|
© M.G.
Bartolini Bussi
|
There is an increasing attention towards the importance
of explanation, justification and proof in mathematics
education. To introduce the discussion, I shall sum up some
of the research studies carried on by my research team in
the last ten years.
In the late eighties, we have started to study
the educational aspects of proof in the geometry setting,
with reference to an original field of experience, i. e. the
field of mathematical machines (physical instruments,
designed during history to solve some geometrical problem by
geometers like Descartes, Newton). Many of them are
linkworks designed for either tracing algebraic curves or
realising correspondence between two regions of the plane.
The parents of these machines are the ruler and the compass
of Euclid's Elements; the last descendant might be
considered a computer with software for dynamic
geometry.
In a research study developed in the early
nineties, we observed a small group of 11th graders
struggling with the production of conjectures and the
construction of proofs related to a particular linkwork, the
pantograph of Sylvester (Bartolini Bussi 1993). What struck
us was the evident presence of the traces of the
conjecturing process in the proving process. The proof
eventually produced by the students (a not trivial proof at
all!) preserved the reference to the physical object, some
of the metaphors used during the conjecturing process and
the typical time inversions present in the wandering phase,
when some property was guessed empirically and tentatively
justified backwards by recourse to the existing knowledge.
However, in spite of this heavy presence of perceptual
events during the whole process, the tension of the students
towards a complete justification within the system of
elementary geometry was evident. In other words, the
students were supposed to work as theoreticians on
perceptual data. Those students, however, could not be
considered novices: they had been involved for three years
in a didactical project where the construction of proofs
(and not only the repetition of proofs) was often preceded
by the production of conjectures; besides they were already
familiar with many properties of elementary geometry.
We felt necessary to shift to the case of
younger students (mainly primary school pupils), put in
suitable learning environments, in order to study the early
emergence of a theoretical attitude towards mathematics and
of proving processes. We hoped to find, on the one hand,
evidences that the theoretical aspects of mathematics are
within the reach of children too and, on the other hand,
models of the teaching-learning processes by means of which
this approach might be realised.
In the same years, the team directed by Paolo
Boero was studying the process of production of conjectures
and construction of proofs with students of the grades
6th-8th. On the base of several studies, from inside and
outside geometry, Paolo Boero and Rossella Garuti formulated
the theoretical construct of 'cognitive unity' as follows
(Garuti, Boero & Lemut 1998). 'During the production of
the conjecture, the student progressively works out his/her
statement through an intensive argumentative activity
functionally intermingled with the justification of the
plausibility of his/her choices. During the subsequent
statement-proving stage, the student links up with this
process in a coherent way, organising some of the previously
produced arguments according to a logical chain'. Then they
transformed this descriptive instrument into an
interpretative and predictive instrument, to study the
students' difficulties in proving, for instance when the
conjecturing phase is omitted and the students are given a
ready made statement to be proved. The construct of
'cognitive unity' was like a flash of light for us and
offered the right language to interpret the data from the
experiment of the pantograph of Sylvester and to design
further experiments.
With primary school pupils we consider only
proving tasks that pass a 'cognitive unity' test, in the
sense that:
1) the pupils are requested to produce a
conjecture (and not given a ready made statement);
2) the pupils are requested to argue for the
conjecture on the base of what they know;
3) the gap between the arguments produced by the
pupils and a 'rigorous' proof within a shared theory is
expected to be small.
The last point deserves a comment: a 'rigorous' proof
exists only within a reference theory, that states what is
postulated within the classroom and which are the rules for
reasoning (see the discussion about Theorems in Mariotti
& al., 1997). However, the translation of the set of
arguments into a logical chain is a matter of social
construction under the teacher's cultural guide, by means of
scripts of proof introduced into the classroom.
We have studied the production of conjectures
and the construction of proofs in three different fields of
experience, more or less related to geometry, as from
primary school:
1) the representation of visible world by means
of perspective drawing (Bartolini Bussi 1996);
2) the kinematics of gears (Bartolini Bussi et al
1999);
3) the geometry of circles (Bartolini Bussi et al to
appear).
