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Comenius University Bratislava
Faculty of mathematics and physics department of
didactic mathematics
Mathematics, Physics and Music
to elaboration of a didactic
situation in secondary school
Doctoral Thesis by
Daniela Galante
Advisor: Prof Filippo Spagnolo
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My doctoral thesis consists into three parts.
In the first part named “The Isometries in the music
history” I deal with geometrical transformations that occur
in the language of music. They are translations, reflections
and rotations. Further, I analyze in detail the work of J. S.
Bach “Musical Offering” and the work of Pierre Boulez
“Structures I” for two pianos from point of view of
presented geometrical transformations.
The second part of the thesis with the name “The way of
the sound” speaks about sound and its characteristics and
about human sound perceiving as well.
In the third experimental part, which is linked with the
research and its evaluation. I made this research in
accordance with the Theory of didactic situations of G.
Brousseau.
The world of music presents two strictly connected
components: the “artistic” one and the “scientific”
one.
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I would like to show how it is possible not only to
see the probable applications of geometrical
transformation but also to listen to the effect they
can have over a melody through the musical aid
which makes the study of geometry more
interesting.
TRANSLATION
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Some transformation of a melody used in the
composition technique quite correspond to a
translation.
If we consider as starting point of our system of
reference, that is y = 0, the height of the
correspondent sound to a G, the following melody:
Can be represented by:
a translation along the x axis
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The transformed melody
is played after a moment
of silence given by the
pause that in this case is
equivalent to the whole
beat value.
a translation along the y axis
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The transformed
melody is played a
fourth higher: that is to
say that all the sounds
have been raised of two
tones and a semitone
reproducing the same
melody in a higher
height.
Frère Jacques
We can see a complete example of a translation along the
x axe through the folk melody Frère Jacques which is
formed by four tune bits and each one of them is repeated
twice.
REFLECTION
Another composition technique frequently
used to develop a melody is the reflection.
 If we look again at the melody exposed before:
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we will have the following reflection:
a) reflection as regards the x axis
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The intervals in the original
melody are played in an inverse
direction: that is to say that the
first interval which is an ascending
tone through reflection becomes a
descending tone. To keep the
distance in the second interval of
the reflected melody unchanged
we had to change the E into E flat.
In music this kind of
transformation is called canon
through a contrary movement or
inversion or also mirror canon
when original and reflected
melody start at the same time.
b) reflection as regards the y axis
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The reflected melody is
composed by the same
notes at the same height
as the original one in a
sequence of sounds
moving backwards: the
melody starts from the
last note of the original
melody to conclude with
the first one.
In music this
transformation is called
retrograde or crab canon.
symmetry as regards to the origin
(composition of the reflection as regards
the x axis and y axis).
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The reflected melody is built
by a simultaneous
composition of the reflection
a) and b) in practice there is
an inversion of the intervals
and the melody moves
backwards.
In music this transformation
is called inverse retrograde.
If we use again the melody Frère Jacques and apply to it
the reflection over the x axe we obtain a melancholic
sensation which is a characteristic of the minor mode
which will not be missed by anyone who will listen to it.
As to the reflection as regarding the y axe we can observe that the
overturned melody has a meaning very different from the original one
and in this case it takes a solemn character as if it were a march,
which is reinforced by the relationship between the rhythmic structure
of the passage and the duration values of the sounds of the melody, in
fact at the beginning of every measure we find the note of highest
value unlike the original melody.
In the end, the reflection from the origin of the two axes
deeply alters the original melody to keep the intervals
unchanged, the major mode becomes minor and the melody
puts on an introspective and intimist character pointed out by
the modulation of the sound sequence which first goes straight
towards the low notes and then towards the original point, that
is the opposite situation which normally occurs in the original
melody.
Perception and sound analysis
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Knowing our body and brain functioning helps us
understanding how to encourage the comprehension
and the storage of teaching experiences during
mathematics teaching – learning activities, and, in this
case, through interaction with musical language.
