piagetian position on cognitive development. While the actual use of strategies is probably best described by Siegler's overlapping waves metaphor.
Therefore, the present study, as well as much of the other cognitive science literature, offers mixed support for the two competing positions ofcognitive development. Perhaps the reason that the debate between the two traditions is ongoing is that both positions are partly correct. This suggests that it may be worthwhile to seek a compromise between the overlapping waves and the staircase metaphors of cognitive development. Itis possible that the variability that Siegler has emphasised occurs around an orderly underlying structural progression compatible with the neo- Piagetian theories.
There are at least two ways of viewing cognitive variability. The first is discussed by Collins (1992) in his critique of artificial intelligence, where he claims that humans are simply unable to reproduce action in a perfectly consistent way. Something about our architecture prevents us from being able to do this. Perhaps children attempt to retrieve their best strategy from memory for any given problem, but since they are unable to follow the same retrieval path consistently, they sometimes retrieve close relatives instead of the strategy desired. Therefore, variability is the
result of our cognitive architecture and occurs despite people's intentions.
The second explanation is offered by the evolutionary psychologists. Here variability in thinking is said to serve the same evolutionary purpose as genetic variation. Since the variability provides some adaptive advantage for individuals, the source of cognitive variability has been selected for over the course of our evolution. This may explain how our less than perfect architecture, described by Collins (1992), occurred in the first place.
Finally, if we accept that the children's conceptual levels determine the strategies that they discover, then the range ofthe strategy arsenal reflects the depth oftheir number concept. Devlin (2000b) points out that higher mathematics is increasingly abstract but not necessarily that much more complex. Therefore, children must master the complexities of arithmetic before they undertake higher mathematics. In other words, the basic arithmetic operations, ofwhich addition may be the foundation, are likely to serve as the infrastructure for all higher mathematical concepts. Since cognitive variability appears to facilitate arithmetic development, children should be encouraged think in novel ways.