Introduction

Mathematics is not just about identifying what is true or what works but also about explaining why it is true or why it works and convincing others that it is true or that it works. That is, mathematics is intrinsically about proof. However, for the past two decades mathematical proof, formal or otherwise, has been conspicuous in the school mathematics curriculum in the U.K. by its absence. With the wide acceptance of the recommendations of Cockcroft report (DES 1982), the learning of formal proofs transmitted by the teacher was replaced by learner engagement in personal investigations and problem solving so as to catalyse process skills such as conjecturing, generalising, and justifying.

Are the self generated justifications of learners meaningful? Do they bear any comparison with formal mathematical proof? Recent research (see, for example, Porteous 1990, Coe and Ruthven 1994) suggests that learners tend to use largely empirical proof strategies to explain rules or generalisations and, according to, Coe and Ruthven, show little evidence of understanding, let alone awareness, of the function and role of proof. Some mathematics educators perceive this state of affairs to be due to defence of particular curriculum approaches:

They indicate the resilience of the epistemic schemas of school mathematics; the continuing triumph of a hidden curriculum over the rhetoric of reform (Coe and Ruthven, 1994, p 52)

Others view the conflict as a result of theory being unable to inform practice:

“It may indicate.......how little contact there is between the work going on in most schools and the research or new developments taking place in the mathematical associations, in departments of education or amongst some teachers.” (Lerman, 1989, p 73)

My belief is that learners prototypical informal proof practices are meaningful and, given sufficient nurturing, can develop to the point where learners become aware of the need for deductive reasoning; that is, learners inductive reasoning may evolve into deductive reasoning in the right environment. The right environment would appear to require that

1. The philosophy of the mathematics department was in agreement with the aims of the Cockcroft Report (DES 1982).
2. The mathematics teachers were actively engaged in INSET and/or curriculum development.

In this article I argue for my case from both theoretical and practical grounds. In the next section I give a framework for proving and justifying the classroom and following that I describe findings from a pilot study of learners proof practices in a progressive mathematics department of a large inner-city secondary school. By progressive I mean that the department has for many years used a published individualised scheme for learning mathematics whose curriculum approach is multifarious - it is in parts Structuralist, Integrated Environmentalist, Problem Solving/Investigational, and Cultural. Furthermore two teachers in the department are actively engaged in writing and re-writing curriculum materials for the scheme. I end with some conclusions and tentative proposals for engendering viable proof practices in the classroom.

Justification and Proof in the classroom.

In the classroom, what is justifying? What is proving? Are justifying and proving different?
According to guidance given to learners in many secondary schools justifying is ‘convincing someone else that you have understood the rule and understand why’.
The most helpful criteria to determine what constitutes proof in the classroom are the answers to three pertinent questions (Simpson 1994)

q1.	What are proofs for?	a1.	For convincing and justifying.
q2.	Who proves things?	a2.	It should be every one.
q3.	How do they prove them?	a3.	Exploring a problem

Consequently, if we accept this description, justifying and proving are identical. Simpson calls the set of three answers at the beginning of the section an ‘extra attitude’ to proof and posits that learners encultured with this extra attitude are more likely to engage with proof because they see that they have a role in doing proofs

In this attitude the learner seems to see proof as part of a developing structure which verifies the theorem and has a role in convincing the mathematical community of the truth of the statement. (Simpson 1994, p 4)

However we must differentiate this description of proof from formal mathematical proofs which necessarily involve a higher order of thinking than those available to many primary and secondary learners. As Semadeni has written

... the concrete-operational child is not capable of hypothetical reasoning, of deduction expressed in words and symbols. (Semadeni 1984, p 32)

The proof-practices of the learner of school mathematics will involve pre-deductive stages. The prototypical proof practices of the learner may be naïve and based on analogy with their real experiences: proving by measurement as in science experiments, proving by weight of evidence, etc.. Such proof practices are legitimate for the learner:

