The long-term cognitive development of reasoning and proof
pp. 137-149
in: Gila Hanna, Hans N. Jahnke, Helmut Pulte (eds), Explanation and proof in mathematics, Berlin, Springer, 2010Abstract
This paper uses the framework of "three worlds of mathematics' (Tall 2004a, b) to chart the development of mathematical thinking from the thought processes of early childhood to the formal structures of set-theoretic definition and formal proof. It sees the development of mathematical thinking building on experiences that the individual has met before, as the child coordinates perceptions and actions to construct thinkable concepts in two different ways. One focuses on objects, exploring their properties, describing them, using carefully worded descriptions as definitions, inferring that certain properties imply others and on to coherent frameworks such as Euclidean geometry through a developing mental world of conceptual embodiment. The other focuses on actions (such as counting), first as procedures and then compressed into thinkable concepts (such as number) using symbols such as 3 + 2, ¾, 3a + 2b, f (x), dy/dx; these operate dually as computable processes and thinkable concepts, termed procepts, in a developing mental world of proceptual symbolism. These may lead later to a third mental world of axiomatic formalism based on set-theoretic definition and formal proof. In addition to charting the development of proof concepts through these three worlds, we use the theory of Toulmin to analyse the processes of reasoning by which proofs are constructed.