WP 6 PILOT-PROJECT TRANSFER IN ITALY
REPORT
This project has been funded with support from the European Commission.
This publication [communication] reflects the views only of the author, and the Commission cannot be held responsible for any use which may be
made of the information contained therein.
INDEX
FOREWORD
page 3
1) TARGETS AND OBJECTIVES
page 3
2)
-
DESCRIPTION OF OBSERVED ITEMS
2.1 Students’ Participation and Involvement
2.2 Learning Context and Objectives
2.3 Didactics
page 6
3)
-
DESCRIPTION OF EXPERIENCES WITHIN SCHOOLS
3.1 Training Offer Plan
3.2 Action Planning in Schools
3.3 IPSAA Persolino and Strocchi Faenza (RA)
3.4 IIS Roberto Ruffilli Forlì
3.5 Istituto Calasanzio Empoli (FI)
3.6 Associazione Osfin Rimini
3.7 Liceo Chierici Reggio Emilia
3.8 Participation and Involvement of School Teachers
page 9
4)
-
CONCLUSIONS
4.1 Proposed Improvements and Strong Points
4.2 The Role of Music
4.3 Teachers and Trainers’ Point of View
page 23
2
Foreword
Doremat© was created in 2007 thanks to a group of authors, among whom Denise Lentini, the inventor.
The method was experimented at Enfap, a regional organization of the Vocational and Educational Training
system of Emilia Romagna, where she works as a director. The method was devised exploiting the analogies
shared by the two disciplines, relating mathematics skills to music, therefore creating a connection
between the two subjects, which are apparently distant but actually very similar: distant if one thinks of the
opposed feelings that they elicit in young people, in fact mathematics is still considered as the most
“difficult” subject by teenagers, while music is their most beloved one. At the same time they are so similar
if one thinks of their roots, which since the times of ancient Greece, have always linked them both
historically and epistemologically.
In order to contrast school dropout and reinforce both cultural and social skills, starting in 2007, at Enfap, a
series of music activities were organized (to play the guitar, the keyboard, the drums and to sing in a choir),
which were very successful with the students.
Doremat© links all the mathematics skills included in the course curriculum, for compulsory education
classes, i.e. from first degree-secondary school to the third year of second-degree secondary school.
Thanks to the EU program Leonardo, we have been able to widen the number of subjects involved,
experimenting the method in the schools of different regions (Emilia Romagna, Tuscany, Latium)
and to also overcome the national borders thanks to a prestigious partnership with the Latvian
Ministry of Education (VISC) and the University of Athens.
1. Targets and Objectives
Before going into the details related to the transfer of the Doremat method, we believe we should
first clarify its targets and objectives.
As far as the latter, the method takes into consideration the mathematics curriculum as indicated
by the Fioroni Decree, on which a program based on skills, i.e. abilities, behaviors, mastery and
competence related to cognitive objectives was organized. The instrument related to cognitive
competence, as well as the abilities related to teaching strategies are called taxonomy.
At the end of the present paragraph you will find Arrigo-Frabboni’s taxonomy table, to which the
following paragraphs will often refer, pertaining to the transfer to every single school.
However, it is not just a question of defining the targeted cognitive objectives per se: on the
contrary they shall tend to a further target, i.e. the personal development of individuals, “the
training of people and of citizens”1. Specifically Doremat’s final target is to educate students to
democracy and to citizenship through education to the scientific method and therefore to
facilitate “the training of young people to face new ideas and choices, with the co-operation on
common and shared commitments”2.
Mathermatical education fulfills this task as far as it manages to involve students in doing
mathematics, it makes them aware of the methods and the procedures, it gives a meaning to
mathematical objects. In the definition of cognitive objectives therefore this further target is to be
taken into consideration.
1
2
As indicated in the General Foreword of Curricula of 1985
U. Vairetti, Organizzazione e qualità della scuola, La Nuova Italia,1995.
3
From a methodological point of view, we promote problem solving as a mathematics learning
method: instead of starting to treat a subject through definitions, statements of theorems or
propositions… we start from a problematic situation, from which mathematical concepts are reinvented, re-discovered and re-built. It is the students that do the work under the guide of their
teachers. Mathematics means building thoughts and learning, it implies the development of
certain skills such as intuition, imagination, hypotheses, deduction and design in order to
understand reality. In order for it to be possible, it is necessary to stimulate the problematization
of real situations. From this point of view the essential role of motivation and involvement of
students is obvious. Needless to say that in our specific case, music acts as a powerful stimulus for
the interest and motivation of students; moreover music lessons create the environment from
which the situations to be problematized will be taken.
Table 1- COGNITIVE OBJECTIVE TAXONOMY TABLE (Frabboni-Arrigo)
I Level Basic Learning
General Objectives
Specific Objectives
1.1 Memorizing
1.1.1 recognizing a certain term and using it
1.1.2 recognizing a certain symbol and using it
1.1.3 reproducing the definition of a certain concept
1.1.4 reproducing the statement of a certain
principle/rule
1.2 Discipline-related Automatisms
1.2.1 executing elementary operations
1.2.2 executing automated procedures
1.3 Cognitive Automatisms
1.3.1 classifying based on a known criterion
1.3.2 performing cognitive sequences
4
II level Intermediate Learning
General Objectives
Specific Objectives
2.1 Describing knowledge
2.1.1 describing facts-procedures
2.1.2 recognizing a concept-principle-rule
2.1.3 summarizing facts-procedures
2.1.4 changing the code (language) to known notions
2.2 Appying and controlling
knowledge
2.2.1 Executing formal rules and/or procedures
2.2.2 Applying knowledge/procedures to other
cognitive contexts
2.2.3 Checking and justifying the knowledge acquired
2.2.4 Estimating the results deriving from acquired
knowledge
III level Higher Superior Converging Learning
General Objectives
Specific Objectives
3.1.1 Analysis
3.1.1.1 Analyzing/decoding
3.1.1.2 Comparing/choosing/deciding
3.1.1.3 Inductive reasoning
3.1.1.4 Foreseeing/estimating converging situations
5
3.1.2 Synthesis
3.1.2.1 Synthetizing/schematizing/contents, methods
3.1.2.2 Deductive reasoning
3.1.2.3 Generating and solving problems
3.1.3 Method
3.1.3.1 Understanding the internal structures of a certain
problematic situation
3.1.3.2 Becoming aware of the mathematical, historical, and
discipline-related way of thinking
3rd level Diverging Higher Learning
Obiettivi generali
Specific Objectives
3.2.1 Intuition
3.2.1.1 Foreseeing/Formulating hypotheses or counter hypotheses
3.2.1.2 Trying solutions (abductive reasoning)
3.2.1.3 Recognizing the key problem
3.2.1.4 Grasping a new concept/principle
3.2.2 Invention
3.1.2.1 Inventing by analogy/procedures/ concepts/principles
3.1.2.2 Extrapolating procedures/ concepts/ principles
3.1.2.3 Formulating new problems/innovative solutions
2. DESCRIPTION OF OBSERVED ITEMS
Hereafter we shall describe the variables observed in the context of all the classes involved in the
transfer of the Doremat method. Some of such variables were hypothesized during the designing
phase of the transfer, others emerged during its development, becoming precious suggestions for
other observations and research hypotheses. The formulation of variables to be observed during
the transfer was suggested partially by the experimentation which took place in the previous years
in the IsFP classes of ENFAP Forlì, and partially from some reflections and researches on the
Didactics of Mathematics. Please refer to the work by Prof. Berta Martini, for what pertains to
Quantitative Analysis (i.e., learning new topics in terms of assessment tests’ results) and
Qualitative Analysis (i.e. the analysis of the components of the “didactic triangle” and their
relations).
