Intuition and Proof:
Reflecting on Fischbein's paper
by
Maria Alessandra Mariotti
Dipartimento di Matematica
Università di Pisa - Italia
Some years ago, in an article entitled "Intuition and
Proof", E. Fischbein (1982) presented the results of a
research project concerning the theme of proof within the
general framework of his reflections on intuitive reasoning.
In the spirit of that discussion, I will develop some ideas
on the same theme.
A basic legacy that is left us by Efraim Fischbein, is
his original approach to educational problems centred on the
complex notion of intuition. The synthesis of this approach
is contained in his book "Intuition in Science and
Mathematics" (1987), where a "theory of intuition" is
sketched and offered to the community of researchers as a
useful tool for the interpretation of educational
phenomena.
In the same way that it is impossible to conceive of a
theory emptied of intuitive meaning, so one cannot conceive
of mathematics deprived of its theoretical organisation:
axioms, definitions and theorems constitute mathematics as
much as its ideas and models. But theory and intuition may
be distant and conflicting poles difficult to reconcile.
Yet, sometimes, contradictory conceptions merge into new
compromise conceptions, a classic example of which being is
the notion of infinity (Fischbein et al., 1979): the dynamic
representation of infinity can be considered as a compromise
between the finite structure of intellectual schema and the
formal infinity itself (Fischbein 1987, p. 205). But not
always compromises are successful, and rather than
compromises harmonisation may have to be sought.
The need for harmonising intuition and
mathematical notions constitutes a basic issue of education
and a contribution to this difficult task comes from those
studies which focus on conflicts and discrepancies and seek
to identify their origins .
Empirical and formal
approach
Theorems constitute the basic chunks of mathematical
knowledge, as organised in a specific theory, and they can
be considered a particular product of the process of
knowledge acquisition. Besides the direct acquisition of
information, mostly related to factual evidence attained
through experience, human culture has developed a complex
way of obtaining information and knowledge, which is not
direct, but mediated through means such as language, logic
and reasoning. As a consequence of this mediation, the
structural unity between cognition and adaptive reactions is
broken: "Knowledge through reasoning, becomes a relatively
autonomous kind of activity, not directly subordinated to
the adaptive constraints of the behaviour of human beings"
(ibid., p.15). In particular, a crucial differentiation
occurs between empirical verification and logical deduction
with the result that their relationship becomes
problematic.
A comparison between truth evaluation in terms of factual
verification and logical validity in terms of deductive
inferences leads one to consider the effect of a factual
confirmation on the validity of a statement. Of course there
are different attitudes which can be described with
reference to an empirical approach, and to a theoretical
approach: despite the fact that a formal proof confers a
general validity to a mathematical statement, further checks
seem to be desirable in order to confirm that validity
(Fischbein, 1982).
Thus, the discrepancy between empirical
verification -- typical of common behaviour -- and deductive
reasoning - typical of theoretical behaviour, is a source of
difficulties, an obstacle to an appreciation of the sense of
proof.
In school practice, it is very common to
confuse these two points of view which can confuse students,
who see 'examples' as playing a basic role in stating axioms
and "discovering" theorems, only to find them forbidden when
they are asked to prove a statement where one or few
examples are not acceptable as a "proof". And what can be
said about the role of counterexamples where one single
example can invalidate a theorem.
Actually, the relationship, crucial for
mathematics, between empirical truth and logical validity,
is a complex and delicate relationship which must be
developed through out education.
The sense of proof is far from common
sense. Although in history it is possible to find
mathematicians who have felt uneasy with a theorem despite
the acceptance of its proof - Cantor is one of the most
famous - it is generally the case that a feeling of general
validity, is what a mathematician attains whenever a theorem
is proven; but that feeling is new and 'strange' with
respect to the natural attitude of mind.
Intuition and theory
Looking more carefully at the relationship between the
intuitive and the theoretical approach leads one to consider
the problem of proof more globally where the unity among
statement, proof and theory must be recognised (see the
notion of theorem introduced in Mariotti et al. 1997). An
analysis of the relationships between theorems (statement,
proof and theory) and intuition can be undertaken in to two
opposite directions.
- On the one hand, a statement expresses the implicit
relationships between the principles assumed in the
theory, and the thesis of the theorem, under the
conditions stated by the hypotheses. Making these
relationships which are implicit at the intuitive level
(Fischbein, 1987, p. 50) explicit, constitutes the first
step towards the construction of an argumentation, which,
in the framework of a theory, can become a proof.
