follow the links below for further information

cognitive psychology (see NOTE ONE)
beyond modularity
Annette Karmiloff-Smith: MIT Press: (ISBN: 9 780262611145)
a developmental perspective on cognitive science

Chapter 4 the child as mathematician

...Piaget's view of the number-free infant and his interpretation of the older child's aquisition of number have been seriously called into question. Yet, the original data are very robust.[the author's postulate is for one to] opt for an integration of aspects of both nativism and constructivism. [Piaget] argued that representation of linear order and seriation(the capacity to represent objects of varying length in correctly ordered sequences) are necessary for number conservation but not sufficient. Also required is a hierarchical system of inclusive relations,in which the class containing only one element is included in the class containing two elelements which in turn is included in the class containing three etc. Thus the child ultimately achieves 'conservation of number' - children of 4 readily accept that 10 objects equispaced in a row with identical rows underneath contain the same number of objects:
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Children under 5 regard the more widely spaced line as having more objects. it is not until the ages of 6 or 7 that children pass this test irrespective of the number of objects involved. Younger children find it easier to conserve number when the objects in two lines are refered to by a term denoting collection rather than by class. Other research suggests that although young children are capable of small number conservation they cannot conserve large numbers. It has been argued that they possess the logical operations for number conservation but can't apply them to numbers larger than 5. Successful conservation does not require knowledge of the cardinality of the set; provided one-to-one corresponence has been established at the outset, you can work with unspecified quantities. The focus must be on what happens during the transformation.