6
2.1 Students’ Participation and (emotional) Involvement
Mathematics is certainly one of those school disciplines which mostly does not elicit any interest in
the students. Many experts in the field of mathematics didactics believe that when mathematics
does not only highlight cognitive aspects, but also metacognitive ones it becomes more complete
and effective. Such hypothesis is supported by several studies and researches3.
For this reason, and for the specific nature and origin of the Doremat method, we believed it
would be interesting to observe the experience carried out on the studies of the so called
emotional factors. Particularly, the observation in such sense was concentrated (partially this
happened spontaneously) mostly on motivational aspects, analyzing participation based on the
manifestation of students’ interest and their involvement in the proposed activities.
Motivation plays a central role in mathematics learning and specifically in problem solving. On this
behalf in fact Zan (2007) observes that: “the importance of the motivational component in
learning is unanimous (…) for a long time motivational aspects, and more in general those related
to emotions, were separated from cognitive ones. In more recent studies on meta-cognition, on
the other hand, motivational aspects are considered strictly related to the selection of strategies
and control processes. The link between the development of meta-cognitive skills and the reasons
pushing a subject toward learning are highlighted”.
A number of investigation and observation tools were used such as: video-recording, satisfaction
tests and essays produced by the students. As far as their participation and involvement, we
considered the following indicators, i.e. questions asked during lectures, proposed solutions to
problems, active participation and initiative during the proposed workshops. Also group activities
in class were monitored, trying to observe the students’ behavior: i.e. whether the assigned task
were discussed by the group members, their relation to the work content and to the other group
members. All such indicators reveal not only students’ satisfaction for the Doremat method, but
also their attitude toward mathematics taught this way. We shall describe such indicators within
the context of methodological transfer.
Bibliography
Cobb (1986). Contexts, Goals, Beliefs, and Learning Mathematics. For the Learning
of Mathematics, n.6, p.2-9.
Di Martino P. (2001). Emozioni e problem solving: un confronto tra bravi e cattivi solutori.
In: Livorni E., Meloni G. & Pesci A. (a cura di), Le difficoltà in matematica: da problema di
pochi a risorsa per tutti. Atti del Convegno Nazionale n.10 su Matematica e Difficoltà,
Pitagora Editrice, Bologna, p. 89-96.
McLeod D.B. (1992). Research on Affect in Mathematics Education: a Reconceptualization.
In D.A.Grouws (Ed.) Handbook of Research on Mathematics Learning and Teaching, New
York: MacMillan, p.575-596.
Zan R. (1996). Un intervento metacognitivo di “recupero” a livello universitario. La
matematica e la sua didattica, n.1, p.65-89.
Zan R. (2007). Difficlotà in matematica- Osservare, interpretare, intervenire, Springer.
3
To quote just a few: Cobb (1986); Di Martino P. (2001); McLeod D.B. (1992); Zan R. (1996) and Zan R. (2007);
although bibliography here is quite wide.
7
2.2 Learning Context and Objectives
“Actually learning is one of the human activities which less needs external manipulations. Mostly learning is
not the result of education, but of a free participation within a significant environment” . Descolarizzare la
società. Una società senza scuola è possibile? , I. Ilich, edited by Perticari P., Mimesis, 2010.
With the description of a learning context, we would like to describe the experience of transfer
into several different situations. For this reason we believe it is useful to introduce the description
of each single school involved and of the classes (both as groups of individuals, and as the
environments where the lectures took place). The context is important because it will help us
better understand what happened, which were the choices of teachers and trainers as far as the
method’s application, as well as interpreting as well as possible the observed variables. We shall
see in fact that sometimes the choice of a didactic strategy is conditioned also by the behavior of
students or by their specific difficulties in relation to a certain mathematical topic. As the schools
are different and the contexts are too, the application of the Doremat method acquires sometimes
different “nuances”, with strategies and cognitive objectives which are not always the same (for
instance being able to calculate of to find an “original” solution to a problem are different kinds of
learning), still maintaining the same educational and training objectives (i.e. the full development
of the individual’s personality, an increase of autonomy and effectiveness, an emotional and
cognitive interpersonal self-awareness).
2.3 Didactics
Based on the objectives formulated for each school and on the learning context, different
teaching/didactic strategies were adopted. Doremat uses mostly workshops, which constitute the
heart of the method, as a privileged teaching tool and environment; it is there in fact that the
students experiment new concepts and discover across-the-board aspects, it is there that they
create and build up knowledge).
Examples of such workshops are: inventing a rhythmic sequence which fulfills certain restrictions
given in mathematical form, inventing an equation or a linear system which might represent a
rhythmic sequence in music, a duet or a quartet. All such activities are then supported by a group
discussion led by the teachers. The discussion then leads to the problematization of a situation
created by the students themselves. The workshops therefore can foresee simple music
experiences finalized to the understanding and assimilation of a certain concept (e.g. equivalence
between music figures and fractions) and/or problem-solving activities, where one starts from a
music concept as a problematic situation. The choice between those two activities is conditioned
by the addressed content and by the cognitive stimulated objectives.
We shall now describe synthetically the workshops to which we shall refer often in the description
of transfer in each school:
- Equivalence among rhythmic figures and equivalence among fractions. Divided into two or
three groups the students clap their hands following several rhythmic figures at the same
time (e.g. a group shall clap the quarters and another the sixteenths). The purpose is to
teach students the basics of music language, experiencing the association between
rhythmic figures and fractions and acquiring through a movement and a sound the
equivalence between fractions..