Each of the experiments lasted at least a couple of
years, even if the total amount of school time was not very
large (about 20 hours x year). For the details we must refer
to the quoted papers.
In this intervention we quote just a small
example of a critical event in the overall process from the
experiement on circles. 5th graders (from a classroom with a
very low sociocultural extraction) have been given the
problem of constructing a circle with a 4cm radius tangent
to two already given circles (with radii 2cm and 3cm),
explaining well the method and justifying why it works. They
know how to use the compass to draw circles. In the 3rd
grade they had designed also non-standard compasses,
including the 'flat' compass given by a rotating segment.
They know (theory) that the condition for two circles being
(externally) tangent is that the distance between the
centres is equal to the sum of the radii. They have worked
individually by trial and errors, verifying later on the
produced drawings that the condition of the sum of radii was
verified.
Yet in the discussion that follows the
individual solution, there is the shift towards the
statement of the standard Euclid's method for this kind of
construction problems, i. e. from a practice oriented to a
theory oriented use of the compass. In the former the
compass is used as a precision tool to draw objects with
round shapes. In the latter, the compass is used as a
geometry tool to select the points of the plane that are at
a given distance from a given point. This use orients the
definition of circle towards the solution of construction
problems.
EXCERPT (5th grade).
The teacher (Mara Boni) introduces the theme of
discussion. Each pupil has a copy of Veronica's protocol.
Veronica has produced a 'right' drawing by trial and errors
(as she says); she has justified the correctness of the
product by referring to the known theory; she has tried to
explain the process of finding the solution as follows:
[After having found the distances between
the centres S and R of the given circles and the centre T
of the circle to be drawn] I have given the right
"inclination" to both segments, so that the radius of the
circle was 4cm in every case.
Teacher : Veronica has tried to give
the right inclination. Which segments is she speaking
of ? Many of you open the compass 4cm. Does Veronica
use the segment of 4cm ? What does she say she is
using ?
Veronica's text is read again
Jessica : She uses the two segments
...
Maddalena : .. given by the sum of radii
Teacher : How did she make ?
Giuseppe : She has rotated a segment.
Veronica : Had I used one segment, I could
have used the compass.
Some pupils point with thumb-index at the segments on
Veronica's drawing and try to 'move' them
They pick up an ideal segment as if it were a stick
and try to move it
Francesca B. : From the circle B have
you thought or drawn the sum ?
Francesca is posing clearly the question about which
referents Veronica has used : an ideal (thought) referent
or a physical (drawn) referent.
Veronica : I have drawn it.
Giuseppe : Where ?
Veronica : I have planned to make RT
perpendicular [to the base side of the sheet]
and then I have moved ST and RT until they touched
each other and the radius of C was 4cm.
Veronica claims to have drawn but to have allowed
herself to move the static drawing.
Alessio : I had planned to take two
compasses, to open them 7 and 6 and to look whether
they found the centre. But I could not use two
compasses.
Alessio states the link between the rotation of the
segments (either thought or drawn) and the compasses that
are nothing for him but materialised segments. But he had
only one.
Stefania P. : Like me ; I too had two
compasses in the mind.
Veronica : I remember now : I too have
worked with the two segments in this way, but I could
not on the sheet.
All the pupils 'pick up' the segments on Veronica's
drawing with thumb-index of the two hands and start to
rotate them. The shared experience is strong enough to
capture all the pupils.
Elisabetta [excited] : She has
taken the two segments of 6 and 7, has kept the centre
still and has rotated : ah I have understood !
Stefania P. : ... to find the centre of the
wheel ...
Elisabetta : ... after having found the two
segments ...
Stefania P. : ... she has moved the two
segments.
Elisabetta and Stefania together by words and gestures
repeat the procedure
Teacher : Moved ? Is moved a right
word ?
The teacher encourages the correction of an ambiguous
word
Voices : Rotated .. as if she had the
compass.
Alessio : Had she translated them, she had
moved the centre.
Andrea : I have understood, teacher, I have
understood really, look at me ...
Andrea too has understood and shows it by gesturing.
The pupils continue to rotate the segments picked up with
hands
Voices : Yes, the centre comes out
there, it's true.
