Douek N. (1999) Argumentative aspects of proving of some undergraduate mathematics students' performances. PME XXIII, Haifa, Israel. Volume 2, pp. 273-280.
	Abstract This paper examines conjecturing and proving in mathematics through analysis of texts written by undergraduate mathematics students. These students reported their reasonings while trying to generalize a property concerning natural numbers and prove the generalized property. Important reference knowledge remained implicit, and non-standardised, appropriate representation of explicit reference knowledge played an important role in students' performances. Referring to semantically rooted arguments was crucial for many students. Subordinating the proving process to the requirements of proof as a final product had negative consequences for some students.
© Nadia Douek

1.Introduction During this century, the specificity of mathematical proof has frequently been an object of heated debate among mathematicians and philosophers. Particularly, keeping mathematical proof (and, more generally, mathematics) free from recours to "meaning" has been upheld as a possibility or even a necessity by someone (see Whitehead, 1925: 'Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about'). By contrast, others opposed it as an illusion or even a danger (see Hardy, 1929:"A formal proof is a kind of X-ray picture of an actual or possible piece of reasoning, revealing the bones [the form] but making the flesh [the content, the meaning] invisible."). Cognitive aspects of mathematical proof were not so extensively investigated. And in mathematics education it was only in the eigthies that a systematic effort was made to establish links between epistemological, cognitive and educational perspectives while tackling the specificity of mathematical proof in relationship with argumentation (see Balacheff, 1988; Hanna, 1989; Duval, 1991).    The study reported in this paper is part of a personal research project concerning the comparison between argumentation and mathematical proof and its implications for teaching. In Douek (1998) I sought to outline some possible guidelines for this kind of investigation, mainly considering a modern-day mathematician's reflection about his own work (Thurston, 1994) and Duval's analysis of the cognitive functioning of formal mathematical proof (Duval, 1991). In doing so, I considered the distinction between ordinary mathematical proof and formal mathematical proof (i.e. proof reduced to a logical calculation); and the distinction between the process of proof construction and its product (the final text of proof- see section 2. for more details).I sought to support the following position: In spite of the undeniable epistemological and cognitive distance between ordinary argumentation and formal mathematical proof, argumentation and ordinary mathematical proof have many aspects in common, both as processes and as products.    In particular, I sought to show analogies between argumentation and ordinary mathematical proof, especially as concerns the use of both implicit and explicit reference knowledge, its dependence on social (and historically evolutive) constraints, and the need for semantically rooted arguments. Concerning the processes (arguing and proving), I sought to show how both are generally built up through 'transformational reasonings' (Simon, 1996) and heuristics. The analyses were mainly based on "evidence" from the history of mathematics, mathematicians' testimony or from what usually happens in school. The aim of the research reported in this paper is to analyse in depth the mathematical activity of conjecturing and proving by exploiting a corpus of texts written by Italian undergraduate mathematics students; they wrote their reasonings while trying to generalize a property concerning the system of natural numbers and then prove the generalized property. In particular I will try to seek for the ways students exploited and represented their mathematical knowledge. 2. Theoretical Background In this paper I will analyse students' protocols concerning production of conjectures and construction of proofs in an open-ended problem; in addition I will study how students exploit their mathematical and meta-mathematical knowledge in this activity. For these purposes the theoretical construct of Theorem, by M. A. Mariotti, seems to be appropriate. According to her (see Bartolini et al., 1997) a "theorem" is a statement, its proof and the reference theory - distinguishing between axioms, definitions and theorems of the specific theory in play, on the one hand, and general meta-knowledge about proving and theorems, on the other. In the same perspective I will consider "Cognitive unity of theorems": this theoretical construct of Garuti's (Garuti et al., 1998) concerns the links that exist between the activity of conjecturing (especially as concerns the production of arguments for the plausibility of the conjecture) and the activity of proving.    