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 (Abstraction), in which patterns or relationships could be recognised, extended and described but there was no attempt to explain, justify or deduce them; and stage 3 (proof), in which an informal, but acceptably complete, deductive argument was given, or a full empirical check of the set of possible cases; and if these could not be achieved, there was an awareness that the argument was incomplete. Of the sample of 160 grammar school girls aged 11-18 who were tested, stage 1 (or below) contained about 70% of the 11-13 year olds (years 1 and 2), whereas only 23 % of the older pupils (years 3-7) were at this stage. Stage 3 was attained by small numbers of pupils spread through the age range, amounting in all to 10% of the sample. About 50% of the pupils were classed in stage 2, which was a transitional stage and less easy to define. It covered deductive arguments ranging from relevant but fragmentary remarks to almost complete arguments, and empirical checks ranging from the consideration of one or two cases only to the testing of a variety covering most of the significantly