properties, but is also about awareness of properties... Attending to proof can force a student to examine what they do know and what they do not know about something... I believe it is the attention to properties which helps develop someone's learning in mathematics, because that is what mathematics is about.”
Mike Ollerton (private correspondence) agrees that proof is fundamental to learning mathematics: "We certainly need to distinguish between repetitive demonstration and proof though I am not sure I understand what 'formal' proof is or what it means. For me I see proof as deeper conceptual understanding; the ultimate process and something that goes beyond generality. Having said that I want to see proof as part and parcel of what it means to "think and behave mathematically" at any age."
As does Alan Bell: “It might be said that this is the essence of mathematics - the art of dealing with the general by working with the particular.”
John Mason suggests that learning about proof has wider (social and personal) implications: “Children have been exposed to a variety of reason-giving situations from a variety of types of authorities… For young children, authority is external, expressed and warranted through adults who know more, or implicit within the structure of 'how things are'.
However, belief alone is inadequate as a warrant in mathematics. Mathematics offers an entirely different form of warrant for belief, and source of authority. One of the features of mathematics is that it lends itself, indeed it depends upon, a shift from external authority to the authority of one’s own reasoning. However this is not an easy shift to make…
Algebra is much more than the principal language of (mathematical) reasoning, because it is also the study of the structure of that reasoning itself, and of the structures within which that reasoning is fruitful. As such I consider it to be both within the reach of, and vital for, every active citizen...
To be responsible means literally to be able to spond from the Latin, meaning ‘to offer explanation’. Thus to become responsible is to be able to justify your actions to others.
Learning to be your own authority in mathematics, with being responsible through being able to justify, with learning how to convince yourself, a friend and a sceptic, probably reflects, or could reflect, perhaps even presage the same processes within a broader social context. Thus mathematics could speak to the experience of students at all ages.”
What constitutes a proof?
John Mason suggests that the learning of mathematical proof starts with convincing others: “Being convinced oneself is quite different from convincing someone else, particularly when that person is being sceptical. Learning to be sceptical oneself, to locate gaps and jumps in other people’s reasoning is important not just in mathematics, but as an involved citizen. As my remarks have suggested repeatedly, convincing, reasoning, proving, justifying are all emergent and developing processes, not rigid formats to be learned or followed."
Nitsa Movshovitz-Hadar broadly agrees with this view: “... mathematics knowledge is in principle not different than any other kind of knowledge, although, of course, the nature of the discipline is different. What, then, is knowledge? An answer, given by philosophers to this question, is that in order to be one’s knowledge a proposition must comply with three necessary (albeit not sufficient) conditions: (i) It must be true. (ii) one must believe it, and (iii) one must have justification for believing it."
She quotes Bertrand Russell: "'Minds do not create truth or falsehood. They create beliefs.' Given a statement p of a mathematical theorem, a learner should be able to relate to two basic questions: (a) 'Do you believe that p?' and, provided the learner’s answer to (a) is yes, (b) 'Why do you believe that p?'"
However, Hanna suggests: “It is imperative to make a clear distinction between a correct proof and a heuristic argument…"
She offers two conceptions of proof: "(1) Formal proof: proof as a theoretical concep