13. How children, under instruction, develop mathematical understanding

The relationship between the development and institutionalisation of mathematical understanding across millennia and its development for an individual child is the starting-point for this chapter. Greatly influenced by the writings of Hans Freudenthal, a position is taken in opposition to the theory propounded by Jean Piaget. The counterposition emphasises that a child can only be said to acquire any but the most elementary mathematics under more or less formal instruction and other forms of social and cultural interactions. The perennial debate about the relative weights that should be afforded in school mathematics to procedural competence and deep understanding is also related to the historical development of mathematics, particularly in relation to conceptual restructuring. This relationship is illustrated by the progressive enrichments of what is meant by ‘number’ and the basic arithmetical operations. The expansion of mathematical modelling from physical phenomena to the complexity of human interactions remains to be adequately addressed in school mathematics. And the question ‘What is mathematics education for?’ should be constantly revisited.