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1. Introduction. Geometry
Reality and the theoretical dimension
1.1 Geometry and
reality
The privileged relationship between elementary Geometry
and physical reality must be recognized. History accounts
for Geometry considered as a science, deeply rooted in
common experience and based on empirical data, from which it
takes the feeling of certitude and the guarantee of
consistency. Despite of this glorious past, recent history
witnessed a deep change in the attitude of mathematicians
towards Geometry. After the 'discovery' of non-Euclidean
geometries a deep crisis characterised the last century and
a new conception of Geometry originated. The link with the
physical world, as a source of truth and coherence, was cut
out once and for all, so that the discussion of the nature
of axioms and their congruence with empirical facts was
suddenly overcome. Complete autonomy of the axioms from
their interpretation in reality was stated and a clean
separation was accomplished between the logical and the
psychological level.
"[...] the choice of the basic elements
of Geometry is not determined "a priori": the simplest
elements are selected with respect to psychological
intuition . That is, they are those elements, the
notion of which is already formed in one's mind as the
content of the concept of space, for instance, the point,
the line and the plane. Geometrical properties which are
the basis of the axioms gather around a number of
notions, unquestionable, but intuitively understandable
by themselves" (Enriques, 1920, p. 3).
Thus because of the impossibility of unequivocally basing
its principles, Geometry lost its status of corpus of truths
about the physical world that Euclidean Elements had
maintained through the centuries. But from this tradition
the main characteristic of the deductive approach was
maintained.
The theoretical aspect of mathematics, and
in particular of Geometry, was better stated, but at the
same time, a break occurred in the relationship between the
formal and the intuitive level, which in the past was much
less clearly defined. The formal distinction between truth
and validation that Mathematics states, does not solve the
problem at the educational level. Managing the complex
relationship between intuitive and theoretical dimension
constitutes one of the main difficulties of geometry
teaching/learning.
A more attentive analysis could be useful
to understand the mental processes involved in geometrical
reasoning, in particular the nature of the so called
"geometrical figure". The theory of figural concepts
provides a powerful tool and offers a theoretical framework
for our analysis.
1.2 The theory of figural
concepts
As far as geometrical reasoning is concerned, a
particular kind of mental object is involved, these are
commonly referred to as geometrical figures. A
geometrical figure - as it is used in geometrical reasoning
- is neither a pure image nor a pure concept.
In principle, geometrical objects - and
this is true with regard to mathematics in general - can be
considered as pure mental entities and can be treated at a
purely mental level. In principle, a geometer may confine
his exploration entirely to a mental activity, that is to
say a material exploration is not strictly necessary to
increase develop geometrical knowledge. However, Geometry
maintains its own specificity: what is particular about
Geometry is that it conserves in the reasoning process
an objective, pictorially representable property of reality
which is
space.
In this sense and for this reason,
Fischbein (1993) introduces the notion of
figural concepts,
referring to geometrical figures as mental entities which
possess, simultaneously, both conceptual and figural
properties. According to the theory of figural concepts, any
simple geometrical reasoning deals with a
"mixture of two independent, defined entities
that is abstract ideas (concepts), on the one hand, and
sensory representations reflecting some concrete
operations, on the other" (Fischbein, 1993, pag. 140).
For instance, reasoning on two congruent triangles, one
refers to the concepts of angle, side, point, triangle, but
also to figural information or figurally represented
operations, such as superimposing the two angles, the sides
delimiting the angle and the like. At the same time one
refers to mental objects which are ideal, abstract and
general as the concepts are, and to visual images, which
retain a property of the objective world which concepts do
not posses, that is, the property of space.
As far as geometrical reasoning is
concerned, the dynamics of Figural Concepts is based on the
hypothesis that the interaction between the two aspects -
figural and conceptual - is the triggering element
(Mariotti, 1992, 1996).
1.3 The relationship
between empirical facts and Theorems
In principle, there is a complete harmony between the two
aspects, so that the conceptual system and images perfectly
match, and productive reasoning is achieved. But, as
previous studies show, harmonising the two components of
figural concepts is neither spontaneous nor simple
(Mariotti, 1991, 1993, 1995) and many difficulties in
geometrical reasoning can be interpreted in terms of a
rupture in the fusion between figural and conceptual
aspects.
The contribution of the conceptual aspect
in the general context of spatial thinking raises the issue
of the specificity of geometrical concepts. Thus, assuming
that mental images are conceptually controlled, raises the
question of the congruence between spatial cognition and
Geometry, i.e. abstract mathematical space.
Our results confirm the possible discrepancies between the
two systems. Even though the concepts of Euclidean Geometry
taught at school derive from the experience of realistic
physical space and, in principle, do not contradict ordinary
beliefs, complete congruence between the two systems is not
always assured.