I will consider argumentative aspects of proving. We cannot accept any discourse as an argumentation. Henceforth in this paper, the word argumentation will indicate two things: the process that produces a logically connected (but not necessarily deductive) discourse about a given subject (from the Webster Dictionary: " 1. The act of forming reasons, making inductions, drawing conclusions, and applying them to the case under discussion"); and the text produced through that process (Webster: " 3. Writing or speaking that argues"). On each occasion, the linguistic context will allow the reader to select the appropriate meaning. The word "argument" will be used as "A reason or reasons offered for or against a proposition, opinion or measure" (Webster), and may include verbal arguments, numerical data, drawings, etc. So, an "argumentation" consists of one or more logically connected "arguments".    Argumentation is frequently opposed to formal proof, i.e. a proof reduced to a logical calculation. According to Duval (1991), in argumentatative reasoning, "semantic content of propositions is crucial", while in deductive reasoning "propositions do not intervene directly by their content, but by their operational status" (defined as "their role in the functioning of inference").    But what are the relationships between formal proof and what has been in the past and is today recognized as mathematical proof by people working in the mathematical field (for this reason, I will refer to it as "ordinary mathematical proof")? My research work has been strongly influenced by the position of Thurston (1994): "We should recognize that the humanly understandable and humanly checkable proofs that we actually do are what is most important to us, and that they are quite different from formal proof. For the present, formal proofs are out of reach and mostly irrelevant: we have good human processes for checking mathematical validity." In the analysis of students' protocols I will distinguish between the process of proof construction (i.e. "proving") and its product (as a socially acceptable mathematical text): for a discussion, see Douek (1998, Section 4). We may remark that ordinary mathematical proof can be considered as a particular case of argumentation.    Argumentation and proof use references, and I will analyse how students do it. The expression reference knowledge will include not only reference statements but also visual evidence, etc. assumed to be unquestionable (i. e. "reference arguments", or, briefly, "references", in general). In Douek (1998, Section 4.1.) I have discussed the necessary existence (in ordinary argumentation as well as in ordinary mathematical proof) of references which are not made explicit. 3. Method 3.1. The educational context I study written production of conjectures and their proofs in a task related to elementary number theory. The output in question was produced by 43 university students over four consecutive years (from 1995 to 1998) while completing their undergraduate studies in mathematics at the Genoa University. At this level, the students are capable of mastering the mathematical knowledge and the rules of algebraic calculation they must deal with. They are following a mathematics education course and work under a contract (explicitly established with their teacher) that requires them to write down every idea that come to them during their work, even if they change their mind about its validity or its usefulness. This contract is intented to obtain productions regularly for use by the whole group for didactical and cognitive analyses of problem solving activities. 3.2. The task The students were to generalise a proposition ("The sum of two consecutive odd numers is divisible by four"), then prove the generalised proposition. The fact that they had to build up their own conjectures makes their work very different from ordinary school proving, where students have to gather arguments to support a proposition they might never have thought of before. In our case we may suppose that the act of forming a conjecture fixes the conjecture very firmly in their minds, and the proof can be strongly influenced by the steps that led to the insight of the conjecture (see Garuti et al., 1998: "cognitive unity of theorems"). 3.3. Modes and criteria of analysis of students' performances I considered 14 texts (by the 1997/98 students) in particular detail, and then checked analogies and possible differences with the whole set of 43 texts. Reference will only be made to the 14 texts analysed in detail, but the aspects described are recurrent in the other texts as well. Some excerpts from two texts (by Students [1] and [2]), chosen as representatives of opposite behaviours, are reported(see Annex).    