...we talk of proof because they are recognized as such by their producers (Balacheff 1988, p. 218)

The progression of the proof practices of the learner have been well documented in the literature (Balacheff, 1988; Bell, 1976; Van Dormolen, 1977). Balacheff (1988) posits that there are levels of proof in the classroom. At the fundamental level there is naïve empiricism which is justification of the proposition on the basis of the weight of evidence from a number of cases. Next there is the crucial experiment where the proposition is verified to be true on the basis of showing its validity in a typical case. A higher level of proof practice is proof by a generic example: here the proposition is proved by examining a prototypical case and appealing to the structural properties of mathematics. Finally there is the thought experiment which differs from the proof by generic example in that instead of a prototypical case an abstract general case is examined.

Alongside proofs by naïve empiricism rank ‘visual’ proofs. These appeal to the intuition of the learner and are effective: to convince the learners of the validity of the statement the angles of a triangles add up to 1800 the teacher demonstrates to the learners how to cut a triangle into three and to lay the angles next to each other. The teacher then shows the angles ‘lie in a line’ and so add up to 1800. The visual aspect of mathematical reasoning has been promoted by mathematicians and mathematics educators (Davis, 1993; Semadeni, 1984). Davis argues for

...a mathematical education which interprets the word ‘theorem’ in a sense that is wide enough to include the visual aspects of mathematical intuition and reasoning” (Davis 1993, p 333)

Visual proofs are connected to but not the same as Semadeni’s action proofs (Semadeni 1984). Precisely, action proofs begin with concrete/physical actions to convince the learner of the truth of a proposition (proving Pythagoras’ theorem by dissecting the square on the hypotenuse in a way to make the pieces fit the other two squares) but end with an internalisation of the process to make the concrete action redundant (becoming convinced that the method works for all right angled triangles).

The efficacy of the concept of visual proof and action proof can, perhaps, be gauged by the fact that these proofs methods are used in a many popular published schemes for school mathematics such as HBJ, SMP and SMILE.

According to the National Curriculum opportunities for mathematical reasoning and proof will arise in Attainment Target 1: Using and Applying Mathematics. But this need not be the only area in which these opportunities will arise. For example, Algebra (Attainment Target 2) is intrinsically connected with formal mathematical proof methods. Here are two examples from SMILE.

Example 1: The learner is asked to investigate the truth of two statements:

You always get a larger number when you multiply two numbers together and You always get a smaller number when you divide one number by another. (From SMILE, card number 2061: Convince yourself. NC level 7)

Example 2: The learner is given that the length of the side of a square is 21cm to the nearest cm. In other words, its length lies between 20.5 and 21.5 cm. and so has range 1 cm. The idea of the range of area for this square is demonstrated and shown to be 42cm2. The learner is then asked the following questions:

1. Find the range of area for a square whose side is 16cm to the nearest cm.
2. Can you find a connection between the length of the side of this square (to the nearest cm) and the range of the area?
3. Prove your result for any square. (From SMILE, card number 2167: Range of area. NC level 8)

Findings from the pilot study

. The most prominent feature of the mathematics scheme in use in the school of the study is that it many of the tasks invite the learner to explore, discover and be convinced of the validity of the discovery. One learner had this to say about these type of tasks:

I remember more that way, because if I convince myself I find that fascinating and it will stick in my mind better.

Open ended investigations also featured in the scheme. Investigations do present many problems for the learner and, indeed, for the teacher (Lerman, 1989); to provide guidance many classrooms are adorned with the following flow chart:

[I have added the symbols B1 to B13]

I worked with 22 pupils from different classes but mainly from Year 10 and had conversations with them about their exploratory/discovery mathematics work; naturally investigations were also discussed. The 22 learners used a range of proof practices in their work. All of them performed crucial experiments to convince themselves of the truth of rules or formulae: they subjected the rule to a number n of tests against untypical cases chosen by themselves. As an illustration I describe Sharon's conversation with me about her work on the traversibility of networks. After examining 16 networks and tabulating the results, Sharon makes the conclusion that if the number of odd nodes is 2 or less then the network is traversible.

