Is argumentation an
obstacle ?
Invitation to a debate...
by
Nicolas Balacheff
Laboratoire Leibniz
Grenoble, France
The first diagnosis of the possible sources of difficulty
in the teaching and learning of mathematical proof to be
proposed was the nature of the most natural didactical
contract emerging from the positions of the student and the
teacher with respect to the knowledge in question. Since the
teacher is the guarantor of the legitimacy and
epistemological validity of what is being constructed, it
follows that the student would be deprived of an authentic
access to a problématique of truth and
proof. Surmounting this difficulty, which is inherent to the
nature of didactical systems, can be looked for in
situations permitting the devolution to the students of
mathematical responsibility for what they produce. Such a
devolution implies the disappearance of the teacher in the
process of decisionmaking in the course of solving a
problem, to be replaced by an effort by the students to
construct autonomous methods of proof.
Argumentation,
a problématique resulting from the study of
social interactions
Social interaction among the students has appeared
clearly as one of the powerful levers for encouraging the
process of devolution to the students of mathematical
responsibility for their activity and their productions. To
the point that certain people have come to regard social
interaction as the most excellent response to the problems
posed. The rhetoric of those who maintain such a position is
based essentially on the idea that relegating the teacher to
the role of guide or organizer of learning will make an
opening, simply by this movement of retreat, for an
authentic construction of knowledge.
I studied such situations, as did other
researchers, in the course of the eighties. The works of
this period seem to me to have 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, and
more generally scientific,
problématique of proof by the students.
These processes and behaviors can be assembled under a
single reference theme, that of argumentation. I cited at
the time, in support of the didactical conjecture according
to which for the students a
problématique of argumentation was in
opposition to a mathematical
problématique of proof, the theses
resulting from the work of Perelman, notably:
"Whereas mathematical proof in its most perfect
form is a series of structures and of forms whose
progression cannot be challenged, argumentation has a
nonconstraining character. It leaves to the author
hesitation, doubt, freedom of choice; even when it
proposes rational solutions, non is guaranteed to carry
the day." (Perelman 1970 p.41)
Even if we don't go as far as a conception of
mathematical proof in its most perfect form, which we do in
taking the point of view of the practice of mathematicians,
there remains a fundamental opposition in the terrain of the
contribution of these two types of discourse to a
problématique of validation. This
opposition, as is frequently forgotten, affects not only the
question of proof, but that of refutation. For example, the
ad hoc treatment of counterexamples, as noted in several
different research experiments, suggests that
counterexamples are seen as objections rather
than as refutations indicating a contradiction.
Argumentation,
a problématique resulting from the study
of verbal productions
The relationship between argumentation and mathematical
proof has been an object of past study from a linguistic and
cognitive perspective. There, it is a matter of exploring
the cognitive complexity of each type, the relationship to
knowledge that it involves or favors, basing the study on
the analysis of the text and the uses of language. To
situate the problématique of such
approaches, taking for my own use a formulation of
JeanBlaise Grize: arguing is doubtless a finalized
activity, but it is a discursive activity (discourse being
understood, whatever its subject, to be a social
activity.)
Argumentation and mathematical proof are
distinguished less by the types of the corresponding texts 
Raymond Duval emphasizes that the discursive distance
between them is tiny  than by the status and functioning of
the statements, and thus in the end that of the knowledge in
play. Argumentation, since its functioning appears to emerge
naturally from the common practices of discourse, would not
permit the identification of the modification of the status
and of the functioning of the knowledge required by
mathematical work, and in return the modification of the
discourse itself.
The examination of the relationship of
argumentation and mathematical proof in this approach, which
is centered on the analysis of discourse, seems to me to
support the conjecture of a conflicting relationship between
the two genres when viewed from the perspective of the
learning of mathematics. Raymond Duval concluded of it that
"the development of argumentation, even in its most
elaborated forms, does not open the way to mathematical
proof. A specific and independent learning process is
necessary where deductive reasoning is concerned" (here
using, in my opinion in a faulty way, deductive reasoning as
synonymous with mathematical proof). His conclusion is that
mathematical proof involves "specific and independent"
learning.
