Counting all, counting on, counting up, counting down: the role of counting in learning to add and subtract

In the last twenty years research on children’s acquisition of numerical skills and concepts has been a vibrant topic of enquiry amongst psychologists and educators alike. While there has been a diversity of interest between the different disciplines, there is consensus in their recognition that young children have much more mathematical knowledge and understanding than was once thought possible. One very significant finding, largely precipitated by the seminal work of Gelman and Gallistel (1978) is that children as young as three years of age may be able to count with an implicit understanding of what they are doing. Such counting is not to be thought of as the mechanistic, rote process once so derided by Piaget (1952) but rather as principled knowledge which allows children to make precise quantitative judgements as distinct from exclusively perceptual or qualitative judgements. According to Schaeffer et al, and Gelman and Gallistell, the essence of this knowledge is understanding the ‘cardinality rule’ or the ‘cardinal principle’; of knowing that the last count word in a sequence represents the numerosity of the set of countables. And it is this knowledge, which underlies learning to add (Groen and Parkman, 1972; Fuson, 1982; Secada et al, 1983) and to subtract (Wood et al, 1975; Fuson, 1986). If counting is critical to the development of addition and subtraction, then clearly it is desirable to understand what the relationship might be.