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1. Introduction
It is well established that the mathematical proof
process has a central role in the learning of geometry. For
students in learning situation this is a source of
difficulties; recurrent cognitive and epistemological
obstacles are present as is shown by research literature in
the area (Balacheff,1987,1991,1998; Chazan,1983;
Hoyles,1997; Moore,1994) and the learning situations we have
been following (Gravina,1996). One of the obstacles lies in
the necessary transition from the already acquired empirical
knowledge to the knowledge that is established as axiomatic
geometry. The construction of this knowledge demands
cognitive attitudes that are beyond the spontaneous
ones:
"Axioms, definitions, theorems, and proofs have
to penetrate as active components in the reasoning
process. They have to be invented or learned, organized,
checked, and used actively by student. Understanding what
rigor means in a hypothetic-deductive construction, the
feeling of coherence and consistency, the capacity to
think propositionally, independently of practical
constraints, are not spontaneous acquisitions of the
adolescent. In Piagetian theory, all these capabilities
are described as being related to age &endash; the formal
operation period. As a matter of fact, they are no more
then open potentialities that only an adequate
instructional process is able to shape and transform into
active mental realities" (Fischbein , 1994, pg. 232)
In the overcoming of difficulties, inherent to the
learning process in geometry, dynamical environments have
been attested to be powerful resources (Keyton, 1997;
Laborde & Capponi, 1994; Laborde, 1994,1998; Yerushalmy
& Chazan, 1990). They favor the externalization of ideas
and the necessary conflicts for adjusting mathematical and
individual meanings. The research literature shows positive
results mainly in the construction of concepts, that is, the
agreement of mental images to geometric concepts with
consequent dismissal of prototypical images. The literature
also shows positive response of the students towards
explanation when they experience the conjecturing process
that naturally takes place in those environments; it seems
that students are genuinely more motivated to search for
explanations of compelling invariants that emerge under
"drag actions".
Our main concern in this work is about the
students' attitudes that are necessary for proving
geometrical facts. For the comprehension of deductive
reasoning the first step seems to be understanding that
there are geometrical facts following as consequences of
some preliminary assertions. With the main purpose of
promoting this first attitude we developed an experience
based on geometrical constructions. These geometrical
constructions, being under students' control, evidence that
they did assert some constructible facts and that certain
geometrical relations are implicit in their statements.
The teaching
experiment
The experience was carried out with a group of fourteen
students, aged 18-19, following a one-semester course in
plane Euclidean geometry, part of the core courses in the
pre-service program for mathematics teachers, at Federal
University of RGS &endash; Brazil, in the fall 1999. It is
worth saying that Brazilian secondary geometry curriculum
does not reserve a special place for proof process. The
students work out very few proofs and, as a consequence,
when entering the university they do not seem to appreciate
the difference between empirical verifications and deductive
reasoning. It is even common to find students showing a weak
understanding of geometrical objects and confusing
properties that are instances of a drawing with their truly
geometrical properties (Gravina, 1996).
The course started with an overview of the
dynamical environment Cabri-Géomètre, where
the students were introduced to some of the menus and the
drag function. The students worked in pairs and were
initially invited to produce some free constructions.
Surprisingly enough, only few pairs made figures that were
stable under dragging; their productions, in general, were
freehand childish drawings, despite of their knowledge of
Cabri resources.
The experiment took ten two-hour meetings and was divided
into four stages:
Stage 1: After the
brief period of acquaintance with Cabri, the students
were invited to construct some well know polygons (
triangles, squares, parallelograms...). They were asked
to do as many constructions as possible of the same
polygon under the condition of being geometrically
stable.
Stage 2: For each
construction they were asked to write down the
geometrical procedures they used. Here the resource
"Replay Construction" of Cabri was useful in checking the
correctness of description. The didactical goal was to
improve the geometric language and to begin the control
of stated facts.
Stage 3: The activities
were similar to those of stage 2, with an added task: to
identify geometrical facts that visually come out from
the "moving draws", that were not explicit in their
constructions. At the beginning of this stage the
teacher/researcher, in a group discussion, used one of
the constructions made by the students to make the task
clear (see Fig3). The purpose in this
stage was to make students aware of the differences
between the given conditions and implicit consequences,
i.e, between the "if-part" and the "then-part" of a
geometrical statement .
