
A confluence of findings related to early childhood counting and conservation...Well and good. Piaget's conclusions that development of this awareness typically does not occur before the age of 6 or 7, as children enter the operational stage, have been confirmed numerous times. Even children's ability to match objects in one set to objects in another in order to determine the relative size of each set, a relatively advanced application of onetoone correspondence, is a fairly late accomplishment (Brainerd, 1979). However, programs that have made conscious efforts to delay young children's exposure to number concepts until they can fully grasp the principle of conservation and its sidekicks—integrated formal understanding of onetoone correspondence and rational counting—have overinterpreted Piaget's findings and intent. Research has shown, and Piaget's findings do not refute, that even though young children still will rely largely on visual perception, they can and do make use of reasoned counting during the preschool years (Piaget's preoperational stage). Not only have young children exhibited the ability to count in any order (Aubrey, 1993), and correct counting errors such as skipping numbers or doublecounting even when the numbers used are beyond their ceiling of known numerals or number words (Gelman & Meck, 1983), but they have done so before exhibiting an operational understanding of quantity. Though there is a difference between operational understanding and the ability to develop fundamental implicit understandings of such important concepts as onetoone correspondence (see Early Childhood Numeracy), Piaget would further concur that counting is an excellent exercise for helping children develop preformal number concepts that eventually lead to an enhanced ability to apply onetoone correspondence and rational counting in an integrated and cogent manner, as well as to conserve number. Evidence supports the notion that children who fail Piagetian conservation tasks still often operate quite successfully when making judgment of equivalence between sets with small numbers of objects through counting (Gelman & Gallistel, 1978), to the extent that some suggest that equality of sets based on numbers counted occurs earlier than when based on onetoone correspondence (Thompson, 1989). As a final precaution regarding overanalysis, noteworthy given the backdrop: Even the phrase "number concept" itself is misleading because of the existence of a variety of number concepts, and the usefulness of the idea of onetoone correspondence is no justification for its use as a criterion for judging a young child's grasp of number (Freudenthal, 1973). Aubrey, C. (1993). An investigation of the mathematical competencies which young children bring into school. British Educational Research Journal, 19(1), 2741. Brainerd, C. (1979). Concept learning and developmental stage. In H. Klausmeier and Associates (Eds.), Cognitive learning and development: Piagetian and information processing perspectives. Cambridge, MA: Ballinger. Freudenthal, H. (1973). Mathematics as an education task. Dordrecht, Netherlands: Reidel. Fuson, K., Richard, J., & Brials, D. (1982). The acquisition and elaboration of the number word sequence. In C. Brainerd (Ed.), Children's logical and mathematical cognition: Progress in cognitive development research. New York: SpringerVerlag. Gelman, R., & Gallistel, C.R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press. Gelman, R., & Meck, E. (1983). Preschoolers' counting: Principles before skill. Cognition, 13, 343Ð359. Jedrysek, E. (2000). Number concept development in young children. In S. Vig, & R. Kaminer (Eds.), Early Intervention Training Institute Newsletter (pp. 13). Bronx, NY: Rose F. Kennedy Center. Piaget, J., & Szeminska, A. (1952). Child's conception of number. London: Routledge & Kegan Paul. Thompson, I. (1989). Early years mathematics: Have we got it right? Curriculum, 15(1), 4248.