Bearing in mind the aim of this study and the theoretical framework, each text has been analysed according to the following modes and criteria: A) overall account of student's conjecturing and proving (global effectiveness of their performance, etc.); B) implicit and explicit reference knowledge backing students' argumentation. I distinguished (see Bartolini et al, 1997: "theorems") between: - content reference knowledge; - meta-knowledge about the operations that the task called for (generalising, etc). I also analysed the external representation of explicit reference knowledge. Concerning this issue, our attention focused particularly on personal (verbal, schematic, etc) expressions that would be unusual in a normally acceptable written mathematical production.This kind of analysis was needed in order to explore in depth how these undergraduate mathematics students used their knowledge; C) occurrence of algebraic-syntactic or semantically based steps of reasoning and the relationships between them. This analysis was needed in order to understand better how the two kinds of reasoning are functionally interlinked and connected to the solution of the problem. D) relationships between the proving process and the proof as a product (and the consequences of matching the former to the latter). 4. Students' behaviour 4.1. Overall account of students' work Within the 14 texts, only four (Students [1], [2], [11], [13]) tried to prove something distinctly: two (Students [1] and [11]) prove their conjectures; and Student [13] a partial result of a confused conjecture. Student [2] (see Doc. 2) tries to prove a result that is stronger than the conjecture expressed in words; his proof lacks a fundamental step (justification of the formula used, which derived by generalisation from numerical examples). Let us call these four students the "proof group". But as we can hardly distinguish the processes of construction of conjectures from construction of proofs in the work of the students, we may as well study more texts from the perspective of proof construction. Another important argument to support this shift in the study from proof to conjecture construction is that five students do not achieve their proofs (even though they were on the right track) probably because of a lack of active mathematical practice combined with the unusual situation of having to build their own conjectures. So we can consider the constructive work of nine students (we may call "conjecture group", which includes the "proof group") and take, as comparative examples, elements of the work of the other five ("failure group"). 4.2. Reference knowledge and its representation The task called for elementary content reference knowledge: elementary arithmetics, algebraic language and its rules of calculation. Some students tried to use other reference knowledge such as functions and series. Concerning algebra, we may remark that the process of formalisation (i.e. the passage from content to formula) was not easy for many students, especially when they wanted to write the sum of K odds: for instance, some of them wrote (2n+1) + (2n+3) + ...+ (2n+?) and then stopped; few were able to express ? as 2K-1: see (E) in Doc.1. Writing the result of the sum was not easy either: it demanded a semantically rooted conversion of a known formula (the formula for the sum of the first n natural numbers - cf. Szeredi & Torok, 1998), or the re-construction of an ad-hoc formula: see (F) in Doc. 1    As concerns the external representation of content reference knowledge, I have found many organisations of data and schemas with visual effects that reveal regularities and help to express algebraically some arithmetic relations; we also found symmetries in the disposition of data and formulas, which provide hints for the calculus (see figures in Doc. [1] for two examples).    We may remark that these behaviours are related to knowledge which is not always recognised as an important tool for solving problems, though it is itself constructed knowledge (cf. Briand,1993, for similar remarks concerning counting strategies). We may also remark that in other fields of mathematics (such as numerical analysis or category theory) schemas and organisational schemes are crucial tools.    Meta-mathematical knowledge was made explicit especially when it was almost algorithmic (see Student 2) or referred to the task ("What does it mean 'to generalize'"), but appeared only implicitly when it was complex (actually richer) and nearer to the mathematicians' behaviour (see Thurston, 1994). Summing up the analyses performed, I may say that, concerning meta-mathematical knowledge, shared explicitable knowledge was much narrower than the actual knowledge used globally by the group. I found that eight students referred explicitly to methods for solving problems of this kind, but, to take an example, "organisation of data" was never mentioned even in partial explicitations of methods though it was a key strategy for four students and useful for three of them. Only one of the fourteen students (Student [12]) seemed to have no idea of possible strategies for solving problems of this kind: she seemed lost, mixed up different steps undertaken and produced several unfinished propositions. For Students [1] and [13] ("proof group") I detected very rich implicit meta-mathematical knowledge about how to solve the problem.    