Nonetheless, the naturalistic study of interactions in
the class, as conducted by, for example, Paul Cobb and his
group, suggests the possibility of a mathematical
argumentation to which the students have access by the
practice of discussions ruled by sociomathematical norms
which would emerge from the interactions between teacher and
student (the teacher being regarded as a representative of
the mathematical community.) In such an approach,
construction of a mathematical rationality (mathematical
proof is not actually taught as such) and argumentation are
tightly connected.
Different theoretical conceptions
of argumentation
The diversity which we may observe among the
problématiques of argumentation and of
its relations with mathematics, notably with mathematical
proof, is in my view basically due to profound differences
in the theoretical research in this domain. I will not make
an analysis here of the different
problématiques of argumentation, but
making use of the synthesis proposed by Christian Plantin in
his Essais sur l'argumentation [Essays on
argumentation], I shall attempt to give an idea of the
importance of taking this diversity into account. Three
authors, by the contrast of their
problématiques and their distance, can be
used to provide a system of benchmarks by reference to which
one can situate works on argumentation: Chaïm Perelman,
Stephen Toulmin and Oswald Ducrot.
Following Perelman, one considers argument
to be characterized less by the mastering of its object than
of the listener. It is finalized less by the establishment
of the validity of a statement than by its capacity to
convince a listener. Taking up the formulation of Plantin, a
statement in this conception has a value of reason, or even
truth, from the moment that an individual accepts it.
Toulmin, on the other hand, relates the
validity of a statement first to that of the structure (the
rationality) of the discourse which defends it and thus
makes this validity depend fundamentally on the premises
within a community (of a domain) of reference from the
moment that this community agrees to these rules.
"[An argument] is the exposition of a
controversial thesis, the examination of its
consequences, the exchange of proofs and of good reasons
supporting it and a well or ill established closure."
Independently of the domains, argumentative discourse is
organized in a compound way which permits passage from the
givens to a conclusion under the generally implicit control
of a "license to infer" (this schema can be augmented by
indicators of strength or restriction making it possible to
take into account a possible uncertainty about the
inference.)
Ducrot places argumentation at the heart
of the activity of discourse. As Plantin emphasizes, in this
problématique "one cannot not argue." The
structure of the sequence of arguments plays a determining
role: the strength of an argument comes neither from
"natural" characteristics nor from rational characteristics,
but from its location in the statement. It is by the
structure that one signifies, that one shows an orientation
which makes it possible to receive "R as the intentional
goal of P", or "R as possibly following from P." The
analysis of conjunctions (connecting words) has a particular
importance for Ducrot because it is they which make the
information contained in a text subject to its global
argumentative intention. The polyphony of the connectors, in
the end, makes it possible to represent in the discourse not
only the speaker but also his potential protagonist. "P but
Q" suggests a subject holding with P to which the speaker
objects Q.
We note that the reference to one or another of these
conceptions of argumentation is likely to make us adopt a
different position with regard to what the argument can
represent in the practice of mathematics, notably with the
intent of teaching and in relationship with mathematical
proof. Following Toulmin, it seems possible to envisage a
solultion of continuity from argumentation to mathematical
proof, and why not consider mathematical proof as a
particular argumentative genre (I do not know whether
Toulmin marked the opposition between argumentation and
mathematical proof that the other approaches emphasize.) On
the other hand, the existence of such a solution appears
doubful if one works from within the framework proposed by
Perelman and Ducrot.
The risks of recognizing a
"mathematical argumentation"
My position at this point in my reflections leads me to
consider that in argumentation there is a double activity of
persuasion and validation. Though one may doubt it in
certain disputes where good faith is not a requirement, one
can, on the other hand, make it a hypothesis in the
scientific perspective excluding trickery and lying (the
ideal position without which our objective loses all
sense.)
 Argumentation seeks to convince a listener, but does
that mean that with Perelman we reduce it to doing just
that?
