Cognitive development, representations and proof

The purpose of this paper is to highlight the different forms ofproof afforded by different types of mathematical representation.The form of proof generally accepted by mathematicians is logicalproof with formal concept definitions and deductions using thepredicate calculus (although there are many subtle differences inacceptability of a proof in the mathematical community). However,the cognitive development of a notion of proof must take intoaccount the differing forms of representation available to thelearner at various levels of sophistication. In particular, there aretwo very different parallel developments of visualisation andsymbolisation with different forms of proof.Introduction“Proof” is regarded as a central concept in the discipline of mathematics. It isimportant for two reasons.(1) (Local) Based on explicit hypotheses, a proof shows that certainconsequences follow logically,(2) (Global) Such logical consequences themselves can be used as“relay results” (Hadamard 1945) to build up mathematicaltheories.In the recent past (eg since the mid nineteenth century in England), Euclideangeometry has been considered as an introduction to both (1) and (2). However,this has fallen out of favour because of the difficulties encountered by children(eg Senk, 1985, showed that only 30% of students in a full-year geometrycourse reached a 70% mastery on a selection of six problems in Euclideanproof). The NCTM standards in the USA suggest that there should be increasedattention on short sequences of theorems and decreased attention to Euclideangeometry as an axiomatic system, favouring (1) over (2). In England the demiseof geometry has proceeded further. The Association for the Improvement ofGeometry Teaching (which later became the Mathematical Association) was