P. L. GALBRAITH ASPECTS OF PROVING: A CLINICAL INVESTIGATION OF PROCESS ABSTRACT. A clinical methodology was used to investigate the perceptions which pupils of secondary school age have concerning modes of mathematical argument which have an agreed status within the world of mathematics. The analysis of data obtained from three extended contexts led to the identification of dusters of characteristic response types. Differences were found to exist between the agreed meaning of some mathematical terms and procedures and the meaning ascribed to them by students. By considering levels of performance it was possible to identify particular components, the presence or absence of which consistently determined the capacity to structure or follow proofs and explanations. 1. BACKGROUND Increased interest in the nature of mathematical proof and its associated skills seems to be one product of an era which has seen the notion of formal proof exorcised from many curricula. Lester (1975), has identified several problem regions within the literature and has drawn attention to conflicting evidence concerning the development of logical reasoning ability. Thus on the one hand studies by Lovell (1961) and LoveU, Mitchell and Everett (1968) following Piaget (1928) were cited as supporting the belief that while children in the age range 11-13 years are able to handle certain formal operations they carmot synthesize a fully exhaustive proof. On the other hand reference to studies by Suppes (1968), following Hazlitt (1930) acknowledged an alternative viewpoint that there is no relation- ship between age and logical reasoning beyond that determined by experience. Bell (1976, 1979) described proof as an "essentially public activity which followed the reaching of conviction, though it may be conducted internally, against an imaginary doubter." He saw the meaning of proof as carrying three senses, (a) verification or justification; i.e., concerned with the truth of a pro- position. (b) illumination; i.e., conveying insight into why a proposition is true (or false), (c) systematisation; the organization of results into a deductive system of axioms, major concepts and theorems. Bell claimed that an understanding of proof grows out of the internal testing and acceptance or rejection which accompanies the development of a EducationalStudies in Mathematics 12 (1981) 1-28. 0013-1954/81/0121-0001 $02.80 Copyright 9 1981 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.