The activity of sound perception and the one of
decoding and analyzing musical language combined
with mathematic language could be analyzed from
different points of view. In this chapter, I’ll try to
analyze it from a neurophysiologic aspect.
As it is shown in the experimental section, in this
work I demonstrate that there are some mathematics
learning ways, in this case some geometrical
transformations, that are not used and that involve
all human body’s senses.
 The general function of the brain is to be informed
about what happens in the body, in the brain itself,
and about the environment surrounding the
organism, so that an adequate and suitable for
survival adaptation could be reached between
organism and environment.
 Perception is our way to explore the external and the
internal world and it is the base of our knowledge.
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Learning process and
recompense
 Before
observing the physiological bases
of learning, let’s try to give a description
of it.
 It’s necessary distinguishing between
memory and learning; the former is a
necessary condition for the latter.
Memory
William James (1842 – 1910) introduced the idea
of two different components of the memory:
Primary Storage (short-term memory) and
Secondary Storage (long-term memory).
 Freud connects emotion and context with memory.
Memories not associated to emotional status are
not memories.
 Musical corporeity is a fundamental component of
learning processes, both for musicians and for
listeners.
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EXPERIMENTAL SECTION
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What has been said till now let us affirm that isometries had
a fundamental role in the development of music language,
from Bach to Boulez, and that application of composing
techniques based on geometrical transformation let the
listener hear and recognize the geometrical transformation
included in the music piece.
Furthermore, people studying a musical instrument moves
constantly his fingers and participates with all his body to
the creation and repetition of melodic-rhythmic cells (scales,
arpeggios, technique development, repertoire) which
contain unlimited geometrical transformations; these are
memorized in the corpus striatum, through the auditory
pathway and the associative perception, and become
acquired linguistic inheritance.
METHODOLOGY AND
THEORETICAL REFERENCE
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Research method is the one of G. Brousseau’s teaching situations.
Experimental phases are:
Teaching problem formulation: we learn geometrical transformations
through geometrical language and musical language; through this we
can see and hear them.
Research aim formulation: favouring a learning process in the students,
through interdisciplinarity, view as a unity of culture in the diversity of
knowledge.
A priori analysis of problem – situation, taking into account:
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the epistemological representation of both mathematical and musical concepts;
the historical-epistemological representation of the same concepts (interfering
time variations)
foreseeable students behaviour regarding the situation – problem.
Work aim
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The aim of this research is to verify if the
constant study of a musical instrument creates
unconscious potentialities which are translated
into strategies and methodologies for the
solution of problems related to isometries
research hypothesis
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H1 In musicians students (music liceo-conservatoire) the
constant study of a musical instrument creates unconscious
potentialities which are translated into strategies and
methodologies for the solution of problems concerning
isometries differently from non musicians students
(pedagogical liceo).
H2 Students possessing a knowledge of the musical
rhythmic structures have a greater ability in recognizing
the rhythm of geometrical forms for the construction of
objects in comparison with those who do not have such
knowledge.
sample
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In order to verify these two hypothesis, we realized an
experimental teaching in Liceo Statale “Regina Margherita”
in Palermo, where two different samples of students have
been identified:
Students from Music Liceo, connected to Conservatorio di
Musica di Stato “Vincenzo Bellini”: 70 students, aged
from 14 to 16 (classes I and II);
Students from Liceo Socio-psico-pedagogico: 70 students,
aged from 14 to 16 (classes I and II).
Furthermore, eight couples of students from the III classes
have involved (four for each course) with the duty to write
their common consideration written after a common
agreement with a interviews protocols registration
the Theory of Situation by Guy
Brousseau
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The experimental research realized in February and
March 2006 has been conducted following the Guy
Brousseau’s Theory of Situations.
First of all, we describe this theory providing an
exhausting analysis of its theoretical assumptions
revised by Filippo Spagnolo.