Besides the possible discrepancies between
spontaneous conceptualisation of space and geometrical
concepts (Mariotti, 1996), what characterises geometrical
facts and distinguishes them from intuitive facts is the way
in which they are made acceptable. A geometrical fact, a
theorem (Mariotti et al., 1997)) is acceptable
only because it is systematised within a theory, with a
complete autonomy from any verification or argumentation at
an empirical level.
1.4 The theoretical
dimension characterises mathematical
knowledge.
The distance between the theoretical and the intuitive
level arises great difficulties, but the theoretical
organisation according to axioms, definitions and theorems,
represents one of the basic elements characterising
mathematical knowledge. This does not mean that mathematics
- and mathematical activities - consist only in formal
deductions, but the tension inherent to a formal
systematisation is one of the main characteristics of
mathematical knowledge.
The deductive approach, primarily set up in the Euclid
Elements came to us with all its power and was the origin of
a method of scientific exposition.
...the construction of a theorematic Geometry,
which was accomplished at around the end of the IVth
century, has theoretical effects of extraordinary
relevance on the complex of Hellenistic scientific
rationality. [... la costruzione di una Geometria
teorematica, che giunge a compimento verso la fine del IV
secolo, ha effetti teorici di straordinaria importanza
per l'insieme della razionalità scientifica
ellenistica.] (Vegetti, 1983, p. 159).
Although it is important to distinguish between the
heuristic construction of knowledge and its formal
systematisation, one must recognise that the deductive
approach has always been inherent to mathematical knowledge.
Even when the accent is put on the heuristic processes it is
impossible to neglect the theoretical nature of
mathematics.
What does mathematics really consist of?
Axioms (such as the parallel postulate)? Theorems (such as
the fundamental theorem of algebra)? Proofs (such as
Gödel proof of undecidability) ? Definitions (such as
the Menger definition of dimension) [...]
Mathematics could surely not exist without these
ingredients; they are essential. It is nevertheless a
tenable point that none of them is the heart of the subject,
that mathematician's main reason for existence is to solve
problems and that, therefore, what mathematics
really consists of is problems and solutions
(Halmos, 1980, p. 519? quoted by Schoenfeld, 1994 )
This confirms the complexity of the
educational problem and the difficulty in Geometry
education; according to the theory of figural concepts, the
main objective is that of developing the interaction between
the figural and the conceptual aspect, but this must be
accomplished within a theoretical framework.
2. Geometry at
school
2.1 Intuitive versus
deductive Geometry
The complex relationship between geometrical concepts and
physical experience corresponds to the ambiguous status of
Geometry teaching. The relationship between the physical
world, most particularly its features generally referred to
as 'Space', and the theoretical domain of Geometry is
generally accepted, but for the most part, the literature
dealing with students' learning of geometrical concepts
reflects the empirical / deductive dichotomy (Schoenfeld,
1986, p. 249). Usually, the blame is focused on the
deductive exposition, and most of the difficulties are
ascribed to a "formal classroom exposure to Geometry in its
deductive-axiomatic form" (ibid., p. 249). The main argument
is that a formal approach to deductive Geometry is
meaningless for those students who lack a deep
intuitive understanding of the reference to the
empirical world. Thus the accent is put on the intuitive
aspects of Geometry and this often determines a radical
shift to an empirical approach; undefined 'criteria of
intuitiveness' affect the choice not only of the basic
elements but also of the methodology of teaching/learning of
Geometry; a generic idea of geometrical
intuition affects the choice of what should be taught
and how it should be taught, both at the general level of
curriculum and at the specific level of didactic.
There is a wide use of 'observation' and
'discovery', but not much attention or care is devoted to
elaborating a coherent system of "geometrical facts".
Traditionally, the move from observation to theory is
considered as being a natural development; unfortunately, as
often pointed out, the gap between spontaneous
conceptualisation of space and Geometry is not easily
overcome.
Actually, a deductive approach to Geometry
is not a generalized situation, and usually it is confined
to high school level. Moreover, it pertains to a certain
tradition and can be considered as peculiar to a few
countries .
Nevertheless, although in the past 'proof'
and generally speaking the deductive approach has been
neglected in the secondary school mathematics curriculum, in
the last years we find an increasing interest among
mathematics educators and many of them definitely claim that
a deductive approach
"deserves a prominent place in the curriculum because it
continues to be a central feature of mathematics itself, as
the preferred method of verification, and because it is a
valuable tool for promoting mathematical understanding."
(Hanna, 1996, pp. 21-22).
2.2 Intuitive versus
deductive Geometry
One of the main points in the comparison between the
intuitive and the deductive approach to geometry is the role
played by "justification", that is by explaining, arguing,
corroborating, verifying.
