There is a vast literature on mathematical proof; this post is an attempt to synthesise some of it into extracts that I have found interesting. It is the first of two posts, the second being on the pedagogical implications of this theory.
Why learn about proof?
Gila Hanna cites many purposes of proofs, including verification and explanation: “But in the classroom, the fundamental question that proof must address is surely ‘why?’. In the educational domain, then, it is only natural to view proof first and foremost as explanation, and in consequence to value most highly those proofs which best help to explain... Mathematicians clearly expect more of a proof than justification; they would also like it to make them wiser.”
Maria Mariotti broadly agrees: “Proof clearly has the purpose of validation - confirming the truth of an assertion by checking the logical correctness of mathematical arguments - however, at the same time, proof has to contribute more widely to knowledge construction. If this is not the case, proof is likely to remain meaningless and purposeless in the eyes of students.”
Guershon Harel & Larry Sowder suggest the process of proof is a search for certainty:
This is roughly in line with Humberto Maturana's view of proof: "If the criterion of acceptability applies, the (re)-formulation is accepted and becomes an explanation... the emotion or mood of the observer shifts from doubt to contentment, and he or she stops asking over and over again the same question."
For Nicolas Balacheff, proof requires a shift in epistemological position: “To construct a proof requires an essential shift in the learner’s epistemological position: passing from a practical position (ruled by a kind of logic of practice) to a theoretical position (ruled by the intrinsic specificity of a theory)." [More about Balacheff's views on the process of proof below.]
Dave Hewitt also talks of a shift, in awareness and attention: “Where attention is placed and of what one is aware are important within any proof. Proof is not only about 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 concept in formal logic, which may be thought of as the ideal which actual mathematical practice only approximates, and (2) Acceptable proof: proof as a normative concept that defines what is acceptable to qualified mathematicians.
Practising mathematicians have long agreed that proofs may have different degrees of formal validity and still gain the same degree of acceptance… The acceptance of a theorem by practising mathematicians is a social process which is more a function of understanding and significance than of rigorous proof… mathematicians undeniably insist on using their judgment when deciding whether or not to accept a new proof… mathematicians seem to adhere to Bateson's view that 'the advantages in scientific thought come from a combination of loose and strict thinking, and this combination is the most precious tool of science'… A proof is valued for bringing out essential mathematical relationships rather than for merely demonstrating the correctness of a result.
The distinction I have made between proofs that prove and proofs that explain is similar to that which Bolzano makes between "making certain" (Gewissmachung) and "building a foundation" (Begruendung).”
Mariotti suggests why this shift from inductive/abductive reasoning based on experience, to deductive reasoning based on logic, is difficult: “The difficulty emerges from controlling the complex relationship between mathematical validation, rooted in the frame of a theoretical system, and common sense validation, rooted in empirical verification."
Besides direct acquisition of information that is mostly related to factual evidence and attained through experience, human culture has developed a complex way of obtaining information and knowledge, which is not direct, but is rather mediated by means such as language, logic and reasoning. As a consequence of this mediation, the structural unity between cognition and adaptive reactions has been broken.
The interest in the case of abduction lies in the fact that, according to experimental evidence, exploration supporting a conjecture is very often accompanied by arguments showing this structure, so that passing from conjecturing to proving would require transformation from an abductive into a deductive structure. This passage presents difficulties that seem to require specific didactical treatment to overcome."
Balacheff explores the difficulties in moving from argumentation to mathematical proof in more detail: "Teachers organize situations in order to “convince” or “persuade” learners (in the vocabulary of Harel & Sowder).
Argumentation seems the best means to this end. It works both as a tool for teaching and as a tool for doing mathematics for a long while. But then learners suddenly face an unexpected revelation: In mathematics you don’t argue, you prove… It must be remembered that at stake is not truth but the validity of a statement within a well-defined theoretical context.
The arguments in such a discussion involve three types of critical considerations: the search for certainty, the search for understanding and the requirements for a successful communication. The complex nature of proof lies in the fact that any effort to improve a proof on one of these dimensions may change its value on the other two. There is no clear standard to decide on the correct balance.
What is produced first is an 'explanation' of the validity of a statement from the subject’s own perspective. This text can achieve the status of proof if it gets enough support from a community that accepts and values it as such; it can be claimed as mathematical proof if it meets the current standards of mathematical practice.
Students have to learn mathematics as social knowledge, they are not free to choose the meanings they construct. These meanings must not only be efficient in solving problems, but they must also be coherent with those socially recognized
Being the product of a choice, actions are considered in terms of validity and the adequateness of effect. Action is related to its aim, and a contradiction becomes apparent when the aim is not fulfilled.
Language must become a tool for logical deductions and not just a means of communication. The elaboration of this functional language requires in particular:
- a decontextualisation, giving up the actual object for the class of objects, independent of their particular circumstances;
- a depersonalisation, detaching the action from the one who acted and of whom it must be independent;
- a detemporalisation, disengaging the operations from their actual time and duration: this process is fundamental to the passage from the world of actions to that of relations and operations."
Balacheff describes the process of proof as consisting of: “...four main types of 'proof': naive empiricism, the crucial experiment, the generic example and the thought experiment."
He gives many examples of these proof types in his papers, and provides the following analysis: “[we] assert the existence of a break between naive empiricism and the crucial experiment on the one hand, and the generic example and the thought experiment on the other. This divide can be characterised as one of passing from a truth asserted on the basis of a statement of fact to one of an assertion based on reasons. We find here an example of an operational cohabitation between empirical pragmatism and logical rationalism...
Although naive empiricism disappears once conceptual proofs are brought into play, the crucial experiment can continue as an ultimate test to guarantee conviction, most noticeably when the assertion has been founded on a generic example.
The crucial experiment means something different in social interaction, where it becomes a means of resolving completely a conflict over the validity of an assertion, or over the choice between two conjectures. We no longer consider it as a means of proof except when it constitutes the refutation of an assertion.
Another connection is that between the generic example and the thought experiment. Passing from the former type of proof to the latter relies on a linguistic construction which involves a recognition and differentiation between the objects and relations involved in the solution of the problem; in other words a cognitive construction."
The question is then: How can we as teachers create opportunities to enable this cognitive construction?
This will be the basis of my next post.