Int	- Would the evidence from 3 networks convince you?
Sharon	- No, I wouldn't be convinced..
Int	- Right, what's the minimum number in your mind to convince you?
Sharon	- About 10. Because the more numbers the more evidence to convince yourself.
Int	- What if that is those 10 were not your own networks or drawings, what would you do then?
Sharon	- Yes, I'd still make sure, yes, I'd still do 10 of my own drawings.....

The fact that all learners interviewed used crucial experiments to convince themselves of the validity of the rule does not necessarily indicate a limited perception of the need for justifying. Some were aware of the possibility of pathological cases as the following extract indicates.

Int	- Can you be absolutely sure you always get an even number when you add two odd numbers?
Daryl	- We can't really be sure, there's millions of ways using loads of numbers. Sometimes it might not be what you expect it to be.

Whereas Daryl is able to perceive the role of proof in removing his doubt about the universal validity of a conjecture, Rachel is able to perceive proof as a catalyst for understanding and convincing.

Rachel	- I can't explain why that rule works. I can only explain how I found the rule.
Int	- How important is it to explain why the rule works?
Rachel	- It's quite important, because you can't.....when you know why something happens, it's easier to understand.

In other words Rachel is aware of the second criterion for justification - understanding why the rule is valid. Fifteen of the 22 learners were aware of the necessity for justification, some it must be said were aware for instrumental reasons “The teacher’s told us we’ll get a higher level if we explain why the rule works”. Five learners had actually used proof by generic example in the work that they had done in the current year or in response to problems I put to them. As an example of non-empirical reasoning in classwork, Carla explains how she tackled a Chinese number problem. In constructing the jig-saw of Chinese numbers from 1 to 35, Carla has recognised the structure of Chinese numbers and from this has been able to extrapolate how other numbers such as 63 may be written:

Int	- OK. Let's look at your next task which is about Chinese numbers. (Carla - Yes). Right, now this is a jigsaw puzzle and you're supposed to make it fit together to form this 5´7 rectangle. (Carla -Yes). After you’ve done that you're supposed to write out 35 and 63 in Chinese, is that correct? (Carla -Yes).
Carla	- 35, this is 35.
Int	- Oh, that's 35, the last number is 35 and that's easy to work out.. OK but then 63 is not in the grid (Carla - No) so how did you work out 63?
Carla	- What you have to do is take the first number, so you know the first line is 1 so you get to 6, which you know is here, and you have to take this - this is 10 - so you have to times it by 10 and then plus the 3.
Int	- So, that's 6 times 10 plus the 3.
Carla	- So you have to just put the 6 at the top, then write 10 then put 3.
Int	- ...... That makes sense. (Carla - Yes). Your reasoning convinces you and it certainly convinces me. OK. Both of us seem to be convinced but could you possibly convince anyone else in the classroom?
Carla	- Yes, quite easily.

While the views of Sharon, Daryl, Rachel, Carla will be good news to their teachers, the views of 7 of the 22 learners may not. These learners were unaware of the function and role of justification as defined in box B12 of the flow diagram. They accepted that their rules and formulae were always true because of the crucial test and occasionally because “It just is....” . This may be uncritical factualism or what Fischbein calls an ‘affirmatory intuition’ that the assertion is true. Fischbein gives a good analogy to describe what this is:

If one asks a child, “Can you tell me what a straight line is?” he will try to draw a straight line or he will offer the example of a well stretched thread. He will not feel the need to add something which could clarify the notion (for instance an explanation, a definition , etc.). (Fischbein, 1982, p 10)

‘It just is’ is an intuition that learners may have developed because their own personal experience in mathematics has led them to a false perception of a well behaved body of knowledge where empirically determined rules are always true. Thus mathematics (or investigations at any rate) consists of merely identifying the correct rule. Here are some statements from these learners expressing these perceptions:

I don't know how to explain it, the only thing is to give evidence to prove it, that should speak for itself.