- On the other hand, a theorem represents a piece of
knowledge and as such must be appropriated by the
learner; in other words, in order that it can be used in
productive reasoning, a theorem should acquire the status
of an intuition. But this can only occur if the unity or
fusion between statement and proof, previously
artificially separated, is restored. Statement and proof
must condense into an intuitive knowledge (Fischbein,
1982). In other words the unity between statement and
proof claims not to be broken: the process of analysis
which led to the proof, has to be recomposed into a
single chunk to acquire that immediacy which makes it
productive.
To summarise, as far as theorems are concerned, intuition
is differently involved at the level of both the statement
and its proof:
- the truth of a statement;
- the structure of the proof: the necessity of a logic as
relationship between (the logical articulation of) the
single steps of the proof;
- the validity (generality) of the statement as a
necessity imposed by the proof.
The articulation between the first and the second level
represents a crucial point in the elaboration of a proof:
uncertainty may trigger the exploration of motivations and
start a process of argumentation.
The second level is the junction between
the first and the third level; in fact, grasping the logical
structure of a proof corresponds to inserting the statement
within a coherent framework of intuitions, that can
guarantee its evidence, necessity and complete
acceptability. It will reach the status of "cognitive
belief" (Fischbein, 1982, p. 11). Finally, it allows a
theorem in its unity of statement and proof, to condense
into a new intuition and to become a productive intellectual
instrument.
" ... The logical form of necessity which
characterises the strictly deductive concatenation of a
mathematical proof can be joined by an internal
structural form of necessity which is characteristic of
an intuitive acceptance." (Fischbein 1982, p. 15)
Interesting to remark that the description of a similar
process can be found in Descartes:
Hoc enim fit interdum per tam
longum consequentiarum contextum, ut, cum ad illas
devenimus, non facile recodermur totius itineris, quod
nos eo usque perduxit; ideoque memoriae infirmitati
continuo quodam cogitationis motu succurrendum esse
dicimus. [...] Quamobrem illas continuo quodam
imaginationis motu singula intuentis simul et ad alia
transeuntis aliquoties percurram, donec a prima ad
ultimam tam celeriter transire didicerim, ut fere nullas
memoriae partes relinquendo; rem totam simul videar
intueri. (Descartes, Regula VII)
[TRAD.]
Implication at the didactic
level
The sense of proof may definitely contrast with the
common behaviour towards the acceptability of a statement
based on factual verification. School practice seems to
neglect or at least undervalue the difficulties related to
the discrepancy between a practical and a theoretical
behaviour, that explains most of the failures of traditional
teaching.
Traditionally at school, students learn theorems which
others have produced and only very late in their school
life, by imitating the products that they learnt, they might
be required to produce a theorem. But confining school
practice to repeating proofs that others have produced, and
doing this moreover for statements that are self-evident and
do not appear to need any justification, is likely to be
useless if students are to construct the complex
relationship between the intuitive and theoretical
attitude.
Students may not develop a correct mental
attitude towards theorems -- they may follow their common
sense and ask for supplementary examples to corroborate
their confidence, so as accept the possibility of
exceptions. These results as reported by Fischbein (1982)
have been confirmed more than once.
Besides the possible discrepancies between the
theoretical and the intuitive approach, intuition can
constitute an obstacle: when the immediacy of a statement
inhibits the process of analysis of the implicit links and
thus the construction of the analytic structure that
constitutes a proof. In this case, it becomes impossible to
understand the meaning of proof because the self-evidence,
immediacy and the feeling of certitude that characterises
intuitive statements can inhibit any kind of argumentation,
i.e. the elaboration of the analytic structure, "step by
step", which constitutes a proof. The process is blocked and
so is the path to proof.
A suggestion immediately follows, which is that pupils'
introduction to theorems would benefit by facing situations
where there are no self-evident solutions.
The basic point concerns the process of
production of theorems, so in this case a comparison with
the common practice in the mathematicians community is
illuminating. A mathematician has direct experience of
producing theorems and can always profit from this
experience when he/she relates to any theorem, in contrast
to students who do not have such opportunities.
