Children's algorithms as schemes

Abstract

Arithmetical knowledge is viewed as simply the coordinated schemes of actions and operations the child has constructed at a particular point in time. A fundamental distinction is drawn between operative and figurative schemes in the context of additive schemes. Numerical extension, an operative scheme, is distinguished from intuitive extension, a figurative scheme. The basis of the distinction resides in (1) the anticipatory nature of the schemes, (2) the adaptability of the schemes, and (3) the nature of the concepts which serve as a basis for what triggers the schemes. This fundamental distinction is crucial in the mathematical instruction of the child for the child necessarily has to construct his or her own mathematical reality even if that reality eventually becomes compatible with the reality of the social group in which the child has to operate.