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Mathematics, Physics and Music to elaboration of a didactic situation in secondary school Doctoral Thesis by Daniela Galante Advisor: Prof Filippo Spagnolo Comenius University Bratislava Faculty of mathematics and physics department of didactic mathematics

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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.

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The world of music presents two strictly connected components: the “artistic” one and the “scientific” one. 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.

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TRANSLATION Some transformation of a melody used in the composition technique quite correspond to a translation.

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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:

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a translation along the x axis 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.

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a translation along the y axis 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.

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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.

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REFLECTION Another composition technique frequently used to develop a melody is the reflection. If we look again at the melody exposed before: we will have the following reflection:

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a) reflection as regards the x axis 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.

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b) reflection as regards the y axis 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.

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symmetry as regards to the origin (composition of the reflection as regards the x axis and y axis). 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.

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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.

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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.

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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.

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Perception and sound analysis 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.

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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.

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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 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.

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METHODOLOGY AND THEORETICAL REFERENCE 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: 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.

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Work aim 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

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research hypothesis 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.

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sample 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

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the Theory of Situation by Guy Brousseau 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.

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The testing 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.

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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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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).

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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).

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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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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).

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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.

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Fourth questionnaire 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.

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Non-musicians sample

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Pianists

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Conclusion 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

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