## References

Denvir, B., & Brown, M. (1986a). Understanding of number concepts in low attaining 7-9 year olds: Part 1. Development of descriptive framework and diagnostic instrument. *Educational Studies in **Mathematics, 17*, 15-36.

Denvir, B., & Brown, M. (1986b). Understanding of number concepts in low attaining 7-9 year olds:Part II. The teaching studies. *Educational Studies in Mathematics, 17*, 143-164.

Fuson, K. C. (1992). Research on whole number addition and subtraction. In D. A. Grouws (Ed.), *Handbook of research on mathematics teaching & learning*. (pp. 243-275). New York: MacMillan.

Gelman, R., & Gallistel, C. R. (1978). *The child's understanding of number*. Cambridge, MA: Harvard University Press.

Steffe, L. P. (1992). Learning stages in the construction of the number sequence. In J. Bideaud, C. Meljac & J. Fischer (Eds.), *Pathways to number: Children's developing numerical abilities*. (pp. 83-88). Hillsdale, NJ: Lawrence Erlbaum.

Steffe, L. P., Cobb, P., & von Glaserfeld, E. (1988). *Construction of arithmetical meanings and **strategies*. New York: Springer-Verlag.

Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). *Children's counting types: **Philosophy, theory, and application*. New York: Praeger.

Thompson, I. (1995). The role of counting in the idiosyncratic mental calculation algorithms of young children. *European early childhood education research journal*, 3(1), 5-16.

Wright, R. J. (1991). The role of counting in children’s numerical development. *The Australian **Journal of Early Childhood*, 16(2), 43-48.

Wright, R. J. (1994). A study of the numerical development of 5-year-olds and 6-year-olds. *Educational Studies in Mathematics*, 26, 25-44.

## Acknowledgements

The descriptions of the counting stages used in solving addition and subtraction problems had its genesis in the work of Professor Les Steffe and Professor Paul Cobb. The current organisation of the early arithmetical strategies owes much to the initiative and ongoing contributions of Professor Bob Wright. Their contributions and the contributions of thousands of students and teachers to our current understanding of counting as a problem solving process are gratefully acknowledged.