8
-
-
Invention of rhythmic sequences and arithmetic calculation. The students, whenever they
invent a rhythmic sequence count and organize rhythmic figures. Making them reflect on
such activity is important because it makes them aware of the mechanisms that they are
using and it makes them better evaluate their answers (i.e. the results of the arithmetic
operations that they are performing), have a better control of their calculating activity, and
understand the meanings of mathematics symbols.
Invention of equations of linear systems which can represent in music either a duet or a
quartet. In this activity the students work in groups of 4 or 5. It would be more convenient
to start from the invention of rhythmic sequences in a quartet, understanding which are
equal (deducing it from other similar activities done previously), choosing some unknown
figures, transforming them into a linear system and then solving them. The procedure is
not banal: in class some students start their assignment from mathematical writing, and
then after several attempts (i.e. results which do not fulfill exactly a 4/4 bar) they realize
that their strategy is wrong and that they should start from music. Or, it can happen that,
starting from a music context, they will manage to solve an undetermined system, which is
impossible, as in music there is a solution. At this stage it becomes necessary to
understand where the strategic mistake lies and correct it.
Workshops, based on their target or on the moment when they are proposes, are used to
understand or deepen a mathematical concept, which shall be built and acquired. They lead to the
awareness of what “is being done”.
Sharing is often one of the final stages of workshops and certainly one of the most important: the
student (or group of students shows to the rest of the class their own achievements.
Workshops can be carried out, based on the context and on the subject treated, first individually
and then in groups, in small groups or involving the whole class since the very beginning.
Sometimes also frontal lessons were experimented, either to recall some preparatory notions to
certain activities, or to “institutionalize” and formalize mathematically the concepts grasped
during the workshops.
.
3. DESCRIPTION OF EXPERIENCES WITHIN SCHOOLS
In the following paragraphs we shall recall all the variables observed during the transfer
experience, which were commented in the previous paragraphs, as well as the formulated
objectives.
A short paragraph on the educational offer and a synthesis of the school planning which took
place before the transfer shall introduce them..
9
3.1 Action Planning in Schools
Following the workshops held in Bologna, during which the teachers of the various partner schools
had the chance to get to know the Doremat method and its related mathematics topics, a series of
meetings were organized with the teachers of the schools involved. Such meetings took place in
the months of June and July 2013, at the partner schools and they were meant to organize,
together with the teachers of the classes involved in the methodological transfer, all the topics (or
teaching units) to be addressed, action times and places, beside all the necessary material. As far
as the topics and times, we shall report hereafter a summarizing table:
WP 6 – TRASFER IN ITALY
MEETING SUMMARY
SCHOOLS
MEETING DATE
EXPERIMENTATION
PERIOD
October 2013/February
2014
IIS RUFFILLI – FORLI’
10/06/2013
ISTITUTO CALASANZIO –
EMPOLI
13/06/2013
October 2013/ February
2014
LICEO CHIERICI – REGGIO 03/07/2013
EMILIA
October 2013/ February
2014
PERSOLINO STROCCHI –
FAENZA
04/07/2013
October 2013/ February
2014
OSFIN – RIMINI
10/07/2013
October 2013/ February
2014
CONTENTS
One first-year class
Fractions
Expressions
First Degree Equations
One first-year class
Expressions -First Degree
Equations
One second-year class
1st Degree Linear Systems
Introduction to Geometry
(translations)
Two first-year classes
Expressions with Fractions
Direct and Indirect
Proportionality
One first-year class
Expressions with Fractions –
Geometry (straight lines,
polygons)
One second-year class
Inequalities – Linear Systems
Two first-year classes
Fractions
Expressions
As you can observe from the table, the topics were different in different schools. However, for
each class fractions and expressions with fractions were discussed although sometimes not
directly4, as they are preparatory for the approach to all other topics (beside the physics of sound,
which during this experience was not taken into consideration anyway). As we shall see in the
following paragraphs regarding the experience in each single school, sometimes such contents
became much wider, through workshops devised during the lectures themselves therefore
allowing a deeper and more flexible discussion.
4
Si veda, per esempio, l’Istituto Calasanzio di Empoli.
10
As far as the environment, the most suitable roomss were found: sufficiently wide and located in
areas where the noise would not disrupt other lessons. As far as the instruments used (drums,
bongos and glockenspiel) we found them on the spot in order not to have to move them around
daily. Other materials were simply a board and notebooks.
The lessons were held by the trainers with the presence of the teachers of the schools involved.
A big part of the first and of the last lessons were dedicated to the assignment of a pre-test and of
a post-test, finalized to the quantitative analysis of the Doremat method.
.
3.2 I.P.S.A.A. PERSOLINO E STROCCHI – FAENZA (RA)
Trainers: Antonio Bianchi, Nicola Magnani, Rachele Vagni, Giovanni Curti
Teacher: Maria Montanari (Mathematics)
Experimental Groups:
I TUR – 25 students (19 girls, 6 boys)
II TUR - 29 students (20 girls, 9 boys)
I TUR
- Description of the class environment
The class is wide and well-lit, suitable for the Doremat lectures.
The students, in general are very lively: they talk to each other and only after the teacher rebukes
them, they become silent. During the school year moreover two students were suspended.
The teacher herself explains to the trainers that the class is unruly and not interested in
mathematics. In fact, to the initial questions asked by the trainers, such as, for instance “do you
like mathematics? And music?”, “do you believe that mathematics is useful?”, the students
answer negatively as far as mathematics and positively as far as music. The students were
impressed by the presentation of the Doremat method, and, most of all they were incredulous and
intrigued.
Four students were born in other countries; and beside one of them, the rest were very silent and
they did not interact much with the rest of the class.
None of the students played a music instrument or had a deep knowledge of music, their only
experiences with it dated back to fist-degree secondary school where they learned how to play the
flute and how to read a music score. The majority of them had soon forgotten all such
competence.
.
-
Didactics
Certain workshops foresaw the presence of students at the board involving at the same time those
who remained seated at their desks, others required students to work individually, basically on the
invention of rhythmic sequences and expressions. However, also those last workshops foresaw a
moment of sharing of the students’ work with the rest of the class, developing this way a group
11
discussion as well as music experience performed in group. Other workshops, for instance, the one
regarding the equivalence between rhythmic figures and fractions, involved the whole class from
the very beginning.
Workshops with different groups simultaneously were avoided in order to be able to better
manage activities: working individually (or with their deskmates) the students were more involved
in the activities proposed, at the same time, with the presentation of their work to the rest of the
group we also had a moment of sharing and discussion.
Moreover, some frontal lessons were also held, which were necessary both for the difficulties of
calculation presented by the students and for the formalization of some of the concepts addressed
(also musically speaking).
