Argumentation and
mathematical proof:
A complex, productive, unavoidable relationship
in mathematics and mathematics
education
by
Paolo Boero
Dipartimento di Matematica
Università di Genova
Italia
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 lowlevel
sociocultural environments shows that while some
Perelmantype argumentations spontaneously develop in
children, the development of Toulmintype and Ducrottype
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
Toulmintype argumentation (and even for Ducrottype
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 nonlinear 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 nonparallel 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, 1141
Bartolini Bussi, M.; Boero,P.; Ferri, F.; Garuti, R. and
Mariotti, M.A.: 1997, 'Approaching geometry theorems in
contexts', Proceedings of PMEXXI, Lahti, vol.1, pp.
180195
Boero, P.; Garuti, R. and Mariotti, M.A.: 1996, 'Some
dynamic mental processes underlying producing and proving
conjectures', Proceedings of PMEXX, Valencia, vol. 2, pp.
121128
Boero,P.; Pedemonte, B. & Robotti, E.: 1997,
'Approaching Theoretical Knowledge Through Voices and
Echoes: a Vygotskian Perspective', Proc. of PMEXXI, Lahti,
vol. 2, pp. 8188
Douek, N.: 1998, 'Some Remarks about Argumentation and
Mathematical Proof and their Educational Implications',
Proceedings of the CERMEI Conference, Osnabrueck (to
appear)
Douek, N.: 1999, 'Argumentative Aspects of Proving:
Analysis of Some Undergraduate Mathematics Students'
Performances', Proceedings of PMEXXIII, 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 PMEXX, Valencia, vol. 2, pp.
113120
Garuti, R.; Boero,P. & Lemut, E.: 1998, 'Cognitive
Unity of Theorems and Difficulties of Proof', Proceedings of
PMEXXII, vol. 2, pp. 345352
Simon, M.: 1996, 'Beyond Inductive and Deductive
Reasoning: The Search for a Sense of Knowing', Educational
Studies in Mathematics, 30, 197210
Thurston, W.P: 1994, 'On Proof and Progress in
Mathematics', Bull. of the A.M.S., 30, 161177
Reactions?
Remarks?
The reactions to the contribution of Paolo Boero will
be
published in the September/October 99 Proof Newsletter
©
P. Boero 1999
