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in an impressively vast study of how pre-school children learn to count Gelman and Gallistel demonstrated that early counting is much more than rote learning and that although children make errors as they count their efforts are constrained by a set of counting-relevant principles. One-to-one correspondance is important and can be used by infants for small numbers but when toddlers subsequently learn to count it constrains their efforts. Though errors may be numerous one-to-one correspondance is rarely violated. The second principle involves stable ordering. If a child counts "one, three, seven, ten" when counting a group of four objects as long as each tag is unique and the ordinal sequence is the same for each counted set then Gelman considers the child to be counting according to number-relevant constraints and it is these constraints that dictate the way in which the child eventually learns the conventional sequence of number words. Three other counting principles constrain the way children count: item difference; order difference; cardinality. the principles of order difference and item difference stipulate that any type of item can be counted and that the order in which different items in a set are counted is irrelevent ofits cardinal value: one can start counting a line of objects at either end or from the centre as long as each is counted once only. The cardinality principle states that only the final count term of of any particular trial represents the cardinal value of the set.
Siegler has shown that children do not automatically realize that a strategy thoroughly practiced in one situation is also relevant to another. Flexible strategy choice takes time to develop. When asked to distribute chocolate bars comprising different, equally segmented lengths so as to ensure equitability 4-year-olds tended to use the distribution procedures irrespective of the number of segments in each bar. Once the different sizes were colour-coded they soon realized that an equal nuber of actions will not necessarily ensure equal division. 4-year-olds require explicate external marking such as colour-coding.
It may be that cardinality grows out of the co-ordination of simpler principles.