-
Learning and Objectives
On the specific arithmetic contents, such as fractions and arithmetic expressions with fractions,
the following cognitive objectives were defined by the trainers:
o Reproducing the statement of a principle/rule. Performing elementary operations and automated
procedures (elementary learning). Through music language and experience, fractions, equivalences
among fractions and fraction operations are addressed. Music helps students understand the
concepts of fraction and equivalence, conferring to fractions different semiotic representations (it
also shows different aspects of the concept of fraction); in the calculation of expressions, because
by performing rhythmic sequences the students realize that they are actually summing fractions.
o Applying knowledge /procedures to other cognitive contexts (intermediate learning).
o Recognizing a concept-principle-rule. Changing code (language) to known notions (intermediate
learning). Fractions can be associated in a natural way to rhythmic figures and their sum and
succession of figures; you can think of the equivalence among rhythmic figures as an equivalence
among fractions. Therefore knowledge and procedures belonging to mathematics can be
transferred into a music context and viceversa. Starting from the music experience, also through
the invention of rhythmic sequences, also the property of equivalence relation is introduced.
As far as geometry, straight lines and polygons were just mentioned, as the arithmetic part was
already quite full. Moreover, in order to discuss straight lines it is necessary to introduce the
concept of harmony in music, which would require a longer time as opposed to the time available.
The hypothesized cognitive objectives were the following:
o Reproducing the definition of a certain concept (elementary learning)
o Classifying based on a known criterion (elementary learning)
o Changing the code (language) to known notions (intermediate learning)
Music uses a language taken from geometry in order to describe motions (straight, parallel,
contrary…) which takes us to the reciprocal positions of two straight lines on a plane. The
groupings of rhythmic figures takes us to the concept of regular polygons. Using a metaphor
between a musical concept and a mathematical concept means finding invariants characterizing
such concepts (or objects), therefore learning to know them better..
12
Because of the difficulties shown by the students, and encountered by the trainers and by the
teacher as far as arithmetic calculation is concerned and due to the short time available ( a total of
15 hours and an hour per lesson) we decided along the way to concentrate on arithmetic,
deepening the topics foreseen and expanding on them (we also addressed percentage
calculation).
At the end of the action the students showed a better mastery of calculation rules and algorithms
(also proved in the results of the Quantitative Analysis); moreover, it seemed that they could use
more easily expressions as an instrument and that they could interpret similarities as a symbol of
relations..
-
Motivation to Learning
Since the very beginning the students have shown an interest in the method. Although they were
still very lively, they worked on the activities proposed, often reacting enthusiastically. Particularly,
they were successful in inventing rhythmic sequences and arithmetic expressions, through which
they could express their creativity and at the same time exercise a rational logic. As indicators of
their participation we analyzed: the questions asked by the students during the lectures, their
willingness to do the assignments and solve mathematics and music problems written on the
board, presenting their own work to the rest of the class and to the teachers. All of those
behaviors revealed their participation, their willingness to try and the will of doing mathematics.
As far as the satisfaction tests, assigned to the students, positive answers were given, specifically
to the following questions: “do you think that the Doremat method helped you better understand
mathematics?”, “and music?”, Do you believe it would be useful to continue learning mathematics
through music?”, “the teachers Nicola Magnani and Antonio Bianchi have managed to: - involve
the class; - expose topics clearly;- relate suitably to the class”.
Moreover, in their final comments, some students manifested their involvement (“I think this
project was useful because it has managed to involve us more”) and the wish to continue and to
be able to play other instruments beside drums.
.
3.3 I.I.S. ROBERTO RUFFILLI - FORLI’
Trainers: Antonio Bianchi, Nicola Magnani, Rachele Vagni, Giovanni Curti,
Teacher: Silvia Golfarelli
Experimental Group: I A
20 students (8 girls, 12 boys)
- Descriptions of the Class Environment
The class is wide and well-lit, suitable to host the Doremat lectures.
Students were very unruly: they spoke to each other, they stood up from their seats, they shouted
and only after having been rebuked repeatedly by the teacher they became silent. During the
school year two students were suspended and one of them started to miss several classes.
13
The teacher herself explained to the trainers that the class was unruly and not interested in
mathematics. Actually, to the starting questions asked by the trainers, “do you like mathematics?
And music?”, “do you believe that mathematics is useful?”, the students answered negatively as
far as mathematics and positively as far as music. The students were impressed by the
presentation of the Doremat method, and, most of all they were incredulous and intrigued, while
others were very skeptical. In general also their performance with mathematics (in terms of
working and teacher assessment) were insufficient, exception made for some students,
particularly for one student with good grades and correct behavior. There were three students
who were not born in Italy, who seemed to be well integrated in the class. Two students had been
assigned a special teacher due to learning difficulties.
None of the students played a music instrument or had any advanced knowledge of music; the
only experiences they had ever had with music were those related to first-degree secondary
school, where they were taught to play the flute and to read a music score. Such rudiments had
been soon forgotten by all the students.
Didactics
Some workshops were proposed trying to involve the whole class, for instance the one on the
equivalence between rhythmic figures and fractions; this way the class’s attention would be
focused on the proposed activity, with all the students involved, avoiding chaos.
Other workshops required students to go to the board involving at the same time the others at
their desks, generally those based on the invention of rhythmic sequences and expressions.
However also those last workshops foresaw a time when the students’ work was to be shared with
the rest of the class, in order to elicit group discussions and group music performances.
Workshops foreseeing the presence of different groups were avoided in order to be able to better
manage the: working either individually (or with a deskmate) the students were more focused on
the proposed activity, and at the same time the trainers could monitor the students by walking
among their desks and interacting with them, therefore avoiding dispersive and chaotic situations.
Moreover some frontal lectures were held, necessary both due to the calculation difficulties
presented by the students and to formalize some concepts (also related to music).
.
Learning and Objectives
As far as fractions and arithmetic expressions are concerned, the trainers foresaw the following
cognitive objectives:
o
Repeat the statement of a principle/rule. Perform elementary operations and automated
procedures ( elementary learning). Through language end music experiences, fractions,
equivalence among fractions and operations with fractions are addressed. Music helps students
understand the concepts of fraction and equivalence, giving to fractions different semiotic
representations (and it also shows different aspects of the fraction concept); in the calculation of
expressions, because by performing rhythmic sequences the students realize that they are actually
adding fractions..
