This post follows on from what is proof, and why is it important? and explores how we might 'teach proof' in the classroom (if such a thing is possible). Again I will present excerpts that I think might be useful when planning to explore mathematical proof in the classroom.
A culture of proof in the classroom
I will start as I did in the previous post with Gila Hanna, who suggests that proof can be considered as: "... an activity in mathematics education which serves to elucidate ideas worth conveying to the student. A first step in shifting to the use of explanatory proofs (i.e. proofs that explain) is for educators to stress that mathematical understanding is the goal. There is no infidelity to the practice of mathematics if in mathematics education we focus as much as possible on good mathematical explanations, highlighting for the students in our proof of a theorem the important mathematical ideas that lead to its truth."
Nitsa Movshovitz-Hadar suggests the use of what she calls 'transparent pseudo-proofs': "A transparent proof is a proof of a particular case which is ‘small enough to serve as a concrete example, yet large enough to be considered a non-specific representative of the general case'. One can see the general proof through it because nothing specific to the case enters the proof.’
Because a transparent proof is not a completely polished proof, this kind of ‘proof’ was later re-named a Transparent Pseudo-Proof... we strongly advocate, wherever it is appropriate, the use of transparent pseudo-proofs as a pedagogical tool, as it was shown to support both the development of one’s belief in the truth of mathematical statements and of one’s ability to justify this belief."
I also like Movshovitz-Hadar's idea that (nearly) all proofs contain a 'surprise' which can be exploited by the thoughtful teacher: "I argue that all school theorems, except possibly a very small number of them, possess a built-in surprise, and that by exploiting this surprise potential their learning can become an exciting experience of intellectual enterprise to the students… I claim that every mathematics theorem is a statement of some non-trivial discovery.
It is the mathematics teacher's responsibility to recover the surprise embedded in each theorem and to convey it to the students. The method is simple: just imagine you do not know this fact. This is where you meet your students. Let them examine their expectations, and make them realize that they get new and very unusual results in every theorem.
Hanna promotes the use of a modelling approach when teaching mathematical proof in order to create a need for proof: "Pupils need to have their own experience with the process of establishing a theoretical perspective. The need to state assumptions in relation to answering the question 'why?', and the need for a reasonable empirical base for these assumptions is part of the 'theoretical physicist' approach [that is] indispensable in the development of a sense of proof... Students might be introduced to a theoretical perspective that emerges as a way of describing and explaining experienced phenomena through a modelling process."
She challenges teachers to try to: "exploit the excitement and enjoyment of exploration to motivate students to supply a proof, or at least to make an effort to follow a proof supplied."
However, she offers a note of caution: "Students need to be taught that exploration, useful as it may be in formulating and testing conjectures, does not constitute proof." More on this in the next section.
Daniel Alibert observed a similar requirement to motivate a need for proof: "Many students have no interest in proof as a functional tool. Proof is only a formal exercise to be done for the teacher, there is no deep necessity for it. It is true, of course, that the problems which the teacher's proofs resolve have not usually been appropriated by the students."
This brings to mind this brilliant paper by Nicolas Balacheff in which he conducted an experiment in which students were asked to: "...write a message which will be given to other pupils of your own age, which is to provide a means of calculating the number of diagonals of a polygon when you know the number of vertices it has."
Balacheff explains: "The problem was given to the pupils without providing them with definitions of polygon and diagonal. This choice was made in order to allow the possibility of observing the (re-)construction of these concepts in the course of the solution, particularly with regard to refutations. In this way, we tried to place ourselves as close as possible to the focus of Imre Lakatos’ study which, along with the problem of proof, also examines that of the construction of mathematical knowledge.
Alibert changed the 'customs' of his classroom in order to motivate need and create opportunities for students to develop their own proof arguments: "In our model of 'scientific debate' the proof arguments made by a student are not addressed to the teacher but to the other students. We distinguish between 'proofs to convince', in which arguments are produced to convince someone (such as another student) of something that is not already a part of his institutionalised knowledge and 'proofs to show', where the aim is to show someone (such as the teacher) that we have reached some knowledge that he already possesses."
