A study of pupils' proof-explanations in mathematical situations

Viewed internationally, the proof aspect of mathematics is probably the one which shows the widest variation in approaches. The present French syllabus adopts an axiomatic treatment of geometry from the third secondary school year (age 14), Papy's Mathdmatique Moderne is axiomatic from the age of 12, early American developments based primary school number work on the laws of algebra. In England, proofs of geometrical theorems have been steadily disappearing from O-level syllabuses for thirty years, and 'it continues to be the policy of the SMP to argue the likelihood of a general result from particular cases'. (Preface to Book 5). Underlying this divergence in practice lies the tension between the awareness that deduction is essential to mathematics, and the fact that generally only the ablest school pupils have achieved understanding of it. The purpose of the work described in this paper is to analyse pupils' attempts to construct proofs and explanations in simple mathematical situations, to observe in what ways they differ from the mature mathematician's use of proof, and thus to derive guidance about how best to foster pupils' development in this area. In a previous paper (Bell, 1976), I have shown that pupils' attempts at making and establishing generalisations, and at supporting these with reasons, can be interpreted in terms of a number of identifiable stages of attainment which are loosely related to age. Two of these stages were fairly well-defined - Stage 1