Stage 4: After the
students realized the distinction between the "givens"
and the "consequences" clearly, a new aim was set: to
find reasons that would explain the relationships
detected in their constructions. The students were
challenged towards attitudes of argumentation. Having
found that it was not quite natural for them to grasp the
meaning of hypothetical-deductive argumentation, the
teacher/researcher intentionally made a meta-knowledge
intervention: a discussion about the axiomatic nature of
geometry. Primitive notions and relations, axioms and
definitions were introduced, as well the "game rules" -
the arguments should be based on axioms and properties
(theorems) that had already been obtained through
deductive reasoning. It was stressed that the progressive
construction of knowledge should be in accordance with a
well-defined social-mathematical consensus and, as a
consequence, even visually evident properties should be
explained. With this in mind and the purpose of throwing
light upon the mathematical subject under study, the
teacher/researcher discussed, collectively, the well
known property "a point P lies in the perpendicular
bisector of segment AB if and only if PA=PB", as follows:
the "givens" were constructed, the consequences emerging
without explicit actions were identified and a deductive
argumentation was produced. After that, the students
carried out similar interplay between construction and
proof process.
The students'
production
Based on class observations and collected material (Cabri
files and written papers) we identified in the students'
productions:
• About the
geometrical constructions (stage 1, see Fig1)
:
a) different constructions of polygons
(starting by side, by diagonal, by circle, by center
and vertex, by side midpoints) were produced.
b) particular constructions instead of general ones
showed up (parallelograms with constant proportional
sides, rhombus with angles measures constant,
isosceles triangles with congruent height and
base).
c) some freehand drawing mixed with geometrical
construction were detected.
• About the
description of the construction statements (stage 2, see
Fig 2) :
a) adequate use of geometrical language.
b) cases of inclusion of facts that were not declared
in the construction ("visual facts").
• About the
identification of implicit consequences (stage 3, see
Fig 4 ) :
a) in general, they were well-identified.
b) there were still cases of taking implicit facts as
stated facts.
• About the very
first argumentations (stage 4, see Fig
5, Fig 6 and Fig
7 ) :
a) "givens" and "consequences", in general,
were under adequate control
b) explanation using measures and visual resources
were detected.
c) cases of difficulties for progressing with
argumentation were present.
d) some of the pairs presented well-developed
deductive arguments, but it was quite frequent to
offer correct proof for a property that was not the
one aimed at.
Discussion of the teaching
experiment
The development of tasks in progressive difficulty helped
students in the construction of meaning of the geometrical
statements "if ...then". The conscious thought supporting
the explicit action of selecting menus in a sequential
procedure of construction and the resulting feedback on the
screen favored a progressive control and organization of
declared facts (control of hypothesis); with this control
and the ability of dragging the constructions they realized
that compelling visual invariants were consequences of
stated facts (control of thesis), which could and should be
explained. Distinct construction procedures for the same
polygon showed that an adequate choice of preliminary
assertions was up to them and that "the consequences" to be
explained, in each case, would be different. They were
developing the feeling for producing conjectures.
It is important to notice that in the
initial productions the construction procedure and the
description of its steps did not always match: implicit
properties that did show up visually stable were taken as
given by construction. For instance descriptions like "line
parallel to line r passing trough A and B" instead of the
construction step "line parallel to line r passing trough
A", being subject to argumentation "passing trough B" (see
appendix, fig2). But gradually the students improved this
control and become very confident in identifying sequential
construction and implicit consequences.
In their first argumentations (in the
context of very elementary properties), the students
frequently forgot the "game rules" using measure and visual
resources to validate properties. For instance, for some of
them it was quite natural to argument that "if a circle
constructed with center O passing trough A 'fits' in B then
OB=OA". The transition from empiric verifications (visual
and measures aids) to deductive argumentation was not
immediate, but it took place.