 Argumentation introduces an object, the validity of a
statement. But 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 that measure, I would like
to say that the theoretical frameworks of Toulmin and
Ducrot, in a fashion nonetheless less radical than
Perelman, still give a central place to social
interactions and relations (though Ducrot might well
protest that point.) Now, if we postulate the sincerity
of the speakers 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 whatever the problématique
that one wishes to associate with it.
I would maintain thus that there is no mathematical
argumentation in the frequently suggested sense of an
argumentative practice in mathematics which is characterized
by the fact that it escapes certain of the constraints
present for mathematical proof. This does not mean that all
discourse in mathematics aimed at establishing the validity
of a statement has always had and can always have the
characteristics of a proof. It is a richness of latin
languages which permits us to make a distinction between
"preuve" and "démonstration" [which, lacking that
richness, we have translated respectively as "proof" and
"mathematical proof"! (V.W.)] imposing on the former the
requirements linked to the construction of an
oeuvre of knowledge, without submission to the
requirements of form of the latter.
If there is no such thing as mathematical
argumentation, there does nonetheless exist argumentation in
mathematics. The resolution of problems, in which I would
like to say that there are no holds barred, 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. I will give a capsule description of the place
I think possible for argumentation in mathematics, using the
sense of the concept of cognitive unity of theorems coined
by our Italian colleagues:
Argumentation is to a
conjecture what mathematical proof is to a
theorem.
A consequence that some may consider catastrophic is
that, as it is the case for problemsolving, argumentation
flatly refuses any attempt to be taught directly (I am not,
of course, confusing the teaching of argumentation with the
teaching of rhetoric.)
Argumentation,
epistemological obstacle to the teaching of mathematical
proof
At the conclusion of this short essay, from the
perspective of teaching and learning, I arrive at supporting
neither the thesis of continuity nor that of a rupture
between argumentation and mathematical proof (or proof in
mathematics), but at proposing the recognition of the
existence of a relationship which is complex and is part of
the meaning of both: argumentation constitutes an
epistemological obstacle to the learning of mathematical
proof, and more generally of proof in mathematics.
To understand mathematical proof is first
to construct a particular relationship with knowledge as the
goal of a theoretical construction, and then to give up the
freedom which one could give oneself as a person in the play
of an argument. Because this movement towards mathematical
rationality can only be accomplished by becoming effectively
aware of the nature of validation in this discipline, it
will provoke the double construction of argumentation and of
mathematical proof. Argumentation in common practice is
spontaneous, as is emphasized by those who study discourse.
Forged in familiar exchanges, in the playground, in multiple
and frequently insignificant circumstances, a student's
argumentative competence is in the image of familiar
practices: it goes on its own. 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, by a sort
of Jourdain effect, that they are capable of producing a
mathematical proof when all they have done is argue.
Finally, and this has to do with a point which I have not
approached but which must not be forgotten, the major point
separating argumentation and mathematical proof is the
necessity of the latter to exist relative to an explicit
axiom system. Possibly because of bad memories left by the
New Math, the idea of linking mathematical proof and axiom
systems seems often to provoke anxiety, if not downright
opposition, and yet: in the long run, can one do otherwise
without reducing mathematical proof to a particular rhetoric
or mathematics to a language game?
Some readings
Ducrot O. (1980) Les échelles
argumentatives. Paris : les éditions de
Minuit.
Duval R. (1991) Structure du raisonnement déductif
et apprentissage de la démonstration. Educational
Studies in Mathematics 22(3) 233261.
Duval R. (1992) Argumenter, démontrer, expliquer :
continuité ou rupture cognitive ? Petit
X 31, 3761.
Perelman Ch. (1970) Le champ de l'argumentation.
Bruxelles : Presses Universitaires.
Plantin C. (1990) Essais sur l'argumentation.
Paris : Editions Kimé.
Yackel E, Cobb P. (1996) Sociomathematical norms,
argumentation, and autonomy in mathematics. Journal for
Research in Mathematics Education 27(4)
458477.
Reactions?
Remarks?
The reactions to the contribution of Nicolas Balacheff
will be
published in the July/August 99 Proof Newsletter
©
N. Balacheff 1999
Free
translation Virginia Warfield