According to a so called 'intuitive
approach' to Geometry, pupils intuitively 'discover' certain
facts, most of them with a high degree of evidence. Often,
the teacher introduces geometrical facts through a
justification, but with the specific aim of convincing
pupils of the evidence of those facts, rather than of
providing pupils with a basis for a deductive method. Pupils
are not required to justify, they must simply know the
'fact'. Usually pupils are never asked to justify their
knowledge, the truth of which is considered immediate and
self-evident, i. e. intuitive (Fischbein,
1987). As a consequence, in pupils experience, justifying
pertains to the teacher, and has the aim of convincing one
of the 'evidence' of a certain fact. When a certain
knowledge is attained a justification is no longer
necessary.
Sometimes this phenomenon is particularly
evident: for instance, in the case of the Pythagorean
theorem. It is considered basic knowledge, teachers use to
introduce it through justification, for instance one or two
of the traditional 'visual proofs'; nevertheless, a few
months later, everybody knows the Pythagorean Theorem and is
able to apply it, but very few of them can remember any
'proof'! Once the formula is learned, there is no need to
remember any justification. As soon as a fact reaches the
status of evidence, any justification becomes useless and
ready to be forgotten.
According to its nature intuition contrasts the
very idea of justification. Thus stressing intuitiveness may
constitute an obstacle to developing a need for
justifying.
On the other hand, a deductive approach is
deeply rooted in a practice of justification; a deductive
approach to Geometry means the construction of a system of
geometrical facts, coherently related through appropriate
argumentations. In particular, it means constructing a
system, which is based on stated primitive statement, often
related to an intuitive interpretation (= axioms), and can
be enlarged by introducing new facts, related to the
previous ones through a proof
(= theorems).
Where proving is concerned, it is commonly
accepted that arguing and proving do not have the same
nature; arguing has the aim of convincing, but not always
does the necessity of convincing somebody coincide with the
need of stating the theoretical truth of a sentence.
"... une très grande distance cognitive
entre le fonctionnement d'un raisonnement qui est
centré sur les valeur épistémiques
lieés au statut théoriques des
propositions." ..." Passer de l'argumentation à un
raisonnement valide implique une décentration
spécifique qui n'est pas favorisée par la
discussion ou par l'interiorisation d'une discussion."
... " Le developpement de l'argumentation même dans
ses formes les plus élaborées n'ouvre pas
une voie vers la démonstration."(Duval, 1992-93,
p. 60)
The distance between these two modalities explains the
reason why, often, arguing can become an obstacle to the
correct evolution of the very idea of proof (Balacheff,
1987, Duval, 1992-93)
Actually, when a deductive system is
concerned, there are two interwoven aspects: on the one
hand, the idea of proof and on the other hand,
the idea of theoretical system (both local and
global theorisation may be considered). Proof makes sense in
respect to a theory and vice versa; thus, a deductive
approach presents two problems of sense which are
interrelated: the sense of proof and the sense of a
theory .
According to the previous analysis, and taking into account
the previous discussion about figural concepts we may
summarise: one of the main points in geometrical reasoning
consists of the fusion between the figural and the
conceptual aspects, thus the problem is that of maintaining
this harmony, but fostering the theoretical control of the
conceptual component.
As a consequence, it is possible to point
out two objectives of geometrical education. The former
concerns the necessity of developing a flexible
interaction between images and concepts . The latter
concerns the development of complex conceptual schemes,
controlling the meanings, the relationships and the
properties of a geometrical figure. From the
educational point of view, the first point is related to
fostering the evolution of particular mental processes, the
second point is related to constructing mathematical
concepts within a theory.
The following discussion aims to face
these crucial educational points and presents the choice of
a specific "field of experience" (Boero, 1995): geometrical
constructions within a software environment (Cabrì-
géomètre microworld)
3. Geometrical
constructions
A geometrical construction consists of a procedure which,
through the use of specific tools and according to specific
rules, produces a drawing. A construction is considered
correct if the tools have been used according to the stated
rules.
Despite the fact that there is a concrete counterpart of a
geometrical construction which can be accomplished on a
sheet of paper, geometrical constructions have a theoretical
meaning, which overcomes the apparent practical objective.
The history of the classic impossible problems,
which so much puzzled the Greek geometers, tells us about
the fundamental theoretical importance of the notion of
construction (Henry, 1994). The tools and their rules have a
counterpart in axioms and theorems of a theoretical system,
so that any construction corresponds to a specific theorem.
This theorem validates the construction: the
relationships between the elements of the drawing produced
by the construction are stated by a theorem concerning the
geometrical figure represented by drawing. History witnesses
that the relationship between a geometrical construction and
the theorem which validates it, is very complex (Heath,
1956, pp. 124-31); as the following example clearly shows,
it is also not immediate for students.