The implicit problem-solving methods I could detect globally were: change of representation; interpreting calculations in words and vice versa; visually organising data and calculations, up to a geometrical regularity. I could also detect changes of mathematical frames: arithmetic, algebra, series, etc: this is common in the process of proving for mathematicians. 4.3. Algebraic-syntactic or semantically based steps of reasoning I have listed numerous breaks during calculations, which were needed to re-interpret the mathematical content of calculus in words. This can be seen as a sign of the primacy of semantical content over algebraic calculation during the process of conjecture and proof construction. As an example, we can consider the need of Student [1] to express algebraic propositions in words when seeking to recognise possible conjectures. This attitude displays the search for a semantically consistent grasp of the algebraic signs. We can interpret it by saying that constructive work in mathematics cannot evolve only within formal expression.    On the other hand, if we observe the students who did not express the results of their calculation in words richly ( five of the fourteen students), three (Students [3], [4], [12]) are in the "failure group" of five students and two ([2] and [13]) in the "conjecture group" of nine students. So the majority of the "conjecture group" (seven out of nine) needed semantic interpretations to pursue their work. I recall that Student [2] did not recognise the strong result obtained, and that [13] was confused in expressing his conjecture - it was not clear to this student what was proved by the calculation. 4.4. Proof as product and proof as process Let us compare two examples that are representative of some others in the whole group of 43: in the first, proof as a product is close to proof as a process, while in the other the distance is very great.    Student [2] is considered skillful (good notes, etc), but sticks very closely to her explicit method and her presentation is very close to that of a final presentation. This approximation to a formally correct mathematical text (cf Hanna, 1989) seems to bear negative consequences on the productivity of the student's work: her research is linear and no change of strategy is found at any level. There are long repetitive arithmetic calculations, quite astonishing for the only student in the group who usually managed algebraic tools very well; more remarkably, the student arrives algebraically at a strong conjecture and interprets it in words as much weaker. And finally she does not produce a complete proof.    Analysing the text of Student 1, we can observe frequent changes of strategy, organisation of data and calculations, as well as a frequent effort to interpret in words. This variety, this need for change might help technically, but these were also "interpretation" efforts. They helped understanding and often stimulated the developement of new ideas. This could be called a "transformational reasoning attitude" (see Simon, 1996; Harel and Sowder, 1998). Some of these very useful forms disappeared in the final draft of the proof (P), where the logical link between the propositions became a priority. In addition, justification of the research method disappeared from the product (while examples of the interwoven presence of meta-mathematical arguments in mathematical reasoning were frequent in the construction stage). Her conjecture is strong and her proof is almost complete. 5. Conclusions We have seen that important reference knowledge remained implicit in the students' proving processes and that some of the different references concerned the content, while others related to the meta-knowledge about the activity to be performed. We have also seen how non-standardised, appropriate representation of explicit reference knowledge had an important role in the students' performances. We have seen that when elaborating a productive process many students found syntactic arguments insufficient, and so semantically-rooted arguments became critical. Finally, we have collected some experimental evidence about the negative consequences of subordinating the proving process to the requirements of proof as a final product.    As concerns the educational implications of the analysis performed in this paper, it can be argued that formal proof (which is sometimes imposed or proposed to students of any school level as a rule of construction of mathematical proof: see Hanna, 1989) is very distant from the effective activity of conjecturing and proving. This is true even for undergraduate mathematics students facing a new, challenging situation. Furthermore, the effectiveness of their activities seems to depend on intellectual qualities that are fully developed even during ordinary, demanding argumentative activities other than proving. References Balacheff N. (1988) Une étude des processus de preuve en mathématiques, thèse d'état, Grenoble Bartolini Bussi M., Boero,P.; Ferri, F.; Garuti, R. and Mariotti, M.A.: 1997, 'Approaching geometry theorems in contexts', Proceedings of