Argumentation and mathematical proof:
A complex, productive, unavoidable relationship
in mathematics and mathematics education

I recognize the importance of N. Balacheff's contribution to the issue dealt with in the last Newsletter on Proof, especially as concerns the discussion of different conceptions about argumentation and its complex links with (mathematical) proving.

I would like to start with some local remarks concerning the coherency between the first and second parts of NB's contribution, and wish to consider two points in particular.
   NB says: "Argumentation in common practices is spontaneous". This statement needs to be related to the specific kind of argumentation. Widely shared experience in Italian classrooms situated in low-level socio-cultural environments shows that while some Perelman-type argumentations spontaneously develop in children, the development of Toulmin-type and Ducrot-type argumentations calls for very strong teacher mediation.
   NB speaks of "freedom that one could give oneself, as a person, in the play of an argument". Once again it seems to me that this comment is inappropriate for Toulmin-type argumentation (and even for Ducrot-type argumentation).

Let us now come to the main issue in the second part of NB's contribution (pages 3 and 4): the role of argumentation in the approach to mathematical proof, particularly the fact that argumentation might be an epistemological obstacle in approaching mathematical proof.
   Here I must say that a significant difference exists between the perspective more or less explicitly indicated by NB and our own perspective ("our" refers to the research group I lead in Genoa). This difference may help to understand why I do not enter the discourse about argumentation as proposed by NB, but focus on other aspects. The difference mainly consists in the fact that, from our perspective, the distinction between "proving" as a process and "proof" as a product is a major factor in discussion about the role of argumentation in mathematical activities concerning theorems. What's more, the nature of these activities is also considered differently.
   According to our perspective, the approach to mathematical proof belongs to a more general cultural and cognitive apprenticeship &endash; i.e. entering the culture of theorems (and mathematical theories). Here I allude to the definition of theorem provided by Bartolini et al (1997) as "statement", "proof" and "reference theory".
   In that framework, entering the culture of theorems means developing specific competencies inherent in producing conjectures and proving the produced conjectures by taking elements of theoretical knowledge into account. Epistemological and cognitive analyses are needed in order to select peculiar, essential elements in the production and proof of conjectures and the management of theories that students will face in their apprenticeship. In this way, entering the culture of theorems will be accessible and meaningful (from the mathematical point of view) for most of them. For instance, the crucial role of dynamic exploration (cf. Boero et al, 1996; see also Simon, 1996) of the problem situation in producing and proving conjectures must be taken into account; this can help in selecting "fields of experience" and tasks where such dynamic exploration is "natural" for students. In addition, the phenomenon of (possible) continuity between the production of a conjecture and the construction of its proof (see "Cognitive Unity of Theorems": Garuti et al, 1996, 1998) must be considered, in order to select appropriate problem situations where this continuity works smoothly. Another crucial issue concerns the fact that theorems (i.e. statements, proofs and theories) belong to scientific culture (in the sense of Vygotsky, "Thought and Language", Chapter VI). Appropriate mediation by the teacher is called for in all those aspects where there is a significant rupture with everyday culture: the shape of statements, the structure of mathematical proofs as texts, the nature of allowed reasonings, the peculiar organization of mathematical theories, etc.

In the framework outlined above, when dealing with the role of argumentation in mathematical activities concerning theorems we must take different aspects of those activities into account. I shall describe them as "phases" in the activities of conjecture production and mathematical proof construction (although they cannot be separated and put into a linear sequence in mathematicians' work - see later):

I) production of a conjecture (including: exploration of the problem situation, identification of "regularities", identification of conditions under which such regularities take place, identification of arguments for the plausibility of the produced conjecture, etc.). This phase belongs to the private side of mathematicians' work. We may remark that the appropriation of a given statement shares some important features with this phase (exploration of the problem situation underlying the statement, identification of arguments for its plausibility, etc.);

II) formulation of the statement according to shared textual conventions (this phase usually leads to a publishable text);

III) exploration of the content (and limits of validity) of the conjecture; heuristic, semantic (or even formal) elaborations about the links between hypotheses and thesis; identification of appropriate arguments for validation, related to the reference theory, and envisaging of possible links amongst them (this phase usually belongs to the private side of mathematicians' work);