Justify my result in an investigation? I usually just put the rule of it.

I don't need to look at the justifying box (viz B12), but maybe other people do, and they can ask me, can you justify all these things? and I don't know how I'd go about actually explaining it to them unless just giving them more evidence.

Conclusions

I contend that in this particular investigation the opportunities for learners to explain or justify are at the atomic and generic level as indicated above. The non-empirical proof practice that will be firstly accessible to Key Stage 3 and 4 learners is generic proof (Balacheff 1988) where an explanation using mathematical reasoning is made with reference to particular examples: this is precisely what Aiysha has done above. In the investigation an accessible proof opportunity arise immediately: learners are asked to identify the biggest number that cannot be made by sums of positive multiples of 3 and 7. All learners said it was 11 without offering any justification primarily because the guidance flow chart makes no mention of justifying or explaining patterns. However learners could feasibly offer such an explanation (based on the facts that 10 can be made as well as the numbers 12 to 21) and they could do so with each pair of coprime numbers. By making such accessible invitations for proof frequently in flow chart for investigations (and, indeed, in the course of their everyday mathematics) learners may be better equipped to prove their rules and generalisations.

In order that frequent accessible invitations for proof may occur I suggest that the flow chart be amended as follows:

The failure of learners in the ‘postage stamp’ investigation to justify their rule is due to an inability to identify the structure that leads to the rule. By focusing on small manageable and, initially, concrete situations such as those that might arise in the modified flow diagram above it is hoped that learners can be more successful in explaining or justifying. It is unimportant if the situations are ‘trivial’, what is important is that they re-direct and focus the learners thinking towards deductive rather than inductive reasoning, towards critical rather than uncritical factualism. The way that Naomi, a Year 9 learner, convincing herself of a well known conjecture bodes well for possible future algebraic arguments:

Int	- If you are sure (that the sum of every pair of odd numbers is an even number), then you have some kind of explanation or some way of justifying.....
Naomi	- Yes, I do. If you take 1 away from that odd number it will be even, so if you add the 2 numbers left over together that makes an even number and three evens make an even number.

References

Balacheff, N (1988), Aspects of Proof in Pupils’ Practice of School Mathematics, in Pimm, D (ed), Mathematics, Teachers and Children, Hodder and Stoughton. (216-235)
Bell, A W (1976), A Study of Pupils’ Proof-Explanations in Mathematical Situations, Educational Studies in Mathematics, 7, (23-40)
Coe, R and Ruthven, K (1994), Proof Practices and Constructs, British Educational Research Journal, 20, (41-53)
Davis, P J (1993), Visual Theorems, Educational Studies in Mathematics, 24. (333-344)
Department of Education and Science, (1982), Mathematics Counts. Report of the Committee of Inquiry into the Teaching of Mathematics, London, HMSO.
Fischbein, E (1982), Intuition and Proof, For the Learning of Mathematics, 3 (2), (9-18 and 24)
Lerman S, (1989), Investigations: Where to Now? in Ernest, P. Ed., Mathematics Teaching; The State of the Art, Falmer Press, London, 1989.
Porteous, K (1990), What do Children Really Believe?, Educational Studies in Mathematics, 21, (589-598)
Semadeni, Z (1984), Action Proofs in Primary Mathematics Teaching and in Teacher Training, For the Learning of Mathematics, 4 , (32-34)
Simpson, A (1994), Student Attitudes to Proof, plenary presentation to the Advanced Mathematical Thinking Group, PME 18, Lisbon
Van Dormolen, J (1977), Learning to Understand What Giving a Proof Really Means, Educational Studies in Mathematics, 8 , (27-34)

© 1996 The Author