ABSTRACT
In Chapter 6 I questioned whether the Piagetian notions of number conservation, transitive-asymmetrical relations, and class inclusion necessarily represent what children know, either when they are doing the associated tasks, or when they use number. It can be argued that knowing about number does not take the form that Piaget attributes to it; instead it involves entering into the social practices involved in using numbers. Thus the possession of knowledge about number does not necessarily have to be shown by demonstration of the possession of certain logical ideas which conceptual analysis leads Piaget to propose as the necessary and sufficient conditions for having that knowledge. Instead, knowing about number is a matter of knowing how to act appropriately in situations in which number is used, that is to say, understanding and sharing in what other people understand and expect in mose situations, and therefore knowing how and when to use numbers. The first section of this chapter considers research by Donaldson and her co-workers on the linguistic, non-linguistic, and social demands of Piagetian tasks and the argument that these create difficulties for children with the result that their true logical and numerical competence is masked. The second section widens the discussion to include the use of numbers in arithmetic, while the third concentrates on work by Steffe et al. (1982, 1983), which illustrates the importance of understanding the context of ‘doing arithmetic’ and its associated social practices.