This also brings to mind Yackel and Cobb's work on establishing 'sociomathematical norms'.
Deborah Ball suggests three domains of work for the teacher in developing a classroom culture of mathematical argumentation and proof: "A first concerns the selection of mathematical tasks that create the need and opportunity for substantial mathematical reasoning. The second domain of teachers’ work centres on making mathematical knowledge public and in scaffolding the use of mathematical language and knowledge. Making mathematical knowledge and language public also requires moving individuals’ ideas into the collective discourse space. A third domain of work concerns the establishment of a classroom culture permeated with serious interest in and respect for others’ mathematical ideas. Deliberate attention is required for students to learn to attend and respond to, as well as use, others’ solutions or proposals"
Easier said than done, but this is essential work towards building a classroom culture, and not at all specific to the learning of mathematical proof. As Hanna describes: "The evolution of a mathematical culture in the classroom is a long-term process, requiring specific strategies of intervention that begin very early and develop over a long period."
From argumentation to proof... and back again
Many of the strategies described above suggest that we 'teach proof' through argumentation and appeal to students' 'natural' ability to reason inductively.
However, there is some research to suggest an 'informal' approach might create obstacles to learning mathematical proof. As Nicolas Balacheff suggests: "It may be good to start from the recognition that mathematical ideas are not a matter of feeling, opinion or belief."
Dave Hewitt warns against a purely inductive approach that involves testing lots of examples, collating this data and looking for patterns. He suggests: "...it is still on the level of trusting what someone else says is true, rather than a student knowing that it must be true... Proof can only lie in looking at the situation itself. Attention needs to be turned away from the numbers and onto the situation where those numbers came from. Yet so many 'investigations' take the form of asking students to collect results from individual cases, put them into a table, and look to see whether they can find a rule... There are lots of formulae (not equivalent to each other) which will fit any set of numbers, but only one formula (or its equivalents) which represent a particular situation and a particular interpretation of that situation.
John Mason shares this concern: "…in my view, inductive generalisation has indeed predominated, while structural deduction is rarely found. Hoyles & Healy found that most students (their study was with nearly 2500 children aged 14-15 in 90 schools) base their confidence in the truth for a finite number of cases - an empirical perspective of proof.
Mason asks: "What is the psychological experience of moving from convincing through ‘arguing’ to convincing through appeal to accepted principles and facts? How might teachers be supported in supporting this transition in their students?"
Alan Bell also identifies the need for movement between 'invalid' to 'valid' mathematical proofs, and offers some practical conclusions from his research: "The critical question for the curriculum is how higher levels of deductive thinking may be encouraged...
1. One prerequisite is clearly that the concepts being dealt with should be familiar; learning new concepts is incompatible with rigorous establishment of relationships involving them. Thus there is a conflict for the teacher between teaching his pupils more advanced concepts and developing their deductive skills.
2. One of the steps towards deduction may be the acquisition of a taste for certainty; this may be acquired through problems based on small finite sets of possibilities where exhaustion is a feasible strategy.
3. There are suggestions among the results that the strategies of reversal, systematic classification and ordering are capable of development, possibly helped by some explicit teaching.
The main conclusion to be drawn is probably that the distinguishing of valid from invalid informal proofs requires judgements of relevance and logical completeness which need more time or more intensity of teaching - or more general mathematical maturity."
Raymond Duval highlights student 'dysfunctions' in valid (mathematical, theoretical) reasoning due to a their lack of awareness of how deductive reasoning differs from linguistic (cognitive, semantic) argumentation: "What is at stake first of all in proof learning is to discover that reasoning in mathematics does not function in the same way as reasoning in discussion that aims to convince other people, outside mathematics... The issue here is not to oppose mathematical and cognitive points of view is mathematics education, but to articulate them. One can learn to prove only in mathematical situations."