14
o
Applying knowledge/procedures to other cognitive contexts (intermediate learning).
o
Recognizing a concept-principle-rule. Changing code (language) to known notions
(intermediate learning): Fractions can be associated naturally to rhythmic figures and their sum to
a sequence of figures; one could think of the equivalence among rhythmic figures as an
equivalence among fractions. Therefore knowledge and procedures belonging to mathematics can
be transferred to a music context and viceversa. Starting from the music experience, also through
the invention of rhythmic sequences, moreover the properties of equivalence relation can be
applied, therefore introducing algebraic thought. An unfinished rhythmic sequence becomes a
linear equation whose solution is nothing but the missing figure needed to complete it.
Afterwards, the rhythmic sequences are put in an equivalence relation and operations are
performed on them, introducing this way the principles of equivalence for equations..
Students showed big gaps in arithmetic calculation, therefore we insisted more on that aspect.
Workshops therefore were aimed at calculation activities supported by music: when they invent a
rhythmic sequence students sing and organize rhythmic figures. It is important to make them
reflect on such activity because it will make them aware of the mechanisms they are using and it
makes them better evaluate their answers (i.e. the results of the arithmetic operations that they
are performing), and to have a better control of their calculation activity, understanding the
meanings of mathematic symbols. In the specific context, this activity is not easy to perform:
group discussions on such aspects were often lost due to the difficulty in managing the class.
At the end of the action, however, the majority of students showed an increased mastery of
calculation rules and algorithms (also proven by the results of the Quantitative Analysis).
.
Motivation to Learning
Since the very beginning some students have manifested, an interest for the method, including
those who are generally not interested in mathematics and demotivated, while others were
skeptical. Particularly the workshops related to the invention of rhythmic sequences and
arithmetic expressions were particularly successful, as the students were able to express their
creativity and at the same time to employ a rational kind of logic. During said workshops some of
the students worked very hard, showing an increased motivation. For other students, the
experience did not seem to be so effective, but we believe that with more time available it could
have produced more satisfying results from this point of view. Actually educational objectives (i.e.
our goals) are in general “long term” goals and pursuing them requires time, especially in
“difficult” contexts as several variables enter into play (social context, family, cultural background
etc..)..
As far as the satisfaction tests, assigned to the students, positive answers were given, specifically
to the following questions: “do you think that the Doremat method helped you better understand
mathematics?”, “and music?”, Do you believe it would be useful to continue learning mathematics
through music?”, “the teachers Nicola Magnani and Antonio Bianchi have managed to: - involve
the class; - expose topics clearly;- relate suitably to the class”.
Some students answered that they would like to repeat the experience.
15
3.4 ISTITUTO CALASANZIO – EMPOLI (FI)
Trainers: Antonio Bianchi, Nicola Magnani, Rachele Vagni, Giovanni Curti, Matteo Ignesti
Teacher: Carla Bianchi Cioni
Experimental Class: 1st year of LICEO, 15 students (6 girls, 9 boys)
Trainers: Antonio Bianchi, Nicola Magnani, Rachele Vagni, Giovanni Curti, Matteo Ignesti
Teacher: Massimo Amato
Experimental Class: 2° year of LICEO, 13 students (5 girls, 8 boys)
1st year of LICEO
- Description of the Class Environment
The class is wide and well-lit, suitable for the Doremat lectures.
The students were calm but not passive: they listened, took notes and participated in the lectures,
answering questions when asked and sometime the asked questions themselves. During the
school year one student missed many classes and the teacher told us that he had some problems,
without going into details: he appeared to be shy and spoke very little.
In general, it seemed that the students did not experience any particular difficulty in mathematics
in terms of their achievements (i.e. marks), but many of them did not seem really interested or
fascinated by this discipline, while others clearly did not like it. This became clear when the
trainers asked them, “do you like mathematics?” and music?”, “do you believe that mathematics is
useful?”, “do you believe that mathematics is beautiful?”
During the presentation of the Doremat lectures, some students were impressed and intrigued.
None of the students played a music instrument of had any deep knowledge of music; the only
experiences they had dated back to fist-degree secondary school, where they learned to play the
flute and to read music score. Those notions had been however soon forgotten by the majority of
students.
Didactics
We proposed both workshops foreseeing the presence of students at the board and involving at
the same time the students seated at their desks, as well as workshops in which students were
involved in group activities, generally for the invention of rhythmic sequences.
All those workshops foresaw the sharing of the students’ work with the rest of the class, therefore
developing group discussions and group music performances. The moment of sharing was also
useful for the trainers in order to problematize the given music context.
Also some workshops were proposed involving the whole class, for instance the one on the
equivalence between rhythmic figures and fractions.
Moreover some frontal lessons were carried out, in order to discuss the solution of linear
equations and formalize some of the concepts introduced (also relating to music).
16
Learning and Objectives
As far as arithmetic expressions with fractions and linear equations, the trainers decided the
following cognitive objectives:
o
Perform elementary operations and automated procedures (elementary learning). Through
language and music experience, fractions are introduced, the equivalence between fractions and
operations with fractions. Music helps students understand the concepts of fraction and
equivalence, giving to fractions different semiotic representations (it also shows different aspects
of the concept of fraction); in the calculation of expressions, because by performing rhythmic
sequences the students become aware that they are actually summing fractions..
o
Applying knowledge/procedures to other cognitive contexts (intermediate learning).
o
Recognizing a concept-principle-rule. Changing code (language) to known notions
(intermediate learning). Fractions can be associated naturally to rhythmic figures and their sum
can be associated to a list of figures. One could think of the equivalence among rhythmic figures as
the equivalence among fractions. Therefore knowledge and procedures belonging to mathematics
can be transferred to a music context and viceversa. Starting from the experience of music, also
through the invention of rhythmic sequences, one can moreover apply the property of the
equivalence relation, therefore introducing algebraic thought. An unfinished rhythmic sequence
becomes a linear equation whose solution is nothing but the missing figure needed to complete it.
After, the rhythmic sequences are put in an equivalence relation and operations are performed on
them therefore introducing the principles of equivalence for equations..
Some students showed some difficulties with arithmetic calculation, but nothing particularly
significant, it was mostly due to uncertainty on certain concepts, for instance on fractions. The
workshops were therefore aimed more at making the students assimilate certain mathematics
concepts which showed both the creative and the instrumental aspect of mathematics. The
method supports this point of view giving to the students the opportunity of inventing a rhythmic
sequence thinking of fractions, of numbers and of their relations in the music context, of the
organization of rhythmic figures (i.e. fractions). On the other hand an equation can be that
instrument with which to be able to complete an unfinished rhythmic sequence: the students, in a
“music” context will answer correctly without realizing that they are actually solving an equation.
