Title in Portuguese/Spanish (secondary language)programmi.wdfiles.com/local--files/umi/ManifestoUMI.pdf · worlds, but scientific in methods, since it uses logic (Odifreddi, 2000).5

ISSN 1980-4415

DOI: http://dx.doi.org/

Bolema, Rio Claro (SP), v. , n. , p.xxx-xxx, mês. Ano 1

Interdisciplinarity and learning/teaching mathematics

Title in Portuguese/Spanish (secondary language)

Enrico Rogora1*

0000-0002-1268-8653

Francesco Saverio Tortoriello2**

ORCID iD

Abstract

Complex problems need interdisciplinary approaches. Thinking in an interdisciplinary way asks for changes in our views on didactical problems and could provide a new framework for dealing also with disciplinary didactical problems. In this paper we propose a methodology which we have called “globally interdisciplinary laboratories” as an effective way to practice interdisciplinary didactics at high school level. We discuss the effectiveness of this methodology in the learning and teaching of mathematics. Globally interdisciplinary laboratories are designed by a pool of researchers in didactics in collaboration with high school teachers of different discipline and they are delivered in class by the teachers "in compresence". They have been tested in Italy in many classes which are part of the national project “Licei matematici”. Keywords: Interdisciplinarity. Didactics of mathematics. Learning mathematics.

Resumo/Resumen

Problemas complexos precisam de abordagens interdisciplinares. Pensar de forma interdisciplinar pede mudanças em nossos pontos de vista sobre problemas didáticos e pode fornecer uma nova estrutura para lidar também com problemas didáticos disciplinares. Neste trabalho, propomos uma metodologia que chamamos de "laboratórios globalmente interdisciplinares" como uma forma eficaz de praticar a didática interdisciplinar no ensino médio. Discutimos a eficácia desta metodologia na aprendizagem e ensino de matemática. Laboratórios globalmente interdisciplinares são projetados por um grupo de pesquisadores em didática em colaboração com professores do ensino médio de diferentes disciplinas e são entregues em sala de aula pelos professores "em compromisso". Eles foram testados na Itália em muitas classes que fazem parte do projeto nacional “Licei matematici”. Palavras-chave: Interdisciplinaridade. Didática da matemática. Aprendizagem da matemática.

1 Introduction

In this paper we discuss the theoretical framework behind a research project in education

1* PhD Sapienza Università di Roma. Associate Professor Sapienza Università di Roma, Rome, Italy. p.le A. Moro 5, 00185, Rome. E-mail: [email protected]. 2** Title and name of the institution (ABBREVIATION) in which the degree was obtained. Role playing and Institution to which he/she is linked (ABBREVIATION), city, state and country. Complete address for correspondence (Street, number, complement, neighborhood, zip code, city, state, country). E-mail: [email protected].

ISSN 1980-4415



developed since 2014 by the University of Salerno, under the scientific supervision of the

Department of mathematics, which involves researchers from 8 Departments.3 This project

stimulated the creation of the national network “Licei Matematici” (AAVV, 2019), more than

one hundred high schools in Italy where some of the ideas of the projects are practiced. In

particular, in some of this schools, in the province of Avellino, Salerno and Rome, is tested the

idea of building a highly interdisciplinary educational environment where mathematics

provides the main glue for connecting manifold disciplines and making them interact. We refer

to the educational project of the University of Salerno as to the “Mathematics as a Glue for

Interdisciplinary Teaching” (MGIT) Project. MGIT aims at dealing with a nowadays sharp

criticality in teaching/learning processes: the need to provide students with tools necessary to

face complex and highly interconnected problems, typical of contemporary world (climate

emergency, sustainability of models of development, management of migratory emergency,

just to name a few). We believe that this kind of problems cannot effectively be dealt with a

disciplinary or multidisciplinary approach, but they require a radically interdisciplinary one.4