A test was administrated at the 9th grade
level. The pupils (15 -16 years old) attended a regular
class of Geometry, that is they experienced a traditional
course in 'deductive Geometry' (within which construction
problems were not treated); at the end of the academic year
they were able to solve the common simple proof-problems
presented in the textbooks. They were asked the following
question
Is it possible to construct with a ruler
and compass the parallel line through a point to a given
straight line? Justify your answer geometrically.
The following protocol can be considered a good
representative example and shows the difficulty of
conceiving the construction task as a geometrical
(theoretical) problem.
According to the task question, Sara drew
the parallel through P to the given straight line, there is
no description of the procedure but the drawing shows that
she used the compass and the ruler. Then she drew a
transversal line t , marked
the equal angles and commented:
Sara (9th grade 1 D Liceo
Scientifico)
"The two lines are parallel because drawing a transversal
line t we obtain isometric internal
alternate angles, isometric corespondent angles and
isometric external alternate angles."
Sara provides a justification for her construction
resorting to an empirical test; in fact, she draws a new
line and observes that the angles look equal. This is an
empirical verification, although based on geometrical
knowledge , that is based on properties given by a
theorem.
fig. 1
The Geometry learnt at school seems not to have
substantially changed the relationship between a drawing and
a geometrical figure. A drawing is definitely confined in
the world of experience, and the link between the theory and
the drawing is controlled by the standards of empirical
verifications.
In the following the possibility of a
theoretical approach to the construction problem is
discussed; our experimental results show the difficulty, but
also the richness of this type of approach.
4. Geometry in a
microworld
Let us consider screen images produced in a microworld.
They cannot be considered as pure representatives at the
perceptual level, they are drawings with their own intrinsic
logic, depending on the procedure used to make them appear
on the screen. The logic of the 'machine' becomes the logic
of the drawing, understanding and managing this logic is the
privileged, often the only, way to productively interact
with drawings on the screen.
As for Geometry, the mediating role of
screen drawings is differently involved, according to the
contribution of perception and conceptualisation, and their
interaction.
Let us take for instance
the drawing activity in a geometrical environment such as
that provided by "Cabrì-géomètre"
(Baulac, Bellemain & Laborde 1988) (*).
This kind of software produces screen
images, which are logically controlled by menu commands
('create' and 'construct'). This means that both the
perceptual and the logical components are present in each
figure.
Screen images within Cabrì
environment represent the direct external counterpart of
what was called a figural concept at the mental
level. In order to manage screen images both the figural and
the conceptual component must be taken into account. Any
construction task must be performed both at the conceptual
and the figural level, and the concepts involved are made
explicit in the sequence of the different commands of the
construction procedure. The Cabrì-figure is a special
kind of figure (Laborde, 1993, 1994); it has a special
status and in this sense, such a software seems to fit the
hypothesis about the dialectics of figural concepts
(Fischbein, 1993; Mariotti, 1992, 1995) so that activities
in the "Cabrì-géomètre" environment
become particularly useful in order to develop the correct
interaction between the figural and the conceptual
components of geometrical reasoning.
Interaction with the machine is mediated
by the menu commands, which reflect a purpose, both
conceptual and figural: how the image looks on the screen
represents only one aspect of its nature. The figure which
appears on the screen after a construction has its internal
logic. That logic is not directly evident, but appears at
once, when one of the elements of the figure is moved. The
particular "dragging" function permits one to move one of
the elements, whilst maintaining all the geometrical
relationships defined by the menu commands used in its
construction. After the change, the new figure may look
different, but some of its geometrical properties are
preserved.
As Laborde points out, " a further
dimension is added to the graphical space as a
medium of Geometry: movement". (Laborde, 1993,
p.56); in other words, the movement becomes an essential
component of the meaning of a Cabrì-figure.
Figures produced by "Cabrì-géomètre"
are seen , but they must be conceptualised in
order to be managed. On the one hand, the
sense of a Cabrì-figure consists of conceiving a
figure in terms of its own (characterising) geometrical
properties and accepting the dragging function as an
intrinsic defining element of the environment.
4.1 Theorems and
constructions in a microworld
According to the authors' intention
Cabrì-géomètre offers a microworld
which embodies Euclidean Geometry .
"As any microworld, it encourages the learner to
explore the environment, here the world of Euclidean
Geometry." (Laborde & Strässer, 1990, p. 171)
In particular, Cabrì refers to the classic world
of "ruler and compass" constructions, that is intersections
between straight lines and circles, perpendicularly and
parallelism; but, the main characteristic which makes
Cabrì so interesting is the fact that there is the
possibility of direct manipulation of its figures and that
this manipulation is conceived in terms of the logic system
of Euclidean Geometry. Cabrì-figures posses an
intrinsic logic, which is the logic of their construction;
the elements of a figure are related in a hierarchy of
relationships, corresponding to the procedure of
construction.