Making them reflect on this activity is important because it makes them aware of the mechanisms
that they are using and it makes them better evaluate their answers and gives them a better
control of some procedures, helping them understand the meaning of mathematical symbols.
Motivation to Learning
Since the very beginning the students have manifested an interest for the method, including those
not passionate for mathematics. Particularly the workshops in which they were requested to
invent a rhythmic sequence and arithmetic expressions were particularly successful, or those
where they had to invent duets and linear equations, through which the students could express
their creativity and at the same time exercise rational logic. Also for simple activities, like for
instance clapping their hands to reproduce rhythmic figures or to follow a performance, they
participated actively.
As far as the satisfaction questionnaire assigned to the students, to the questions:
“do you believe that Doremat has helped you better understand mathematics?”,
17
“Do you believe it would be useful to continue learning mathematics through music?”,
The majority of answers were positive, only a small part of the students answered negatively,
considering “useless” the experience in order to learn mathematics.
While to the questions: “do you believe that Doremat helped you know music better?” and “the
teachers Nicola Magnani and Antonio Bianchi managed to: -involve the class; -expose clearly the
topics; relate suitably with the class” were all positive..
Moreover, in this class an essay was assigned during the Italian lesson: “Mathematics and I, my
relation with mathematics from primary school until today”. About this we would like to point out
that, although the essay was written during the Italian lesson and the title did not refer explicitly
to Doremat, all students wrote about the Doremat experience, defining it as positive.
3.5 ASSOCIAZIONE OSFIN - RIMINI
Trainers: Antonio Bianchi, Nicola Magnani, Rachele Vagni, Giovanni Curti,
Teacher: /
Experimental Classes:
Tourist Promotion and Hospitality Operator, 20 students (10 girls, 10 boys)
Accounting and Secretary Operator, 23 students (16 girls, 7 boys)
Hereafter we shall report for both schools a single descriptions of the items observed. The choice
is motivated by the many similarities of the two experiences, avoiding repetitions.
- Description of the Class Environment
The class is small and not well lit, we would have needed a more suitable place for our lessons.
Toward the end of our action we were given the possibility to teach in a better place; a less
suffocating class.
The students were very unruly: they talked to each other, raised from their chairs, shouted and
during the lectures their behaviors were not suitable. For instance they used their cellphone or
ate; and after being rebuked several times they still did not change their attitude or behavior. The
Tourist Promotion and Hospitality Operator class was more unruly than the other, but it was also
more active, although both classes were strongly uninterested in mathematics and in general they
are not interested in learning.
In fact, to the initial questions of the trainers, “do you like mathematics? And music?”, “do you
believe that mathematics is useful?”, the students answered negatively about mathematics but
positively about music. They believed that mathematics corresponds simply to calculations and
that therefore it is useless since calculators can do it.
When we presented the Doremat method some students were intrigued while others were very
skeptical. During the pre-test only a few students worked, the majority of them answered
randomly or did not answer to some questions altogether.
18
During the lessons some students did not even have a pen or paper, it was the trainer who had to
encourage them to get some papers and take notes.
None of the students played an instruments of had a deep knowledge of music; the only
experiences they had were those during first-degree secondary school where they learned to play
the flute and to read a music score. Such notions had been soon forgotten by all students.
.
- Didatctics
Some workshops were proposed trying to involve the whole class, for instance, the one on
equivalence between rhythmic figures and fractions; this way the class’s attention would be
focused on the proposed activity, and all the students would be involved avoiding chaos.
Other workshops foresaw the presence of students at the board and trying to involve at the same
time other students seated at their desks; this strategy did not appear to always be winning: some
students listened from time to tine only, sometimes talking to each other. During those workshops
the most uninterested and unruly students were asked to go to the board, while other asked to
participate of their own will.
We believed it useful moreover to propose some “assessment tests” based on the topics of
mathematics-music seen during the previous lessons: during said moments the trainers walked
among the desks in order to help studens and involve them trying to help them reflect on the
tasks assigned.
All the workshops foresaw a moment of sharing in order to develop a group discussion and a
group performance.
We avoided workshops with different groups in order to better manage activities: working
individually or mostly with their deskmate and with the help of trainers the students were more
involved in the proposed activity and at the same time the trainers could monitor them walking
among the desks and interacting with the students.
Moreover some frontal lessons were held, needed both for the calculation difficulties shown by
the students and for the formalization of some concepts (also related to music).
Learning and Objectives
The topics agreed upon were fractions and arithmetic expressions with fractions; however during
the last two lessons also some simple linear equations were introduced, since there was some
extra time available. The trainer decided to target the following cognitive goals:
o Repeating the statement of a principle/rule. Carrying out elementary operations and automated
procedures (elementary learning). Through language and music fractions, equivalence among
fractions and operations with fractions are introduced. Music helps the students understand the
concepts of fraction and equivalence, giving to fractions different semiotic representations (and it
also shows different aspects of the fraction concept); in the calculation of expressions, because by
following a rhythmic sequence the students realize that they are actually summing fractions..
o Applying knowledge/procedures to other cognitive contexts (intermediate learning).
o Recognizing a concept-principle-rule. Changing the code (language) to known concepts
(intermediate learning). Fractions can be associated naturally to rhythmic figures and their sum to
a list of figures; the equivalence among rhythmic figures can be thought of as an equivalence
among fractions. Therefore knowledge and procedures belonging to mathematics can be
19
transferred to a music context and viceversa. Starting from the musical experience, also through
the invention of rhythmic sequences, the properties of equivalence relation can be applied,
therefore introducing algebraic thought. An unfinished rhythmic sequence becomes a linear
equation where the solution is simply the missing figure needed to complete it. Afterwards the
rhythmic sequences are put in an equivalence relation and operations are performed with them,
therefore introducing the principles of equivalence for equations..
-
The students showed big gaps in arithmetic calculation, therefore we insisted more on that aspect.
The workshops were targeted to calculation activities supported by music: when they invent a
rhythmic sequence the students sing and organize rhythmic figures. Making them reflect on such
activity is important because it makes them aware of the mechanisms they use and it makes them
better evaluate their answers (the results of the arithmetic operations that they are performing),
to have a better control on their calculation activities, understanding the meaning of mathematic
symbols. Specifically, this activity was not easy to realize: group discussions on such aspects were
often missed out due to the difficulties encountered in managing the class, so they were possible
only for a part of the students.
At the end of our action, however, a big part of the students showed a better mastery of
calculation rules and algorithms (also proven by the results of Quantitative Analysis)
.