One of the criticalities of teaching/learning models based on disciplines is the splitting of

science and humanities, especially a marked division of their roles. Scientific disciplines for

technical training and humanities for moral education. We believe that mathematics could play

a unique role to link in a fruitful, critical and dynamical way scientific disciplines and

humanities. It has been observed that The cultural role of “corpus callosum” which connects the two hemispheres is played by mathematics, which is humanistic in content, since it describes and creates possible worlds, but scientific in methods, since it uses logic (Odifreddi, 2000).5

The two qualifying points of the MGIT Project that we discuss in this paper is the

specificity of the gluing provided by mathematics, already considered in (Capone R., 2017) and

the proposal of a specific modality for teaching/learning, that we have called “globally

interdisciplinary laboratory”, that is, for us, an important teaching/learning mode aimed at the

creation of fruitful, equilibrated and critical interactions between disciplines. In this mode we

have dealt with transversal topics that do not find their proper space in disciplinary based

education models but that we consider very important in contemporary education. For example,

topics like “observational skills”, “knowledge management”, “choosing in condition of

uncertainty”, “confrontation management”.

3 Mathematics, Physics, Informatics, Chemistry, Humanities, Engeneering, Economy, Education. 4 By multidisciplinarity we mean juxtaposition of disciplines. By interdisciplinarity we mean developing

strict relations between them and condivision of methodologies. 5 Translated from Italian by the authors.

ISSN 1980-4415



On the basis of our experience we got the belief that globally interdisciplinary

laboratories rise educational problems at an intermediate level between those considered by

general educational theories and those considered by disciplinary educational theories. We

believe that considering disciplinary educational problem from the point of view of this

intermediate level allows to elaborate effective teaching strategies which are helpful also for

the learning/teaching of a single discipline, as we shall try to argue in this work for the case of

mathematics.

2 Interdisciplinary teaching

Modern science bases many of its successes on the method of splitting problems into

smaller ones which are easier to tackle. This model has proved to be very efficient to train

people to solve specialized problems. However, many warnings have been raised against the

ultimate implications of this approach, especially in education: knowing more by narrowing

object of a research may lead to pulverization and finally to “knowing everything on nothing”.

Moreover, ultra-specialization in research could make science less effective in dealing with

complex problems and, from a general cultural point of view, induces a dangerous form of

illiteracy of the specialists. Previously, men could be divided simply into the learned and the ignorant, those more or less the one, and those more or less the other. But your specialist cannot be brought in under either of these two categories. He is not learned, for he is formally ignorant of all that does not enter into his specialty; but neither is he ignorant, because he is 'a scientist,' and 'knows' very well his own tiny portion of the universe. We shall have to say that he is a learned ignoramus, which is a very serious matter, as it implies that he is a person who is ignorant, not in the fashion of the ignorant man, but with all the petulance of one who is learned in his own special line (Ortega y Gasset, 1932)

A learned ignoramus is usually not concerned with the social role that a scientist should

assume in relevant debates concerning science and society and never takes the responsibility to

warn against the undesired consequences of choices which may have a very strong impact on

everyday life and on the future of mankind, like climate changing or pursuing obviously

unsustainable politics of unlimited growth of production and consumption, for example. They

tend to limit their role to that of mere consultants on technical matters.

The quest for overcoming the division of disciplines in scientific research and in

education has been fostered by many researchers.

In education, it has been observed that «the core of the idea of discipline is a social

relation between the teacher and his or her disciples. » [Alvarogonzaáles D.,

ISSN 1980-4415



“Multidisciplinarity, Interdisciplinarity, Transdisciplinarity, and the Sciences”, International

Studies in the Philosophy of Science Vol. 25, No. 4, December 2011, pp. 387–403]. The

dynamic of these relations is the object of the science of education.

