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Cognitive aspects of proof, with special reference to the irrationality of √2

England The criteria by which a proof is judged in mathematics seem, on the face of it to be quite different when considered by sophisticated mathematicians as compared with learners. The sophisticated mathematician may concern himself with logical structure, mathematical style, the degree of generality, the aesthetic quality of the proof, and so on. The learner may lack the sophistication to appreciate these criteria fully and may concern himself more with the manner in which the proof explains the result and demonstrates why it must be true, based on his current state of development. A basis for the cognitive development of proof is already available in the psychology of learning in terms of the meaningful learning of Ausubel (1978) or the relational understanding of Skemp (1976). At any stage in development the ideas presented need to be potentially meaningful (in Ausubel’s terminology), which may mean, in the short term, presenting proofs in a radically different form from that ultimately desired. In this paper we will see an example of this phenomenon. The initial proof may be more cumbersome, less aesthetically pleasing, yet prove more meaningful to the learner at the particular stage under consideration. Even so, the long term desire for full sophistication must be kept in mind, yielding two complementary but, at times, conflicting, principles: l. To present the material in a potentially meaningful manner for the learner, 2. To aid the learner in achieving long-term sophistication. These two principles in fact underlie the whole of education. They must be carefully balanced; lack of balance in either direction leads to its own particular problems. One may conjecture that psychologists tend to err by over-stressing the first principle (for example in interpreting Piaget’s theory of stages) whilst mathematicians pay more attention to the second. Over indulgence in the first can lead to stunted mathematical growth whilst the second can lead to less able students having no growth at all. When these principles are applied to the understanding of proof, the first suggests that proofs should be prepared in a manner which is