Editing Jean Piaget (section)

===Psychology of functions and correspondences===
Piaget had sometimes been criticized for characterizing preoperational children in terms of the cognitive capacities they lacked, rather than their cognitive accomplishments. A ''late turn'' in the development of Piaget's theory saw the emergence of work on the accomplishments of those children within the framework of his psychology of functions and correspondences.<ref name = "correspondences">Piaget, J. (1976). On correspondences and morphisms. ''Jean Piaget Society Newsletter, 5''. (unpaged)</ref><ref name = "Piaget, Grize">Piaget, J. Grize, J.-B., Szeminska, A., & Vinh Bang. (1977). ''Epistemology and psychology of functions. Studies in genetic epistemology. Vol. 23''. Dordrecht, Holland: D. Reldel, 1977.</ref><ref name = "Groupings, 1977, recent">Piaget, J. (1977). Some recent research and its link with a new theory  of groupings and conservation based on commutability. In R. W. Rieber and K. Salzinger (Eds.), ''Annals of the New York Academy of Sciences,  291'', 350-358.</ref> This new phase in Piaget's work was less stage-dependent and reflected greater continuity in human development than would be expected in a stage-bound theory.<ref name = "Schonfeld, 1986">Schonfeld, I. S. (1986). The Genevan and Cattell-Horn conceptions of intelligence compared: The early implementation of numerical solution aids. ''Developmental Psychology, 22'', 204-212. doi.org/10.10'37/0012-1649.22.2.204</ref> This advance in his work took place toward the end of his very productive life and is sometimes absent from developmental psychology textbooks.

An example of a function can involve sets X and Y and ordered pairs of elements (x,y), in which x is an element of X and y, Y. In a function, an element of X is mapped onto exactly one element of Y (the reverse need not be true). A function therefore involves a unique mapping in one direction, or, as Piaget and his colleagues have written, functions are "univocal to the right" (Piaget et al., 1977, p.&nbsp;14).<ref name = "Piaget, Grize"/> When each element of X maps onto exactly one element of Y ''and'' each element of Y maps onto exactly one element of X, Piaget and colleagues indicated that the uniqueness condition holds in either direction and called the relationship between the elements of X and Y "biunivocal" or "one-to-one".<ref name = "Piaget, Grize"/> They advanced the idea that the preoperational child manifests some understanding of one-way order functions.

According to Piaget's Genevan colleagues,<ref name = "Inhelder  1974">Inhelder, B., Sinclair, H., & Bovet, M. ''Learning and the development of cognition''. Cambridge: Harvard University Press, 1974</ref> the "semilogic" of these order functions sustains the preoperational child's ability to use of spatial extent to index and compare quantities. The child, for example, could use the length of an array to index the number of objects in the array. Thus, the child would judge the longer of two arrays as having the greater number of objects. Although imperfect, such comparisons are often fair ("semilogical") substitutes for exact quantification. Furthermore, these order functions underlie the child's rudimentary knowledge of environmental regularities.<ref name = "Inhelder  1974"/> Young children are capable of constructing—this reflects the constructivist bent of Piaget's work—sequences of objects of alternating color. They also have an understanding of the pairwise exchanges of cards having pictures of different flowers.

Piaget and colleagues have examined morphisms, which to them differ from the operative transformations observed on concrete operational children.<ref name = "correspondences"/> Piaget (1977) wrote that "correspondences and morphisms are essentially comparisons that do not transform objects to be compared but that extract common forms from them or analogies between them" (p.&nbsp;351).<ref name = "Groupings, 1977, recent"/> He advanced the idea that this type of knowledge emerges from "primitive applications" of action schemes to objects in the environment.<ref name = "Piaget, Grize"/> In one study of morphisms, Piaget and colleagues asked children to identify items in a series of movable red cutouts that could cover a pre-specified section of each of four base cards—each card had a red area and a white area.<ref name = "Piaget, Grize"/> The task, in effect, asked the child to superimpose the cut-outs on a base card to make the entire card appear to be red. Although there were 12 cutouts in all, only three, which differed slightly from each other, could make an entire base card look red. The youngest children studied—they were age 5—could match, using trial and error, one cut-out to one base card. Piaget et al. called this type of morphism bijection, a term-by-term correspondence. Older children were able to do more by figuring out how to make entire card appear to be red by using three cutouts. In other words, they could perform three to one matching. Piaget et al. (1977) called a many-to-one match surjection.