The theory of situations is currently evolving through
several experimental and theoretical works among
researchers on mathematics teaching methodology.
The testing
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I proposed four sets of questions to both samples
examined: the first two are about classical
exercises on geometrical transformations present
in any textbook for the first two years of upper
secondary school; the third one is a problem
regarding the reconstruction of a mosaic through
the identification and iteration of geometrical
figures and finally a last set of exercises regarding
the application of the geometrical transformations
in melodic tune bits.
ANALYSIS OF DATA
In a double entry chart
“students/strategies” for each student I
have shown with value 1 the strategies
used and with value 0 the strategies not
used.
 The collected data where analyzed in a
quantitative way by using the implicative
analysis of the variables of Regis Gras
through software CHIC
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First questionnaire
The first questionnaire has 6 exercises on
geometrical transformations in the plane
without analytic references, present in any
textbook for the first two years of upper
secondary school.
 In general, through the quantitative analysis
is evident that both samples have difficulties
in identifying and recognizing the symmetry.
 Both samples tried to solve the exercises
using only the mathematics language.
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Second questionnaire
The second questionnaire has five exercises
on geometrical transformation with analytic
references of the Cartesian plane.
 In general, through the quantitative analysis
is evident that both samples have difficulties
in identifying and recognizing the symmetry
in the Cartesian plane.
 Again, both samples tried to solve the
exercises using only the mathematics
language.
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Third questionnaire
The third questionnaire is a problem
regarding the reconstruction of a mosaic
through the identification and iteration of
geometrical figures.
 Observing the Similarity Graph musical
and pedagogical liceos
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5
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2
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4
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6
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we find out that they can rebuild the
mosaic (3_1) or they don’t look for a logic
in the drawing and decide to complete the
mosaic through their fantasy (3_6).
3
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3
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5
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6
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2
3
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4
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1
Observing the Similarity Graph, of the sample of Liceo
Socio-psico-pedagogico students, we notice that they
just draw some of the tesseras , not the whole mosaic
(3_2), or they don’t look for a logic in the drawing and
decide to complete the mosaic through their fantasy
(3_6).
5
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2
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3
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4
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1
Observing the Similarity Graph, of the sample of
pianist students of Liceo Musicale, we find out that
they’re able to rebuild the mosaic (3_1) or they
don’t answer.
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4
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2
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Observing the Similarity Graph, of the
sample of instruments students of Liceo
Musicale we find out that they can rebuild
the mosaic (3_1), or they draw the
confused lines, not following the mosaic
sequence (3_3).
3
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Through the quantitative analysis is evident that both
samples identify the recursiveness and the rhythm of
the shape sequence; analysing the samples
separately we see that Liceo Pedagogico students can
identify part of the mosaic, whereas most of the
students of Liceo Musicale identify the recursiveness
of the whole draw.
This happens because the musician, playing a lot with
music rhythm while studying a musical instrument,
fixes in his mind the technique of recursive repetition;
so that, after identifying the starting cell, he’s
automatically able to rebuild the rhythm structure of
the whole mosaic.
Fourth questionnaire
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The set of questions was met with great interest
and enthusiasm by both samples of pupils because
they were made curious by the matching of
geometrical transformation with music.
The students who have elementary music
knowledge preferred to look for solutions in the field
of music rather than in that of geometry, for
example in the first exercise they said there was a
translation because there is a pause.
Non-musicians sample
Pianists
Conclusion
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From the analysis of the answers given to the set of
questions proposed I have been able to find out a different
behaviour, between the two samples taken into
consideration, in facing the solution of problems
concerning the geometrical transformations.
In general, both for musician students and non-musician
ones, a concept mistake between the terms translation and
reflection is present (which we can hypothesize is a
“misconcept”) and this is found also in the first two sets of
strictly geometrical questions.
From a quality and quantity analysis of the sub-group of
pianists has come out that the “misconcept” concerning
the translation-reflection decoding is less present and this
is due to the characteristics of the piano