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Almeida D.
(1996)
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 IntroductionMathematics 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: ìIt 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.î (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
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?
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 ëextra attitudeí 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 naÔve 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 naÔve 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 naÔve empiricism rank ëvisualí 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 ëlie in a lineí 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 ëtheoremí in a sense that is wide enough to include the visual aspects of mathematical intuition and reasoningî (Davis 1993, p 333) Visual proofs are connected to but not the same as Semadeniís action proofs (Semadeni 1984). Precisely, action proofs begin with concrete/physical actions to convince the learner of the truth of a proposition (proving Pythagorasí 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:
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.
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.
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.
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 ìThe teacherís told us weíll get a higher level if we explain why the rule worksî. 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:
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 ìIt just is....î . This may be uncritical factualism or what Fischbein calls an ëaffirmatory intuitioní that the assertion is true. Fischbein gives a good analogy to describe what this is: If one asks a child, ìCan you tell me what a straight line is?î 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) ëIt just isí 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. ConclusionsWhile it was clear that the majority of the learners in the study were aware of the need for non-empirical methods of proving it was just also clear that the majority used empirical methods in establishing their results. There is some evidence from the study that this may be due to the frequency and incidence of invitations to provide proofs and explanations. In general it appears that learners are invited to justify at the end of rather than during a exploratory/investigatory task learners; this may have the effect of springing a surprise on the learner and/or preventing vital preparation for presenting the argument or justification. I illustrate this point by using the extreme example of the investigation commonly known as the ëpostage stampí problem where the learner firstly has to identify the largest number that cannot be made by sums of positive multiples of a particular pair of numbers beginning with the pair 3 and 7. All 18 of the learners whose attempts I examined correctly conjectured the formula, N = xy - x -y, which identifies the largest number N that cannot be made from numbers x and y. None of the learners had attempted to justify the conjecture in any way. However, four learners explicitly stated that the formula is valid only when x and y are coprime and these learners, co-incidentally, were the only ones whose work contained what could be described as mathematical reasoning. One of them, Aiysha, had this to say about the conditions for the formula to work:
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 ëpostage stampí 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 ëtrivialí, 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:
ReferencesBalacheff, N (1988), Aspects of Proof in Pupilsí
Practice of School Mathematics, in Pimm, D (ed),
Mathematics, Teachers and Children, Hodder and
Stoughton. (216-235) |