IV) selection and enchaining of coherent, theoretical arguments into a deductive chain, frequently under the guidance of analogy or in appropriate, specific cases, etc. (this phase is frequently resumed when mathematicians present their work to colleagues in an informal way &endash; or even in public presentations such as seminars: cf Thurston, 1994);

V) organization of the enchained arguments into a proof that is acceptable according to current mathematical standards. This phase leads to the production of a text for publication. We may observe that mathematical standards for this phase are not absolute &endash; they differ when we compare a paper published today with one from the eighteenth century, or a chapter from a mathematical textbook for high school with one for university level;

VI) approaching a formal proof. This phase may be lacking in mathematicians' theorems (although most of them are aware of the fact that formal proof can be reached and some of them might reach it in some cases). Sometimes this phase concerns only some parts of the proof (where formal treatment is easy, or subtle bugs must be identified). However, Thurston (1994) claims that it is practically impossible (and meaningless for working mathematicians) to produce a completely formal proof for most current theorems in mathematics. He writes: "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."

We may note that these six phases are usually interconnected in non-linear ways in mathematicians' normal work. For instance, in the fifth phase a bug may be discovered in the enchaining of arguments, and this may call for renewed exploration of the problem situation and strengthening of hypotheses (first phase) with a new statement (second phase).
   I would also like to stress the importance of the distinction (which emerges from the preceding description of the six "phases") between the statement of a theorem as a product and conjecturing as a process, and between mathematical proof as a product and (mathematical) proving as a process.

Now let us come back to argumentation. In order to deal with argumentation in mathematical activities, especially in conjecturing and proving, I think that it would be useful to elaborate a specific framework for argumentation. Indeed, both Toulmin's and Ducrot's conceptions should be taken into account, but neither of them seem to be satisfactory for the purpose of dealing with the peculiarities of argumentation in mathematical activities: the problem of reference knowledge is not relevant in Ducrot's conception, while the linguistic structure of the sequence of arguments is not considered in depth by Toulmin. In mathematical activities, both reference knowledge and the structure of the sequence of arguments are relevant.
   The Webster Dictionary hints at a possible, comprehensive framework for argumentation as "The act of forming reasons, making inductions, drawing conclusions, and applying them to the case under discussion" and "Writing or speaking that argues". We may note that this distinction between argumentation as a process and argumentation as a product may help interfacing argumentation as a process with (mathjematical) proving, on the one hand, and argumentation as a product with mathematical proof, on the other (see later). The Webster Dictionary defines "argument" as "A reason or reasons offered for or against a proposition, opinion or measure". This definition could be developed into a comprehensive discourse on "reference knowledge" in arguing (and proving). Douek (1998, 1999) exploits these definitions in order to analyse argumentative aspects of (mathematical) proving. Taking her analyses into account, we may consider multiple roles of argumentation in mathematical activities concerning theorems.
   In the first two phases, argumentation concerns inner (and eventually public) analysis of the problem situation, questioning the validity and meaningfulness of the discovered regularity, refining hypotheses, discussing possible formulation(s). In the third phase, argumentation plays three important roles: producing (or resuming from the first phase &endash; "Cognitive Unity of Theorems", Garuti et al, 1996, 1998) arguments for validation, discussing their acceptability according to requirements about their nature (for instance, although empirical arguments may be relevant in the first phase and even in the approach to validation, they must be progressively excluded from this phase on), and finding possible links leading from one to another. I could add that the nature of the whole third phase is argumentative, and the fourth phase is also largely argumentative (especially as concerns the control of argument enchaining). In the fifth phase, argumentation may play a role when comparing the text under production with current standards of "rigour", textual organisation, etc.

The preceding analysis can help when dealing with the problem of approaching mathematical proof in school. In our opinion, two main problems must be faced:

• the nature of arguments taken into account by students as reliable arguments for validation. Students can use empirical arguments (measurements, etc.), visual evidence, body references, etc.; most of these arguments are useful and even necessary in the first, third and (with a different, specific function) the fourth phases of the activity concerning theorems, but must be rejected from the fourth phase on. However, in the last four phases students should also necessarily refer to "theoretical" arguments belonging to reference theory (these arguments become exclusive in the fifth phase);

• the nature of the reasoning produced by students. Frequently, they find analogies, examples, etc. sufficient in order to be sure of the validity of a statement. While these are very useful and perfectly acceptable in some activities concerning theorems (particularly in the first and in third phases and, with a different function, in the fourth phase), they are no longer acceptable in the fifth phase.