But there is something more. The dynamic
system of Cabrì-figures embodies a system of
relationships consistent in the broad system of a
geometrical theory . Thus, solving construction
problems means accepting not only all the possibilities of
the software, but also accepting a logic system within which
to make sense of it.
The Cabrì environment introduces a
specific criterion of validation for the solution of the
construction problems: a solution is valid if
and only if it is not possible to mess it up
by dragging (Noss et al., 1994). This criterion of
validation, does not depend on the perceptive appearance of
the product of the construction, on the contrary it can be
completely independent of it; this means that, inside the
world of Cabrì-figures, the use of the menu commands
although preserving the contribution of the figural aspect
corresponds to a logic intention consistent with a
geometrical system.
In conclusion, as far as Geometry is
concerned, there are two main aspects in the Cabrì
environment, one concerns the correspondence between the
primitives of the software and the basic geometrical
properties, the other concerns the dynamic of manipulating
Cabrì-figures which corresponds to a specific
criterion of validation within a coherent system of those
properties. According to these two basic aspects which link
the Cabrì environment and the Geometry theory, it is
possible to build a correspondence between a construction
and a theorem . In these terms, justifying a
construction corresponds to proving a
theorem.
4.2 The construction task
within the Cabrì environment
Let us now analyse the construction task as it is
presented within the Cabrì environment, the analysis
will refer to an experimental project carried out over the
last few years and still in progress (Mariotti 1996,
Mariotti in press).
In the Cabrì environment, the
construction activity, i. e. drawing figures through the
available commands on the menu, is integrated with the
dragging function, that is, the construction of a figure can
be associated with a control by dragging. Thus a
construction task is solved if the figure on the screen
passes the dragging test.
In this case, the necessity of a
justification for the solution comes from the need for
explaining why a certain construction works (that is, it
passes the dragging test); thus, a justification comes from
the need of validating one's own construction, in order to
explain why it works and/or foresee that it will function
.
As discussed above, from the theoretical
point of view, the idea of geometrical construction is tied
to the idea of theorem: underlying a construction there is a
theorem validating this construction.
Our experimental design aimed to make the
idea of construction evolve into the idea of theorem,
passing through the need of justifying towards the idea of
validating within a geometrical system.
The key point is that what must be
validated is the correctness of the construction; that is,
it is not the product of a procedure that must be validated,
but it is the procedure itself; the necessity of this
validation is mediated by the necessity of explaining why
the Cabrì-figure will not be 'messed up'.
When dragging is used, why do some
constructions work and not others ?
The dragging function is accepted as a validating tool,
but the problem must be shifted from validating by dragging,
to explaining the "proof by dragging" itself.
4.3 The evolution of the
sense of justification
According to the new meaning of constructing
Cabrì-figures, the meaning of justifying
is expected to evolve; the idea of figure acceptable as a
solution of a construction problem, is enriched by the
property of invariance by dragging and is related to the
constructing procedure, i.e. the correctness of a solution
is related to the sequence of Cabrì commands
describing the geometrical relationships of the figure.
According to
the two main aspects involved in the meaning of Theorem -
the need of a justification and the fact that the
justification must be provided within a theoretical system -
the evolution will start from a general need of
justification and move towards the idea of validation within
a theoretical system; this evolution may be articulated into
a sequence of steps:
a) Explanation of the reasoning
followed in the solution. Pupils are invited to report about
all the attempts, even the wrong ones (the solution given
will be the subject of a collective discussion). At this
point the basic aim is that of communicating to the teacher
and classmates one's own reasoning, i.e. the main objective
is that of being intelligible.
b) After the first discussions, when the
verbal reports are used to discuss the 'correctness' of the
construction a new aim arises which is that of making one's
own construction acceptable. This means that it is no longer
sufficient to be intelligible (clear, complete and precise),
the report must show that certain rules were respected.
Pupils accept the principle that a construction must be
validated and try to adapt their justification
to this new request. Now, the main question becomes how to
defend one's own construction; an agreement must be
negotiated about the rules that the whole class has to
accept.
4.4 The construction of
the theory: the role of a Microworld
There are two areas of difficulty, strictly
interwoven.
On the one hand the new idea of validation
must be introduced, on the other hand the rules of
validation must be stated. The acceptance of validation
depends on the meaning of the rules and on the acceptance of
these rules
According to our basic aim of introducing pupils to
'deductive' Geometry, we decided to build a dialectic
relationship between Cabrì constructions and
geometrical theorems. Starting in the Cabrì
environment we guide pupils to enter the geometrical system;
this occurs through the link between the logic of
Cabrì, expressed by its commands, and the Geometry
Theory expressed by its axioms and theorems.