Alessio : It's true but you cannot use two
compasses
Alessio has still his problem : only one physical
compass whilst the two rotation are contemporaneous
Veronica : you can use first on one
side and then on the other.
but Veronica breaks the time of contemporaneity using
the same compass twice.
In the excerpt of the discussion we are real-time
observing the emergence of the theory oriented use of the
compass. The way of using the compass (i. e. the gesture of
handling and of tracing) is the same for both practice and
theory tool, but the senses given by the pupils to the
process (gestures) and to the product (drawings) are very
different. When the compass is used to produce round shape,
its main goal is communication ; when the compass is used to
find the points which satisfy a given relationship, it
becomes an instrument of semiotic mediation (Vygotskij,
1978), that can control - from the outside - the pupil
process of solution of a problem, by producing a strategy
that (i) can be used in any situation, (ii) can produce and
justify the conditions of possibility in the general case
and (iii) can be defended by argumentations referring to the
accepted theory.
The geometric compass, embodied by the metal
tool stored in every school-case, is no more a material
object : it becomes a mental object, whose use may be
substituted or evoked by a body gesture (rotating hands or
arms). The collective construction of the 'mental' compass
is very important in this approach to the theoretical
dimension of geometry with young learners : even if the link
with the body experience is not cut (it is rather
emphasized), the loss of materiality allows to take a
distance from the empirical facts, transforming the
empirical evidence of the drawing that represents a solution
(whichever is the early way of producing it) into the
external representation of a mental process. The realisation
of this learning process (guided by the teacher) is
consistent with the epistemological analysis carried out by
Longo (1997), on the basis of neurological findings, about
the 'geometrical abstraction' : the (geometrical) circle is
not a generalisation of the perception of round shapes, but
the reconstruction, by memory, of a variety of acts of
spatial experiences (a 'library' of trajectories and
gestures).
We can draw some conclusions from this small
piece of a research study:
1) the process of building a theoretical
attitude towards mathematics is quite long and can last
for years;
2) the process is developed under the guide of a
cultured adult (the teacher), who can on the one hand
select the tasks and on the other hand orchestrate the
social interaction towards this aim;
3) gaining a theoretical attitude does not mean to cut
the link with concrete experience, but rather to give a
new sense to 'the same' concrete experience.
Our research studies show that it is possible to design
and implement suitable long term teaching experiments for
young pupils with a very low sociocultural extraction with
the aim of introducing them to the theoretical dimension of
mathematics. This 'proof of existence' (metaphorically a
'constructive' proof, like in the Euclidean tradition) might
throw a stone in the debate about the nature of school
mathematics, that is often biased by ideological
declarations. However to pursue this constructive aim, we
must be equipped with innovative methodologies: the
researcher must monitor in the same time both the long term
process realised by the whole teaching experiment and the
short term processes of problem solving ; both the social
processes orchestrated by the teacher and the individual
processes of each pupil. This complexity surely requires to
find new means for scientific communication.
References
Bartolini Bussi M. (1993) Geometrical Proofs and
Mathematical Machines: An Exploratory Study, in Proc.
17th PME, vol. 2, 97-104, Tsukuba (Japan).
Bartolini Bussi M. (1996) Mathematical Discussion and
Perspective Drawing in Primary School, Educational
Studies in Mathematics 31(1/2) 11-41.
Bartolini Bussi M., Boni M., Ferri F.,
Garuti R. (1999) Early Approach To Theoretical
Thinking: Gears in Primary School. Educational Studies in
Mathematics 39 (1-3), 67-87.
Bartolini Bussi M., Boni M., Ferri F.
(to appear), Construction Problems in Primary School: a case
from the geometry of circle.
Garuti R., Boero P., Lemut E. (1998),
Cognitive Unity of Theorems and Dfficulty of Proof, Proc.
22nd PME, vol. 2, 345-352, Stellenbosch (South
Africa).
Mariotti M. A., Bartolini Bussi M., Boero
P., Ferri F., Garuti R. (1997) Approaching
Geometry Theorems in Contexts: From History and Epistemology
to Cognition, in Proc. 21st PME, vol. 1, 180-195,
Lahti (Finland).
|