Motivation to Learning
Since the very beginning some students manifested an interest for the method, also those not
interested in mathematics and generally demotivated, while others were skeptical. The
participation, commitment and interest of some students were clear considering the attention
they paid in the activities proposed, in their willingness to solve exercises and problems, in their
increased curiosity, that was noticeable fom the questions they asked. Particularly, the workshops
of invention of rhythmic sequences and arithmetic expressions were particularly successful, as the
students could express their own creativity and at the same time use rational logic. During some of
such workshops some students worked very hard, showing an increased motivation. For other
students the experience did not seem to be so effective, but we believe that in the long run they
could have achieved far better results from this point of view. Actually the educational targets (i.e.
the finalities) are to be considered generally as “long term” achievements, and reaching them
requires time, especially in “difficult” contexts, where many variables have to be taken into
consideration (logistics conformation of the class, social context, family, students’ cultural
background…).
As far as the satisfaction questionnaire assigned to the students, to the questions:
“do you believe that Doremat has helped you better understand mathematics?”,
“Do you believe it would be useful to continue learning mathematics through music?”,
some positive answers were collected in the class for Tourist Promotion and Hospitality Operator;
while in the Accounting and Secretary Operator some of the answers were positive, other negative,
because “after a while the experience became boring”.
20
While to the questions: “do you believe that Doremat helped you better understand music?” and
“the teachers Nicola Magnani and Antonio Bianchi have managed to: -involve the group; - expose
the topics clearly; relate suitably with the class” the answers were all positive.
3.6 LICEO GAETANO CHIERICI – REGGIO EMILIA
EXPERIMENTAL CLASSES:
I B, 25 students (20 girls, 5 boys)
Trainers: Antonio Bianchi, Nicola Magnani, Rachele Vagni, Giovanni Curti
Teacher: Francesco Curti
I C, 27 students (22 girls, 5 boys)
Trainers: Antonio Bianchi, Nicola Magnani, Silvia Santunione, Giovanni Curti
Teacher: Stefano Francesconi
Hereafter we shall report for both schools a single description of the items observed. This choice is
due to the many similarities of the two experiences, in particular for what concerns the objectives
and didactics; we would like therefore to avoid an excessive repetition of the items which we shall
describe. In case of need, the differences shall be specified.
- Description of the Class Environment
The class chosen for the lectures (wide and well-lit) is not the one where the students normally
follow their lessons: the students of the experimental group move to this class for the Doremat
lessons.
The classes, although they were both first year classes of secondary schools, showed to be
structurally different from the very beginning. In one of them (I B) it appeared to be much more
difficult to attract attention and to maintain a suitable condition to hold the lecture; the other on
the contrary ( I C) was much more disciplined and responded to the teachers’ stimuli much better.
In general however, the students were lively and resourceful: they responded to the teachers’
incentives actively, they answered correctly, they confronted with each other also about what the
teachers said.
Only one of the students played a music instrument (the guitar, which he had started studying not
long before), but nobody had a deep knowledge of music; the only experiences that they had
dated back to first-degree secondary school, where they had learned to play the flute and to read
a music score.
Didactics
Different workshops were proposed, foreseeing the presence of students at the board and
involving simultaneously those seated at their desks, as well as workshops engaging students in
group activities, generally inventing rhythmic sequences (also in the form of duets) and
expressions.
All such workshops, foresaw a moment of sharing of the students’ work with the rest of the class,
developing therefore group discussion and the performance of group music.
21
Some workshops, particularly the one regarding the equivalence among rhythmic figures and
among fractions involved the whole class.
Experiences and activities that started from arithmetic were also proposed and, emphasizing more
the relation of mathematics symbols (as in the case of equivalence) and on the properties allowed
the introduction of algebra.
Also some frontal lectures were held, aimed at formalizing some concepts (also relating to music)
and some procedures.
.
Learning and Objectives
For arithmetic expressions with fractions and direct and indirect proportionality, the trainers
thought of the following cognitive objectives:
o
Performing elementary operations and automated procedures (elementary learning).
Through language and music fractions, equivalence among fractions, operations with fractions and
their possible relations are studied (proportionality). Music helps students understand the concept
of fraction, the operations with fractions and their possible relations (proportionality). Music helps
students calculate expressions because by performing a rhythmic sequence, the students are led
to the awareness that they are actually adding fractions.
o
Applying knowledge/procedures to other cognitive contexts (intermediate learning).
o
Recognizing a concept-principle-rule. Changing code (language) to known notions
(intermediate learning). Fractions can be associated naturally to music figures, and their sum to a
list of figures; one could think of the equivalence among rhythmic figures as of an equivalence
among fractions. Therefore knowledge and procedures belonging to mathematics can be
transferred to a music context and viceversa. Starting from music, also through the invention of
rhythmic sequences, the properties of equivalence relation can be applied, therefore introducing
algebraic thought..
o
Analyzing/decoding
o
Synthetizing/schematizing contents (higher learning). Music offers a variety of experiences
(sound, movement etc) which become realities to be analyzed. Particularly, an emphasis is placed
on rhythmic figures and their relations, as well as on the pitch of sounds, which can be described
both in terms of numbers and relations among them, as well as in terms of proportions..
The proactive environment also allowed trainers to address other topics, such as percent
calculation.
Some of the students showed some difficulties in calculation, but nothing extremely significant,
exception made for some specific student, but at the end of our action, the students appeared to
be more sure of themselves as far as calculation.
Moreover, the problem solving and discussion of possible strategies to solve a given problem
facilitated the dialogue and the exchange of ideas and allowed the students to adopt personal and
original strategies.
-
Motivation to Learning
22
Many of the students were enthusiastic of the training; they worked hard on the proposed
activities, seriously and responsibly. Particularly, the workshops developed in small groups, when
they had to invent rhythmic sequences and expressions, through which the students could express
their creativity and at the same time use their rational logic, made them realize that they have
good problem-solving skills.
Other indicators of their participation were: the questions they asked during the lessons, their
willingness to do mathematics and music exercises at the board, to present their own personal
work to the rest of the class and to the teachers, to answer positively to the trainers’ stimuli.
All those behaviors reveal participation by the students, their willingness to take a chance, their
willingness to do mathematics..
As far as the satisfaction questionnaire, assigned to students, to the questions: “do you believe
that Doremat has helped you better understand mathematics”,
“Do you believe it would be useful to continue learning mathematics through music?”,
“The teachers Nicola Magnani and Antonio Bianchi have managed to: - involve the class; - expose
the topics clearly; relate suitably with the class”
In general the answers given were positive in the I C class; while in the I B class some of the
answers were positive, others negative, as they considered the experience as “futile” and “not
very engaging”.
While the question: “Do you believe that Doremat has helped you get to know music better?”
mostly elicited positive answers.
