Abstract
Mathematics educators face a significant task in getting students to understand the roles of reasoning and proving in mathematics. This challenge has now gained even greater importance as proof has been assigned a more prominent place in the mathematics curriculum at all levels. The recent National Council of Teachers of Mathematics (NCTM) Principles and Standards document and several other mathematics curricular documents have elevated the status of proof in school mathematics in several educational jurisdictions around the world. This renewed curricular emphasis on proof has provoked an upsurge in research papers on the teaching and learning of proof at all grade levels. This re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification (including dynamic graphic software) has already produced several theoretical frameworks, giving rise to many discussions and even heated debates. An ICMI Study on this topic would thus be both useful and timely. An ICMI Study on proof and proving in mathematics education would necessarily discuss the different meanings of the term proof and bring together a variety of viewpoints. Proof has played a major role in the development of mathematics, from the Euclidean geometry of the Greeks, through various forms of proofs in different cultures, to twentieth-century formal mathematics based on set theory and logical deduction. In professional mathematics today, proof has a range of subtly different meanings: for example, giving an axiomatic formal presentation; using physical conceptions, as in a proof that there are only five Platonic solids; deducing conclusions from a model by using symbolic calculations; or using computers in experimental mathematics. For mathematicians, proof varies according to the discipline involved, although one essential principle underlies all its varieties: To specify clearly the assumptions made and to provide an appropriate argument supported by valid reasoning so as to draw necessary conclusions. This major principle at the heart of proof extends to a wide range of situations outside mathematics and provides a foundation for human reasoning. Its simplicity, however, is disguised in the subtlety of the deep and complex phrases ‘‘to specify the assumptions clearly’’, ‘‘an appropriate argument’’ and ‘‘valid reasoning’’. The study will consider the role of proof and proving in mathematics education, in part, as a precursor for disciplinary proof (in its various forms) as used by mathematicians but mainly in terms of developmental proof, which grows in sophistication as the learner matures towards coherent conceptions. Sometimes the development involves building on the learners’ perceptions and actions in order to increase their sophistication. Sometimes it builds on the learners’ use of arithmetic or algebraic symbols to calculate and manipulate symbolism in order to deduce consequences. To formulate and communicate these ideas requires a simultaneous development of sophistication in action, perception and language. G. Hanna (&) Ontario Institute for Studies in Education/University of Toronto, 252 Bloor St. West, Toronto, ON M5S 1V6, Canada e-mail: ghanna@oise.utoronto.ca
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