However instead of giving pupils an
already-made Cabrì menu, corresponding to an
(Euclidean) axiomatization, we decided to construct our
Cabrì menu step by step; we aimed to make pupils
participate (Leont'ev, 1976/1964) in the construction of an
axiomatization and of the corresponding menu.
The main idea of a microworld
(Hoyles,1993) is that of providing an environment for
solving problems where pupils can experience the constraints
of the underlying mathematical system and in so doing
construct their own mathematical system. Where Cabrì
is concerned, the complexity of the system is too high, the
whole of Euclidean Geometry is available, thus the focus on
the underlining system becomes too complex. Over all,
however, the relationships between the main concepts
available through the geometrical primitives of Cabrì
remain largely hidden.
In fact, because of the richness of the
'geometrical tools' available, it is difficult to state what
is given (axioms) and what must be proved (theorems).
Generally speaking, it is possible to have different levels
of control on a Cabrì-figure. According to the theory
of figural concepts, pupils must achieve the conceptual
control of what they see on the screen. It is just this
control that we want to promote, together with the idea of a
coherent system. The richness of the environment might
emphasise the ambiguity about intuitive facts and theorems
and constitute an obstacle to the choice of correct
hypotheses.
Thus, the basic idea of working inside a
microworld was adapted to our objective. At the beginning an
empty menu is presented and the choice of commands
discussed, according to specific statements selected as
axioms. Then the other elements of the microworld are added,
according to new constructions and in parallel with
corresponding theorems..
In this way the system is slowly built up,
and step by step the complexity increases: the aim is that
of providing a complexity which can be managed by pupils; if
the whole system is present at the beginning, there is the
risk that pupils are not able to control the relationship
between what is given and what is deduced. On the other
hand, if the menu commands are changed too frequently it is
impossible to grasp any systematic order
The analysis of pupils' protocols shows
the slow evolution of the meaning of construction. At first,
a construction is conceived as a concrete process to reach a
drawing, which has its own justification in the
acceptability of the product; then, a construction is
conceived as a theoretical procedure which has its own
justification in a theorem.
On the one hand, the descriptions of the procedure change,
improving in clarity through an increasing mastery of
correct terms; on the other hand, the argumentations
approach the status of theorems, that is the justifications
provided by the pupils assume the form of a statement and a
proof.
5.
Examples
It is not possible to enter into the details of the
protocols' analysis. None the less, the following examples
aim to give an idea of the evolution of meaning that was
described in the previous section.
5.1 The construction of
the angle bisector
Let us consider the Cabrì environment. In analogy
with the Euclidean axioms, besides the primitives of the
"creation menu", in the "construction menu" the commands are
reduced to the intersection of objects, the report of length
and the report of angle. From the theoretical point of view,
this situation corresponds to the disposal of the three
criteria of congruence for triangles. This is what pupils
already have in their theoretical system and what they can
refer to in their justifications. The following task is
presented to the pupils.
"Construct the bisector of an angle.
Describe and geometrically justify your solution."
This is one of the first construction problems proposed
to the pupils; they are grouped in pairs at the computer and
they are asked to provide a common text for the
solution.
Finding the procedure does not present
great difficulty, some of them have already used it at
junior school. On the other hand, carrying out the procedure
is only the first part of the task. Difficulties arise when
the procedure must be described and justified according to
the accepted rules. The following protocol is an
example.
Alex & Gio (9th grade IV
Ginnasio 1993-94)
1 Attempt
We took two points and we made a line pass through them,
then we took another point C, which does not belong to
the first line
We joined the point which doesn't belong to r1 with a
second line. In so doing we determined an angle.
We transferred (abbiamo riportato) a segment AB belonging
to r2 and we transferred the same segment on r1 (AB=AC);
we drew two circles (centre/point) centre in C and point
A and centre in B and point A (puntando in C e apertura
AC e puntando in B con apertura AB.We joined A and D
(line through two points). We took the intersection
between the circle and the line, but
FAILED!
Construction of the angle bisector
fig. 2a (1 attempt)
2 attempt
We drew an angle as we did in the first attempt. We
drew a circle (centre/point), taking a point belonging to
r1.
Construction of the angle bisector
fig. 2b (2 attempt)
This circle gave us the segments AB and AC
belonging to r1 and r2, which are equal because they are
rays of the same circle. We drew two circles
(centre B and C point A). Using the intersection of two
objects (of the two circles) we found the point D that we
joined with A determining the angle bisector.
