Sophian, C. (2007). Rethinking the starting point for mathematics learning. In O. N. Saracho & B. Spodek (Eds.), Contemporary perspectives in early childhood education: Mathematics, science and technology in early childhood education (pp. 21-44).  Information Age Publishing.

The pervasive assumption that counting is the starting point for mathematical thinking and learning is contrasted with an alternative account according to which the concept of number itself depends upon other concepts, among them the concepts of equivalence and unit.  This analysis leads to the claim that mathematical thinking begins not with counting but with comparisons of unenumerated quantities, through which children develop the concepts on which an understanding of number depends. Three lines of developmental research that bear on the contrast between counting-based accounts of early numerical development and the comparison-of-quantities position are summarized: research on infants’ ability to discriminate between different numerical quantities; research on preschool children’s reasoning about numerical relations between quantities; and research on children’s reasoning about continuous quantities, particularly lengths.  An important instructional implication of the comparison-of-quantities position is that difficulties students experience with relatively advanced topics such as fractions are likely to stem from an inadequate grasp of much more basic concepts, notably the concept of unit.  In order to forestall such difficulties, it is important that the design of early childhood mathematics instruction take into account the profound interrelations between basic numerical concepts and more advanced mathematical material.