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This paper sets out to examine the notion of proof in the context of elementary mathematics, as discussed in Andreas J. Stylianides' proposals in 'The Notion of Proof in the Context of Elementary School Mathematics' (2007). Stylianides' argument is constructed around two conceptual sub-categories of proof: what he terms the 'intellectual-honesty principle', and the 'continuum principal'. (Stylianides 2007: p.1). In either case, Stylianides' thesis is fundamentally predicated on the idea that proof is indispensable to the integrity and function of mathematics, and the associated teaching and learning. (Stylianides 2007: p.1). Whilst this may be taken as self-evident, it is argued here that Stylianides' eventual settlement on the hegemony of the teacher undermines the positivist basis of the mathematical validation model. As, in this instance, the didactic role supplants that of the expert academic mathematical community, it is invested with ultimate authority on the question of proof. There is thus an unavoidable trade-off between the intermediate state of mathematical perception in elementary school learners, and their intellectual right to an absolute or positivist outcome. In essence, they end up in a situation where the truth or proven principle is unknowable, as it resides only in the unchallengeable remit of the teacher.
In the first instance, it is necessary to set out the two guiding principles of visualizing proof as proposed by Stylianides. The 'intellectual-honesty principle', he states, is based on the notions of Bruner, whom Stylianides quotes as having said that any subject '….can be taught effectively in some intellectually honest form to any child at any stage of development.' (Bruner 1960: p.33). The 'continuum principle' meanwhile proposes that students' learning of mathematics would be helped by the establishment of a consistent means of apprehending proof: they should not '…develop a conception of proof in elementary school that has to be…unlearned in high school.' (Stylianides 2007: p.4). Stylianides' epistemology is based on the application of these models to practical examples, mediated through four considerations:
Foundation: what constitutes the basis of the argument.
Formulation: the logical deduction or generalization developed within it.
Representation: whether the argument is expressed algebraically, digitally, in text etc.
Social Dimension: the interaction of the mathematical concepts with their social and educational context. (Stylianides 2007: pp.2).
Stylianides focuses predominantly on the attempt of Betsy, a third-grade student, to demonstrate proof that of the fact that the sum of two odd numbers will be an even number. Within the parameters of his own analysis, he characterises the praxis and effectiveness of this as…
Foundation: observable in the definitions of odd and even integers as reviewed in class.
Formulation: a logical deduction from a set of definitions and mutually observed definitions.
Representation: everyday language, in which Betsy's parlance was consistent with her peers' definitions.
Social Dimension: the normative class and didactic arrangement. (Stylianides 2007: pp.9-11).
One of the principal dissonances within Stylianides' interpretation of his data appears to lay in the graduated application of intellectual honesty criteria, which he deems necessary in the elementary class context. Betsy's reasoning was developed by a third grader, achieving a 'definsible [sic] balance between respecting third graders as mathematical learners and honouring mathematics as a discipline…', however her explanation is at odds with exemplar mathematical models. As Stylianides further explains, the intellectual-honesty principle does not require correlation between these two positions. (Stylianides 2007: p.10). However, this only serves to illustrate the difficulties of settling upon an 'ideal type' benchmark of proof within the elementary milieu. As Stylianides concedes, the scenario of an extended system of validation '…is hard to realize.' not only because of the limits of social interactions in the domain of proof, '…but also because students in the early grades are just beginning to interactively constitute…certain rules of discourse...'. This, he further concedes, is a 'long and demanding process.' (Stylianides 2007: p.16).
Essentially, it may be argued that Stylianides' conclusions are predicated on the conflation of two points. Firstly, the that the proving of principles in the wider mathematical community, although never arbitrary, rests on the extensive and discursive validation of the various actors who comprise it. Secondly, that within an elementary mathematics educational context, the intervention of the teacher as a convening authority '…would initiate a situation for institutionalization.' (Stylianides 2007: p.16). In fact, such a mechanism might reflect the teachers' wider societal role in determining mathematical understanding. (Stylianides 2007: p.16). This seems a perfectly reasonable argument, although it does somewhat undermine the entire positivist basis of the discipline, taking into a more phenomenological paradigm. As Tanner and Jones point out, from a practitioner's perspective, it is imperative that learners appreciate that '…satisfying a large number of cases does not constitute proof for all cases.' (Tanner and Jones 2000: p.32). However, it remains the case that exposition in elementary education relies heavily on the visual, verbal and written demonstration of model answers and principles.
Moreover, although these issues undoubtedly impinge on the agreement of an approach to the recognition of proof, they are not constant or unmediated in their nature. As Bell has noted, there are at least two significant barriers to the consolidation of a universal approach to this issue: firstly, cultural contrasts, and secondly, the level of pupils' ability: as he explains, '…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.' (Bell 1976: p.23). Also, as Orton has argued, there is little evidence concerning the ability of pupils to apply a consistent praxis in the establishment of proof across different areas of the mathematics curriculum. (Orton 2004: p.112). In other words, learners might agree upon a successful approach in once curricular area, such as measure and space, but find it unsatisfactory or incompatible with operations such us multiplication and division. The overall didactic context also has to taken into consideration. It is debatable, for example, whether the curricular constraints and outcome-driven standards assessments prevalent in many educational contexts, could be harmonised within such an approach to proof. For example, many school systems and individual institutions are assessed on the results of closed examinations which cannot recognize the achievements of learners in terms of intellectual honesty: only correct answers equate to a mark. As Stylianides concedes, the ideas of Bruner were influential in visualising the 'intellectual honesty' approach. (Stylianides 2007: p.5) However, most contemporary curricular and assessment arrangements in western educational contexts, it may be argued, rely far more heavily on 'staged', age related goals. As such, social constructivist or developmental models are more prevalent than the individual-centred, chronologically seamless schema envisaged by Bruner.
In conclusion, it may be argued -as indeed does Stylianides - that further research is required on this topic, in order to obtain more granular data, and further insights into the role of teachers in cultivating proof amongst students. (Stylianides 2007: p.18) It is also proposed here that such research should consider the problems of establishing such an approach in what is, predominantly, an output-dominated educational culture. As Mariotti points out, social interaction represents '…a basic factor that affects, motivates and fuels a dialectic of validation.' (Mariotti 2006: p.190). Such dialectical arrangements are only achieved through the accretion of both trust, consistency, and achievement, making them more difficult to envisage in an elementary learning context. As in any community, arguments produced to support a particular point have to be compared with arguments '…which are acceptable…that are already stated and shared in the mathematics community…' (Mariotti 2006: p.198). The challenge is therefore to envisage such a community in a schools' system that is already under pressure.