Maria Mariotti describes Duval's position further: "According to Duval, the rupture between the two levels (the semantic and the theoretical) may be irretrievable, so that the conception of proof as a process that aims to convince the interlocutor may conflict with the requirements of a mathematical proof... Such a conflict may become an epistemological obstacle that students have to overcome in order to grasp the very idea of proof in mathematics. In fact, the learner has to make sense of the difference between argumentation and proof, without rejecting one for the other. Argumentation as experienced in everyday practice has to be consciously brought back into the mathematical classroom; but achieving a theoretical perspective means becoming aware of the particular nature of mathematics validation, so that particular argumentative competencies that naturally emerge in social interaction might appear inadequate and, for this reason, are likely to be overcome...
Mariotti's research suggests that we: "...escape the rigid dichotomy setting argumentation against proof. When the phase of producing a conjecture had shown a rich production of arguments that aimed to support or reject a specific statement, it was possible to recognise an essential continuity between these arguments and the final proof; such continuity was referred to as Cognitive Unity."
Nicolas Balacheff might disagree that such continuity is desirable. He studied the role of social interaction in learning mathematics, and proof in particular, and discovered that: "[This work] confirmed the productive and essential character of social interaction, but they also, and perhaps above all, revealed that by its very nature this type of interaction creates social processes and behaviors which run counter to the construction of a mathematical problématique of proof by the students. These processes and behaviours can be assembled under a single reference theme, that of argumentation.
…it seems possible to envisage a solution of continuity from argumentation to mathematical proof. However, argumentation introduces an object, the validity of a statement. The sources of argumentative competence are in natural language and in practices whose rules are frequently of a profoundly different nature from those required by mathematics, and carry a profound mark of the speakers and circumstances… in the field of scientific practice, argumentation must satisfy the conditions for entry into a problématique of knowledge which involves the decontextualization of the discourse, the disappearance of the actor and of the duration. All conditions which run counter to the profound nature of argumentation....
The resolution of problems is the context in which to develop the argumentative practices using means which could be used elsewhere (metaphor, analogy, abduction, induction, etc.) but which disappear in the construction of a discourse acceptable with regard to the rules specific to mathematics…
Argumentation constitutes an epistemological obstacle to the learning of mathematical proof, and more generally of proof in mathematics... Mathematics class is one of the few places where the existence of that practice can be revealed because it suddenly appears inadequate (but situations for creating this awareness are difficult to construct). In my eyes it would even be an error of epistemological character to let students believe that they are capable of producing a mathematical proof when all they have done is argue."
Guy Brousseau finds a compromise between these positions. In his 'theory of didactical situations', Brousseau describes three main types of situations: action, formulation and validation. He describes learning mathematics using the 'metaphor of game', described by Maria Mariotti thus:
"When the pupil is playing, s/he develops strategies; this means that actions are selected from a set of potentialities according to intuitive or rational reasons; the feedback produced by the environment allows the subject to check the effectiveness of her choice and may consequently lead her to accept or reject it. The sequence of interactions between the student and the environment (milieu) constitutes what is called the ‘dialectic of action’.
Continuing in the game and the student passes through what is called the ‘dialectic of formulation’ that consists in 'progressively establishing a shared language', making 'possible the explanation of actions and modes of action'. During this phase, according to Brousseau’s model: '... it can happen that one student’s propositions are discussed by another student, not from the point of view of the language... but from the point of view of the content (that is to say, its truth or its efficacy) ... these spontaneous discussions about the validity of strategies are usually referred to as 'validation phases'.
It is only by entering the ‘dialectic of validation’ that the student is motivated to discuss a situation and encouraged to express his/her reasons, which might previously have remained implicit."
I think the idea of a 'dialectic of validation' is a very powerful one, and suggests a movement back and forth between inductive and deductive reasoning.
However we choose to teach students about mathematical proof, it seems clear that we must provide numerous opportunities, over a long period of time, for students to develop an understanding of what mathematical proof specifically entails and why it is important.