AND OTHERS
3.3 The Overlapping Waves Metaphor of Cognitive Development
According to Siegler (1996), the overlapping waves model of cognitive development is a better reflection of the empirical data than previous staircase models have been. Different waves represent the changing frequency of individual strategy use. The direction of development is
towards greater use ofthe more advanced strategies. New strategies are generated and added to the child's repertoire, becoming new waves in the model. This model represents the variation in thinking that previous models have failed to capture. The emphasis is placed on the variation that occurs throughout development, and not only between the stages. (The notion of cognitive variability is discussedinmore detail in Chapter Four.)
The different waves in the overlapping waves metaphor of development refer to the different strategies that people employ,inthe case ofthe present investigation, simple addition strategies.
These addition strategies, based on those described by Siegler and Jenkins (1989), include the following:
1. Retrieval involves retrieving the answer directly from memory. This implies that arithmetic facts are stored in some kind of memory table.
2. Thesum strategy involves counting each ofthe addends separately, then counting up to the first addend and continuing to count on by the number indicated by the second addend. This is one ofthe more basic strategies and often reflects the way children are initially taught. According to this definition, the completion time could be described for the addition problemx +y by the formula x +y + (x +y).
3. The shortcut-sum strategy involves counting from one up to the total of the two addends. Therefore, the completion time is described byx +y.
4. The count from first strategy involves counting on from the first addend by the number indicated by the second addend and the completion time is described byy.
5. Themin strategy involves counting on from the larger of the two addends by the number indicated by the smaller of the addends. In other words, if the child is presented with the problemx+y and y is greater than x, then the child starts at the
numbery and counts on by the numberx.Ifxis greater thany, then the child starts withxand counts on byy, in which case the strategy (at least for the present study) would be coded as the count from first strategy. The completion time would be indicated by the smaller of x andy.
6. Decomposition involves breaking the problem into more manageable parts.
Decomposition could be described a class ofstrategies since there are many different ways in which problems can be decomposed and recombined. The completion times would probably be best indicated by a number less than the smaller of x andy.
7. Guessingis different to retrieval in that the child explains that he or she guessed. The child makes no attempt to retrieve an answer from memory, but simply provides any number that comes to mind, which implies that the number is randomly generated.
There is evidence, however, that children spontaneously activate the sum of the two numbers, so it seems likely that guessing somehow involves consciously not attempting retrieve an answer.
8. Finger recognitioninvolves putting up fingers to represent each of the addends and the child recognises the total. This is different to the case where the child uses her fingers to aid the execution of the particular backup strategy chosen.
The strategies described above are the strategies commonly used by children who are exposed to the base-ten arabic number system. Children who have learned the roman number system, for example, may develop different addition strategies. The list, however, is not exhaustive. Dixon, Smilek, Cudahy and Merikle (2000) describe the phenomenon of
coloured number synaesthesia.
They have studied a child who perceives numbers as colours. Each numeral has a specific and fixed hue. Thus, simple addition, for some, may involve the mixing of different colours. Consider another example from the addition domain, although not single digit addition. Devlin (2000a)
describes a well known story concerning Karl Friedrich Gauss. When Gauss was a young child, one of his teachers instructed him to add all ofthe numbers from 1 to 100, no doubt to keep him occupied for some time. Gauss realised that the problem could be conceptualised differently. He decomposed the problem into a set of pairs as follows: (50+51)+(49+52) through to (1 + 100). He than took the number 101 (the sum ofeach ofthe pairs) and multiplied it by 50 to arrive at the correct answer in far less time than his teacher had hoped. This anecdote demonstrates that there are many different ways of decomposing addition problems. Therefore, there may be a number ofother addition strategies not listed above. However, the strategies described by Siegler and Jenkins (1989) are likely to include all of the common ones.
Siegler's (1996) overlapping waves metaphor emphasises the variability ofstrategy use, which is in direct opposition to Halford' s (1987) neo-Piagetian model (see Chapter Two), which suggests that there are structural constraints that determine the strategies that the child is able to use.
Siegler's (1996) and Halford's (1987) views, however, are not necessarily mutually exclusive.It is possible that the cognitive variability that is described by Siegler (1996) occurs around structural constraints that are probably best accounted for from the Piagetian tradition. This point will be discussed in Chapter Seven.