The solution is divided into two parts corresponding to
successive attempts, when pupils realise the first failure
they start a new attempt. It is interesting to remark that
the first attempt is considered a failure because it does
not pass the dragging test. The text of the second part is
more accurate in describing the command used for the second
construction, as if, after the first failure, the pupils had
felt the need to be more attentive. At the same time, it
present a first rudimentary trace of a justification.
"This circle gave us two segments AB and
CD belonging to r1 and r2, which are equal because they
are rays of the same circle ".
This is an incidental sentence, within the description of
the procedure, and in this sense it is far from a theorem (a
statement and its proof).
The following example shows a more
developed solution. The protocol does not present any
description of the procedure, but there is a sketch, drawn
with ruler and compass, reproducing the Cabrì-figure
.
Lorenzo (9th grade 1C Liceo
Scientifico 1993-94)
I consider the triangles ABD and ACD. They have side AD
in common and the side AB of the first is equal to the
side AC of the second. In fact, if I take the circle with
the centre in A and point B, it passes through both B and
C. Thus, the sides AB and AC are equal because they can
be considered as rays of a circle. If I also point in D
with the ray DC, the circle passes through both C and B.
Thus, the sides BD and DC are equal for the same reason
as the previous ones
fig. 3
I discovered that the triangles ABD and ADC have
respectively equal the sides; for this reason the 2
triangles are congruent for the 3 criterion of
congruence
If the two triangles are equal, there
is the rule that equal sides are opposite to equal
angles. Thus the angles 1 and 2, which are opposite to
equal sides BD and DC, are equal.
In this case the justification is evolving into a
theorem, although the difficulty in selecting the correct
hypotheses clearly appears. Such a difficulty is also
witnessed by the fact that after the first step, when the
equality of two of the sides is correctly derived form the
construction, the equality of the other sides is obtained by
considering the circle with centre D and ray DB, which does
not pertain to the original construction. Actually, in the
construction D was obtained from the intersection of two
circles ...
It is interesting to remark that in the
sketch drawn on the protocol, the three circles are present-
centre B ray BC, centre C ray CB, centre D ray BD; the
sequence of the operations used in the construction is not
preserved in the drawing, thus the confusion occurs! Even in
the Cabrì-figure the correct order of the
construction cannot be established immediately, when the
figure is moved the mutual relationships among the three
circles are preserved and it is necessary to refer to the
basic points in order to detect the correct
relationship.
The configuration is globally clear, but
the necessary order in the construction disappeared, Lorenzo
faces this obstacle and is not able to keep the logical
control of the geometrical figure.
5.2 An open-ended
problem
Let us consider another example. After the activity on
the angle bisector and that on the construction of the
perpendicular to a given line through a given point, the
following problem is presented.
Given a triangle, is it possible that the
bisector of one of its angles is perpendicular to the
opposite side?
The goal of the task is not a construction but the
explication of a theorem; the solution requires a
construction to be used in the exploring activity. The
openendedness of this situation makes it difficult to attain
the theoretical level and express the intuitions coming from
the exploration in terms of a theorem , that is, formulate a
statement and provide its proof.
Pupils performances show a great variety; some of them give
a correct expression of the theorem, completely
decontextualized from the exploring situation, others
maintain a closer relationship with the exploring phase.
These differences are reflected in different styles of
verbalisation: schematic style, approaching the standard
schema (hypothesis / thesis), and discursive style. The
following protocols may be exemplar.
Francesca (9th grade 1C Liceo
Scientifico 1993-94)
In a isosceles triangle proof that the bisector of the
vertex angle is perpendicular to the opposite side.
Hypothesis: CAH = HAB
AC
= AB
Thesis : AH is orthogonal to CB for the 2 criterion of
congruence of the triangles, which states that given two
sides (CA e CB) and the angle enclosed between them (CAH
= HAB), the triangle that can be constructed is unique.
Guia (9th grade 1C Liceo
Scientifico 1993-94)
Proof: The possibility exists that the angle bisector,
fixed (fissata) in an angle of a triangle, is
perpendicular to the opposite side.
The possibility that the angle bisector
is perpendicular to the opposite side always exists in
equilateral triangles, whilst the isosceles triangles
only when the angle bisector cuts the angle enclosed
between the equal sides. This [is true] for the
second criterion of congruence for the triangles,
according to which two segments and one angle given, it
is possible to construct only equal triangles with the
angle enclosed between the two segments.
fig. 7
The solutions proposed in the two protocols appear quite
different. Although both of them state the properties
characterising the angle bisector, the second protocol
maintains a trace of the exploration process, whilst in the
first any trace disappeared. After a complete
deconstextualization, Francesca expresses the statement and
the proof correctly. In contrast, Guia offers a description
of the different cases which seems to go over the exploring
process again but does not clearly express any general
statement; the argumentation that she gives as a proof is
not the standard form (statement + proof) as for Francesca.