So, when it comes to activities concerning theorems, we may state that there is an important difference between working mathematicians and students: working mathematicians are able to play not only the game of a rich and free argumentation (especially in Phases I and III) but also the game of argumentation under the increasing constraint of the strict rules inherent in the acceptability of final products (especially in Phases II and V); by contrast, students face serious difficulties in learning the rules of the latter game and passing from one game to the other (but we must recognize that they also experience difficulties in free argumentation in mathematics!).
   I feel that both problems must be considered and tackled from the educational point of view.

The nature of arguments (empirical or theoretical, etc) which students refer to not only depends on the culture of theorems developed in the classroom, but also relies strongly on the nature of the task. By their very nature, some tasks induce children to produce and/or exploit empirical arguments (measurements, visual evidence, etc). For instance, the plane geometry tasks that school students are usually set enhance spontaneous recourse to measurements and visual evidence, while appropriate space geometry tasks might prevent it. From these tasks, students could learn (under the teacher's guidance) to exploit arguments belonging to a set of reliable statements ("germ theory") concerning space. An example is presented in Bartolini Bussi (1996): the problem situation concerns a rectangular table with a small ball lying in the center; students have to draw the ball on a perspective drawing of the table and validate their construction by making reference to a "table of invariants" concerning plane representation of space situations. Another example is presented in Boero et al (1996): in this case students have to find out whether (and under what conditions) two non-parallel sticks produce parallel shadows on the ground and validate their solutions by making reference to geometrical properties of sun shadows (particularly, the property by which vertical, parallel sticks produce parallel shadows on the ground).
   As concerns the nature of reasoning, the role of the teacher here becomes even more significant. By making reference to appropriate "models" (or "voices", according to Boero et al, 1997), the teacher should progressively emphasise specific kinds of reasonings. Here again the choice of the task may help: in both of the examples alluded to above, reasoning by examples, considering specific cases, etc. clearly appears to be insufficient to students, and deductively organised reasoning can prove powerful. In such situations, the teacher's task becomes that of helping students to organise the only possible performant reasoning according to some prescriptions and modes defined in the mathematics community.

References

Bartolini Bussi, M. (1996): 'Mathematical Discussion and Perspective Drawing in Primary School', Educational Studies in Mathematics, 31, 11-41

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

Boero, P.; Garuti, R. and Mariotti, M.A.: 1996, 'Some dynamic mental processes underlying producing and proving conjectures', Proceedings of PME-XX, Valencia, vol. 2, pp. 121-128

Boero,P.; Pedemonte, B. & Robotti, E.: 1997, 'Approaching Theoretical Knowledge Through Voices and Echoes: a Vygotskian Perspective', Proc. of PME-XXI, Lahti, vol. 2, pp. 81-88

Douek, N.: 1998, 'Some Remarks about Argumentation and Mathematical Proof and their Educational Implications', Proceedings of the CERME-I Conference, Osnabrueck (to appear)

Douek, N.: 1999, 'Argumentative Aspects of Proving: Analysis of Some Undergraduate Mathematics Students' Performances', Proceedings of PME-XXIII, Haifa (to appear)

Garuti, R.; Boero, P.; Lemut, E.& Mariotti, M. A.:1996, 'Challenging the traditional school approach to theorems: a hypothesis about the cognitive unity of theorems', Proc. of PME-XX, Valencia, vol. 2, pp. 113-120

Garuti, R.; Boero,P. & Lemut, E.: 1998, 'Cognitive Unity of Theorems and Difficulties of Proof', Proceedings of PME-XXII, vol. 2, pp. 345-352

Simon, M.: 1996, 'Beyond Inductive and Deductive Reasoning: The Search for a Sense of Knowing', Educational Studies in Mathematics, 30, 197-210

Thurston, W.P: 1994, 'On Proof and Progress in Mathematics', Bull. of the A.M.S., 30, 161-177

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