Recent results (Boero et al., 1996, Bartolini, in press)
confirm that open-ended problems are very suitable in the
early approach to theorems. Open-ended situations may
generate a feeling of uncertainty which calls for indirect
means for getting knowledge, in particular, those problems
which require the production of a conjecture. Moreover, the
process of production of the conjecture is essential for the
introduction of pupils to argumentation. But, engaging in an
argumentation is not enough (Balacheff, 1987; Duval,
1992-93); the unity among statement, proof and theory shall
not be broken, requiring the construction of the complex
relationship between stated principles and consequences
(Mariotti et al. 1997). The preservation of this unity
maintains the link with the intuitive level, the basic
condition for the autonomous production of theorems, and the
productive use of theorems in mathematical reasoning.
Here again a traditional school practice must be
criticised. When experience is confined to "ready-made"
theorems (formulated and proven by others) the link between
a theorem and its intuitive counterpart can be
underestimated and finally neglected. Of course, from the
point of view of formal logic, any theorem is completely
independent of its interpretation, so that it can loose any
link with intuition. But this cannot be the perspective of
education.
Generally speaking, the main point here is how to
overcome conflicts and construct a correct relation between
intuition and theoretical attitude, i.e. a complementarity
between different forms of knowledge, the intuitive and the
formal, so distant may be, with the aim of making them two
aspects of the same mental behaviour.
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Fischbein taught us to look carefully at
conflicts, incongruent phenomena, in order to
detect deep reasons, which can indicate how to
overcome the obstacles. Mathematical education aims
to harmonise intuition and theory, but keeping in
mind the possible obstacles: there is nothing more
dangerous for mathematics learning than neglecting
the deep discrepancies between spontaneous
thinking, sometimes common sense, and mathematical
thinking.
Maybe the case of proof, is
exemplary, although it is not the only one -- ; in
fact, definitions present similar problems
(Mariotti & Fischbein, 1997). Actually, proving
is an activity characteristic of doing mathematics,
but also an activity which differentiates
substantially mathematics from common thinking and
real life practice.
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Reactions?
Remarks?
The reactions to the contribution of Maria Alessandra
Mariotti will be
published in the January/February 99 Proof Newsletter
©
M. A. Mariotti 1998
Edited
by Celia Hoyles
References
Balacheff, N. (1987) Processus de preuve
et situations de validation, Ed.St. Math.18, 147-76
Bartolini Bussi M., Boni M., Ferri F. & Garuti R. (in
press), Early Approach To Theoretical Thinking: Gears In
Primary School. Ed. St. Math.
Boero, P., Garuti, R. & Mariotti, M.A.(1996) Some
dynamic mental processes underlying producing and proving
conjectures, Proc. of PME-XX, Valencia
Descartes, R. (1701/1964). Rules for the direction of the
mind. In R. Descartes, Philosophical essays (pp.145-236; L.
Lafleur, Trans.). Indianápolis: Bobbs-Merrill.
Duval, R. (1992-93) Argumenter, demontrer, expliquer:
continuité ou rupture cognitive?, Petit x , n°
31, 37-61.
Fischbein, E. (1982) Intuition and proof; For the learning
of mathemarics 3 (2), Nov., 8-24.
Fischbein, E. (1983) Intution and analitical thinking in
Mathematics Education, Z.D.M.2, 68-74.
Fischbein, E. (1987) Intuition in science and mathematics,
Dordrecht: Kluwer
Fischbein, E., Tirosh, D. & Melamed, U. (1979) Intution
of infinity, Ed.St.Math.10, 3-40.
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
Mariotti M.A. & E. Fischbein, (1997) Defining in
classroom activities,Ed.St.Math., 34, 219-248
Mariotti M.A., Bartolini Bussi, M., Boero P., Ferri F.,
& Garuti R. (1997) Approaching geometry theorems in
contexts: from history and epistemology to cognition, Proc.
of PME-XXI, Lathi, pp. I- 180-95.
Nota
bene
"For this [admitting of truths which
are not immediate consequences of first and self-evident
principles] may sometimes be accomplished through such a
long chain of inferences that when we have arrived at the
conclusions we do not easily remember the whole procedure
which led us to them; and thus we say that we must come to
the assistance of our weak memory by means of a certain
continuous process of thought…. Because of this, I have
learned to consider each of these steps by a certain
continuous process of the imagination, thinking of one step
and at the same time passing on to others. Thus I go from
first to last so quickly that by entrusting almost no parts
of the process to the memory, I seem to grasp the whole
series at once." [BACK]
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