.
3.7 Participation and Involvement of School Teachers
In general school teachers were attentive to the lessons and more or less participative to the
activities. Particularly, the teachers of the Chierici School of Reggio Emilia and Calasanzio of Empoli
showed a lot of enthusiasm for the lessons, participating also to the experiences and trying to help
involving their students. We believe that this kind of attitude is very important, both for the
environment and for the possibility to share certain aspects of our methods with school teachers.
The Calasanzio school teachers were then interviewed..
-
Conclusions
3.8 Proposed Improvements and Strong Points
In general the method gives to Mathematics a more complete and wider perspective, as opposed
to regular classes, as it has emerged also from the interviews to the Calasanzio School teachers.
We believe that Doremat allows to see mathematics both from a cultural and an instrumental
point of view, therefore changing the distorted vision that the general public often has of this
discipline, with a positive outcome in terms of learning. As for motivation to learning, the
indicators that we have taken into consideration (questions asked by the students during the
Doremat lectures, their responses to the trainers’ stimuli, their behaviors during both individual
23
and group activities) and the satisfaction questionnaires, all suggest that the majority of students
have enjoyed a positive experience.
Still on the didactic plan, we believe that this method contributes to the construction of the
meaning of some mathematical concepts. We have observed, in fact, that the concepts of
1
mathematics do not exist in physical reality: polygons, numbers, for instance, 1, 3, 2 , operations..,
are all mental constructions of objects and abstract structures. We have representations of
1
mathematical concepts; so for instance” 2 “ and “a half” are two representations (respectively in
the arithmetic register and in the verbal register) of the same concept, in the same way as, in our
case, two (suitable) music figures. This process of abstraction should start from a real experience;
it is a process consisting in seeing similarities, properties, equivalences in order to construct a
concept. But from this principle, we should not believe that teaching mathematics should be
something detached from reality, or an abstract teaching. Mathematics constitutes, for its
flexibility and for the forms of reasoning that it requires, an instrument for the interpretation of
reality; therefore the teaching must start from experience (in order to then proceed to
abstraction), the contents should be as much as possible taken from other disciplines within a
cross-curricular logic, which can detect common structures (and less common ones) to
mathematics and to other subjects. Music, in that sense, can also provide a powerful stimulus.
From a methodological point of view music lessons constitute the ideal environment from which
to problematize situations within problem-solving activities, the most important method to learn
mathematics. One starts from a musical case-concept, from which it is possible to re-discover, reinvent and re-build mathematical concepts; the teachers should lead the lesson without exceeding
in their leadership and allowing the students to become the protagonists of class activities. It is
clear that in this situation the students should be active, participative and very motivated.
From this point of view some lessons did not really exploit all the potential of the Doremat
method, but certainly problem-solving activities are quite engaging and they require a lot of time,
therefore they could not be used for all the topics foreseen by our program.
Finally one of the educational aspects, which we believe should be highlighted, is how Doremat
works on the class as a group of individuals sharing a common target and relating to one another.
In that sense, this experience has allowed students to get to know each other better, to
understand those aspects in others which had remained hidden up to that moment, to integrate
through group workshops. People belonging to different nationalities and cultures have managed
to “meet” thanks to two universal languages, music and mathematics: quiet and shy kids,
experiencing some social difficulties have managed to express themselves.
3.9 The Role of Music
Hardly any student had a knowledge or a competence which could be used during the workshops.
Many of them – about half - showed some quite good rhythmic skills, although they were
unaware of it. In fact they not only did not have the slightest knowledge of music reading and
writing, but they did not think they had, not even intuitively, a rhythmic competence, much less
did they think they could use it.
24
It is therefore just fair that workshops also address the aspect of rhythmic figure reading, as the
semiology link between music figures and fractions is one of the contents on which this activity is
based.
During the transfer and during the lectures in class, we did not really insist in requiring a better
accuracy in the reading of rhythmic sequences and in theoretical music notions, since this would
have required too much time. It seemed however that the lack of music competence could be an
obstacle to the good outcome of workshops, which however could be inspired by a less leading
and more problem-solving inspired model, as we mentioned in the previous paragraphs. As a
consequence, we would need a different articulation of didactic activities and probably more time
at our disposal.
If the method works, starting from an experience of a music concept-case, i.e. from a concrete and
active music experience, workshops must allow the assimilation of the music concept-case on
which the mathematic reasoning is based.
I truly believe that this is the main strong point of the method, i.e. the possibility that music has to
work as a catalyst, allowing to discipline reasoning on the same subject to be put together and to
give a meaning to the experience of that subject through a concrete, active and motivating
activity.
I believe moreover that one of the best developments achievable is represented by musical
invention, which can be declined also in the form of a workshop, for which in fact workshops are
the privileged learning environments. Many analyses of compositions or, to be even more generic,
of sound objects, are carried out using geometric terms, and I believe therefore that it would be
possible to proceed the other way round, i.e. to achieve a didactics of music production through
geometric and mathematic suggestions and reasoning, as it often happened for the music
produced in the last century.
3.10
Teachers and Trainers’ Point of View
During the project’s implementation within the “Leonardo” program, we have had the occasion to
teach using the Doremat method in several classes belonging to different schools.
This has allowed us to observe in different school contexts, how much the Doremat method is
effective.
In the classes where mathematics is a virtually unknown subject, and certainly not really
appreciated, practical workshops in general have bridged the big gaps which many students
showed. In fact up to that moment the majority of them had only taken part in theoretical classes,
and this was for them a chance to reason mathematically from a different perspective, from the
perspective of music.
It is the practicality of the method which is its strong point, moreover, as a musician I strongly
believe in music education in schools, at all levels, at every age. Music is one of the best ways to
express oneself, to grow properly both from a cultural and from a social point of view.
25
The workshops have highlighted also other values: many students showed an interest in the
music-mathematics topics discussed in class, they created their own rhythmic scores based on
their personal taste, which they then transformed into expressions, equations, systems etc.
Another very important aspect of the method is therefore also the freedom given to the students:
they created their own equations, they did not copy them from the board, and all this thanks to
the application of mathematical rules to the music that they themselves had composed! It is
priceless when you hear a student say: “teacher, it is incredible, I would have never believed that
you could do such things with mathematics!”The experimentation in Latvia went very well, as I could see by monitoring of the trip to Riga.
During the demonstration to the Latvian students, my satisfaction in seeing our workshops, those
same workshop that we had invented, were taught at the other side of Europe, was enormous. I
believe however that the workshops we saw in Latvia are just the first step toward the
development of more future co-operations, and it is doubtless that in that case a trainers’ training
program shall be organized.
26