The Cognitive Development of Proof: Is Mathematical Proof For All or For Some?

Proof is considered a foundational topic in mathematics. Yet, it is often difficult to teach. In this paper, I suggest that different forms of proof are appropriate in different contexts, dependent on the particular forms of representation available to the individual, and that these forms become available at different stages of cognitive development. For a young child, proof may be by way of a physical demonstration, long before sophisticated use of the verbal proofs of euclidean geometry can be introduced successfully to a subset of the school population. Later still, formal proof from axioms involves even greater difficulties that make it appropriate for a few, but impenetrable to many. At this formal stage of development, I will identify two different strategies that students adopt to come to terms with formal definition and deduction. Either strategy may be successful, but both are cognitively demanding and prove difficult for many to achieve. This leads to the observation that formal proof is appropriate only for some, that some forms of proof may be appropriate for more, and that, if one allows the simpler representations of proof such as those using physical demonstrations, perhaps some forms of proof are appropriate for (almost) all.