A Further Test of the Ordinal Theory of Number Development

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Abstract
Two questions about recently reported findings that the natural number concept derives from a prior understanding of ordination rather than a prior understanding of cardination were examined: (a)Do the findings hold when Piaget's criterion of natural number competence (number conservation) is substituted for the mathematician's criterion (addition and subtraction of integers)? (b)What accounts for the later emergence of cardination relative to ordination and natural number? Concerning a, it was found that the sequence of emergence of number concepts in five- to seven-year-olds is ordination → natural number → cardination when number conservation is the natural number criterion. Concerning b, it was found that roughly half the subjects did not possess a concept which is both logically and empirically necessary for cardination: viz., the concept of number-as-class. The concept of number-as-class—as measured by a card sorting task—also was found to emerge after ordination and concurrently with natural number.

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