It was quite frequent to find an intended
proof replaced by proving something else, i.e.,
mathematically the deductive argument was correct, but it
did not prove the aimed property. It was not easy for the
students to become aware that "they missed the point", that
a detour had been made. Sometimes such behavior was caused
by poor comprehension of geometrical concepts; in others it
seemed that such behavior could be caused by the
intellectual satisfaction in producing a logically coherent
argumentation up to the point of missing the proof initially
aimed at.
As the experiment progressed (stage 4),
the students gradually started to distinguish the nature of
the statements they were making during argumentation and did
stress it through a discourse like "I am saying this because
it is part of the construction" (control of preliminaries
assertions), "I am saying this because I see it" (but aware
of the empirical information, when they did not succeed in
proving a stable visual fact) or "I am saying this because I
proved it" (plain control by deductive reasoning).
The material analyzed showed a meaningful
progress towards the comprehension of what a proof is about.
The very first competence skills for producing proofs were
acquired: the control of declared facts; the understanding
of progressive restriction for imposing certain facts which
then should be proved; the awareness of the distinction
between arguments based on empirical evidence and arguments
based on deductive reasoning.
Final remarks
The teaching experiment reported in this paper is part of
a wider on-going research project that aims at investigating
the potential of dynamic geometry environments in the
learning of the proof process. To be able to engage in a
proof process, with all its complexity (identifying
assumptions, conjecturing, looking at special cases, to name
but a few the aspects), the students need to be aware of
what producing a mathematical proof means. In the teaching
experiment developed, simple geometrical constructions were
proposed for attaining such awareness; to explain why a
geometrical property came out as consequence of some
declared properties was the purpose rather than to be
convinced that a (obvious) relationship was true. The
environment made the interplay between geometric
construction and proof process possible; the great precision
of drawings and the geometric invariant facts, instead of
dismissed attitudes for deductive reasoning, made the
students even more alert to implicit facts that should be
proved.
So far our main concern was to foster the
students understanding of the hypothetical-deductive nature
of geometry. Our next research point is how to help the
students learn the classical results of geometry (triangle
middle base, inscribed angle, Tales theorem, Pythagoras
theorem, ...), more efficiently by using dynamic
environments. To have students engaged in the richness of
the learning process that can be promoted in such
environments, a new treatment for classical theorems must be
thought of. Inductive evidence, in general, is not enough
for insights towards proving. Deductive reasoning plays an
important role, but mainly for organizing the arguments of a
successful proving process. To produce proof, an insight
must come up; the question we have is how to design
didactical situations where students would experience "forms
of thought" which could empower them in creating proofs. In
this general direction, Simon (1996, p.198, p.207) suggests
a third type of reasoning, not inherently inductive or
deductive, as part of the students' mathematical
explorations and justifications: "(...) very often what the
students are seeking is a sense of how the mathematical
system in question works. Such knowledge is often result of
'running' the system (...) I call this transformational
reasoning (...) Transformational reasoning involves
envisioning the transformation of a mathematical situation
and the results of that transformation(...) is often a sense
of understanding how it works". Also Goldenberg (1995)
suggests to approach theorems as functions; the theorems
should not be static statements, but they should come up as
functions defined in a class of geometrical objects and
would be represented through a dynamic process. 'Theorems
like functions' (even if vaguely defined by Goldenberg)
might be an interesting approach for recasting static
theorems (at least some of them) as dynamic theorems and
'transformational reasoning' might be one of the cognitive
attitudes to be developed for generating insights on the
proofs of theorems, using dynamic environments.
References
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à la lumière de l'intelligence artificielle
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Appendix
Fig 1
Geometrical constructions with points A and B as initial
objects
(production of Car &Cla, Mar &Rog, Ana &
Dan)
   
Fig 2
Inclusion of facts not declared in the
construction(bolded)
(production of Ana & Dan , Car &Cla, Mar &
Nat)
  
Fig 3
Support for the teacher /researcher intervention in
stage 3
Fig 4
Identification of implicit facts
(production of Ana & Dan)
Fig 5
Deductive argumentation using 'visual fact'
(bolded)
(production Ana & Dan)
Fig 6
Deductive argument with detour (bolded)
(production of Mar & Nat)
Fig 7
Deductive argumentation with quite efficient
control
(production of Mar & Nat)
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