Although all the elements required for the proof are
present, the verbalisation is still involved
5.3 The construction of a
parallel line
For the last examples I would like to come back to the
problem of construction with which we started: the
construction of the parallel to a line through a given
point. It is interesting to analyse the solution provided by
pupils of experimental classes and compare them with the
example of Sara discussed in the section 3. Consider the
following example.
Adriana 1A Liceo
Scientifico
The compass opened at will I place the point at P, I
trace a circle which intersects the line at the points A
and R; I join them with P and obtain an isosceles
triangle, because PA and PR are rays of the same circle.
I place the point of the compass in R, open it to AR and
find the point E. Now I place the point of the compass at
E, open it to RP and draw an arc, I place the point of
the compass at R, open it to RP and draw an arc and I
find the point L.
The line LP is parallel to the line r
because:
the two triangles ARP and PRL are equal
(because they have PR in common, they have equal bases,
because they rays of the same circle, and AP = RL,
because they are rays of the same circle).
Thus, P1=R2 the two alternate angles of
the line r and PL cut by the line PR are equal, thus the
lines are parallel.
fig. 8
The construction is correct and actually the line PL is
parallel to the given line; but the justification is
incorrect: the two triangles considered do not have
equal bases ; in fact AR and PL are equal but
not because they are rays of the same circle, rather because
of the particular construction accomplished. Actually a
correct justification could be the following: the triangles
APR and RLE are equal (because of the third criterion of
congruence) then ARP = ERL and consequently, knowing that
the sum of the interior angles of triangles is 180 and that
A, R and E are collinear, PRL =RPA.
Apart from the mistake, the protocol is
very interesting because it is exemplar of a good
interrelation between the description of a construction
procedure and the justification of that procedure. Some
characteristic elements can be highlighted. Firstly, the
separation between the justification and the description of
the construction. Some traces of their confusion are still
present, for example the initial remark about the isosceles
triangles. The hypotheses of the theorem validating the
construction are listed explicitly, that is, all the
equalities (correct or incorrect) obtained by construction,
and the equalities obtained consequently. Finally the
central implication is highlighted; it is based on the
theorem about parallel lines cut by a transversal.
Let us consider one more example .
Matteo 1 A Liceo Scientifico
1994-95
I consider the line r and the point P external to it. I
draw (traccio) the line PH perpendicular to r. (that is
the distance from the point P to the point H of the line
r)
The angle H is 90
If the line passing through P is
parallel to r XPH is 90 (Criterion of parallelism) the
line passing through P is parallel to r it is the angle
bisector of the stright angle QPH (the line QPH is ^
(orthogonal) too). I consider QPH (180 ) and I construct
the angle bisector passing through P because the angle
bisector of a (piatto ?) angle divides the angle into two
angles of 90 if one considers the two lines the two
internal (conjugate) angles are equal for the criterion
of parallelism the two lines are parallel.
fig. 9
Matteo's solution presents a conversational form,
following the steps of the construction, but with a clear
intention to maintain the logical thread of argumentation
(see the use of the arrows). The inversion between the
description of the construction and its justification shows
traces of the heuristic process: for instance, "I consider
... and I construct" and " If the line passing through P
...". Although not all of them find their theoretical
systematisation, the elements required by the 'theorem'
validating the construction are present. Despite the
intrinsic difficulties of the problem, Matteo shows that he
grasped the theoretical meaning of a geometrical
construction.
6.
Conclusions
The examples discussed, show that the possible evolution
of a justification in a proof as well as the fact that this
evolution is not expected to be simple and spontaneous.
The basic modification we were interested
in concerned the change of the status of justification in
geometrical problems. This modification is strictly related
to the passage from an 'intuitive' geometry as a collection
of facts submitted to empirical verification, to a
'theoretical' geometry, as a system of relations among
statements, validated by proof. According to our basic
hypothesis the relation to geometrical knowledge is modified
by the mediation of the computer.
Our results confirm that the specificity
of the Cabrì environment is determinant in order to
make sense of justification evolved from an empirical
verification towards a theoretical proof. Coming back to the
main educational problems previously pointed out, it is
possible to state the following conclusions. Geometrical
constructions within Cabrì-géomèrtre
provide a rich field of experience where the harmony between
the figural and the conceptual aspects can be achieved
together with the development of a sense of theory.
The construction task may develop a
theoretical meaning in relation to the logic of the software
environment and provide a powerful means for introducing
pupils to Geometry.
footnote
(*) I will discuss the example
of Cabri-géométre, but other software, may be
used as well, providing screen images controlled by
geometrical logic. [back]
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