PME-XXI, Lahti, vol.1, pp. 180-195 Briand, J.: 1993, L'énumération dans le mesurage des collections, Thèse LADIST, Un.Bordeaux-I Douek, N.: 1998, 'Some Remarks about Argumentation and Mathematical Proof and their Educational Implications', Proceedings of the CERME-I Conference, Osnabrueck (to appear) Duval, R.: 1991, 'Structure du raisonnement déductif et apprentissage de la démonstration', Educational Studies in Mathematics, 22, 233-261 Garuti, R.; Boero,P. & Lemut, E.: 1998, 'Cognitive Unity of Theorems and Difficulties of Proof', Proceedings of PME-XXII, vol. 2, pp. 345-352 Hanna, G.: 1989, 'More than formal proof', For the Learning of Mathematics, 9, 20-23 Hardy, G.H.: 1929, Mathematical Proof, Cambridge University Press Harel, G. and Sowder, L.: 1998, 'Students' Proof Schemes', in E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research on Collegiate Mathematics Education, Vol.III, pp.234-283, A.M.S. Simon, M.: 1996, 'Beyond Inductive and Deductive Reasoning: The Search for a Sense of Knowing', Educational Studies in Mathematics, 30, 197-210 Szeredi, E. & Torok, J: 1998, 'Some Tools to Compare Students' Performances and Interpret their Difficulties in Algebraic Tasks', Proc. of the CERME-I Conference, Osnabrueck (to appear) Thurston, W.P: 1994, 'On Proof and Progress in Mathematics', Bull. of the A.M.S., 30, 161-177 Whitehead, A.N.: 1925, Science and the Modern World, Cambridge University Press   ANNEXE. DOC. 1 Excerpts from the text of Student [1]; it contains seven large, spatially organized pieces, like the two reported below, and many arrows, connecting lines and encirclings. "I have some difficulties in understanding in what direction I must generalize. It might be: 'by adding two odd or even consecutive numbers I get a number divisible by 4' [she performs some numerical trials]. This does not work. I shall try to generalize in another way: I was looking for something that could help me [...] but I got nothing. [other trials, with a rich spatial organisation: two consecutive even numbers, two consecutive odd numbers - here she gets divisibility by 4; then three, four, five, six, seven consecutive odd numbers. By performing calculations, she gets the following formulas: 3(2K+3); 8(K+2) 10K+25=5(2K+5); 12K+36=12(K+3); 14K+49=7(2K+7)]. Is the result of the addition of n consecutive odd numbers (n odd) divisible by n? (2K+1)+(2K+3)+..(2K+    What must I put here? (E) [she performs an unsuccessful trial by induction; then she considers n numbers in general] n numbers: (2K+1)+(2K+3)+...(2K+(2n-1))=2nK+1+3+5+...+(2n-1)=2nK+ (I am thinking of the anecdote of "young Gauss": (F) it makes 2n.n/2=n2) = 2nK+n2= n(2K+n) OK!! [Trials performed by applying the preceding formula 2nK+n2 in the cases n=2, n=4, n=6, n=8: she gets: 4K+4 divisible by 4; 8K+16 divisible by 8; 12K+36=12(K+3); 16K+64=16(K+4) divisible by 16]. Then if I add n consecutive odd numbers (n even), I get divisibility by 2n. Let us try a proof: (P) (2K+1)+(2K+3)+...(2K+2n-1)=2nK+(1+3+....2n-1)=2nK+(2n.n)/2=2nK+n2 [....]= 2n(K+n/2); n even implies that n/2 is an integer number: so I get divisibility by 2n. [...]       DOC. 2: Excerpts from the text of Student [2]; spatial organization is almost linear, like that in the following trascript. Student [2] starts her work by checking (on numerical examples: 3+5; 5+7; 101+103) the validity of the given property, then proves it. Then she writes: "When I must tackle a problem, I try to see how it works in particular cases and then I generalize, as I have done in this case - although I knew the solution. I reason in this way because the particular case allows me to understand better how I can reach the solution of the problem in general (and this method works even when I do not know the solution). Thinking in arithmetic terms and then in algebraic terms helps me to solve the problem. For the original property the generalization comes fairly automatically, because [she explains why in detail]. What does it mean 'to generalize' ? It means considering a property in which there are some closed variables (two odd numbers, or divisibility by 4) and getting a property in which variables are open. I change the number of odd consecutive numbers to add. For instance, I consider 3 [crossed out] 4 consecutive odd numbers 2n+1, 2n+3, 2n+5, 2n+7 and make the addition: 2n+1+2n+3+2n+5+2n+7=8n+16=8(n+2)=4(2n+4). Then I find a number that is divisible by 8, so it is divisible by 4. I perform the addition of 6 consecutive odd numbers [similar calculations]=12n+36=6(2n+6). Then I find a number which is divisible by 12, so it is divisible by 6. I try with 8: [similar calculations]=8.2n+64=8(2n+8) Then I find a number that is divisible by 8, so it is divisible by 4. Following my reasoning, for an even number K of odd consecutive numbers I get: 2n+1+2n+3+....+2n+15+....=K(2n+K)=2K(n+K/2); but K is an even number, so it is divisible by 2 and (n+K/2) is an integer number. Then 2K is divisible by 4 (because K is odd). So I have found that the given property is still valid if I add up an even number of odd consecutive numbers.