In this paper, as a decisive step towards interdisciplinary education, we consider “social

relations between teachers, considered as a cohesive entity, and their classes”. These relations

may vary within an ample spectrum of gradation from “additive” to “globally interactive”. An

example of additive relation is when at least two teachers in a classroom discuss how to

coordinate the presentation of parts of their program in order to make students aware of their

connections and appreciate them. For example, the presentation of cultural and social reasons

behind the emergency of perspective in renaissance painting can be very well coordinated with

the presentation of the mathematical theory of vision, rediscovered and further investigated and

advanced in those same years. This may be called multidisciplinary education.

On the other extreme of the spectrum, an example of globally interactive interaction

between teachers and their classes, which provides an example of interdisciplinary education is

that of “globally interdisciplinary laboratories”, that we are going to discuss in chapter 4.

Figure 1: Disciplinary, Multidisciplinary and interdisciplinary education

3 The role of mathematics in interdisciplinary learning/teaching

Learning/teaching mathematics is difficult and very important in our society. In many

countries, and we believe also in Italy the problem of a good teaching of mathematics has

become a national priority. Most people believe that mathematics is a strategic matter for today's economy and certain studies claim, for example, that mathematics contributes for about 15% of the gross national product of France. We live in a period of extraordinary accelerated changes: new knowledges; new technologies; new ways of communication - all directly connected with mathematics. Quantitative information, once reserved to specialists, is nowadays largely spread by media. The need to understand and use mathematics in every day’s life and working has never been so urgent and it does not stop increasing. In a rapidly and continuously changing world, students need a solid background in mathematics. Mathematics provides essential instruments for an active citizenship. Mathematics is necessary for democracy since it favors autonomy and innovation. (Torossian, Villani 2017)

ISSN 1980-4415



Because of its fundamental role, a basic understanding of Mathematics can't be left to

mathematicians alone and mathematicians must not forget their role and responsibilities in

shaping today's world. This is why an interdisciplinary, mathematically centered, education is

in our opinion so important. Moreover, we believe that stressing the capacity of Mathematics

to provide links to disciplines may help to find new effective ways to teach it. Mathematics, for

its ability of abstracting and modeling interactions, provides a specific glue for interdisciplinary

teaching and learning (Capone R., 2017).

Mathematical theories are enlightened by their applications and, in turn, a purely

theoretical development in mathematics creates unexpected connections which provide new

lenses for looking at real world and approaching its understanding. Recognizing the essential

role of mathematics for shaping our views of the world and for building useful models for

developing technologies is a fundamental preliminary step on which our educational project is

based. We assign a paramount role to developing critical attitudes toward mathematization. In

a mathematical model a virtual reality is built which behaves according to the mathematical

rules that only approximately describe reality. A precise understanding of these limits is crucial

and is both a philosophical and a scientific issue. Understanding these limits addresses a

democratic emergency. Interdisciplinary education plays a fundamental role in this (Rogora,

Ancora su INVALSI, test di apprendimento e modello di Rasch., 2014) (Rogora, Ancora su

INVALSI, test di apprendimento e modello di Rasch, 2014) (Rogora, Evaluating and choosing:

the role of mathematics, 2014) (De Marchis & Rogora, 2017).

Our project does not intend to upset methodologies and contents of traditional teaching,

but it considers the effectiveness of supplementary activities, above all Globally

interdisciplinar laboratories, described in the next section. These Laboratories aim at

promoting, through an interdisciplinary approach, the exchange and the comparison of ideas,

detecting connections and relationships between different disciplinary structures and the

reciprocal integration of fundamental concepts.

4 Globally interdisciplinary Laboratories

Globally Interdisciplinary Laboratories (GILs) are designed to tackle complex topics

with strongly interdisciplinary character from a strongly interdisciplinary perspective. They

constitute, in our opinion, the most innovative element of the MGIT Project.

ISSN 1980-4415



Examples of issues that we have dealt with these laboratpries are:

“educate the sight”; “deciding how to choose in situations of uncertainty”; “knowing

how to organize the knowledge”; “knowing how to argue”, etc.. To illustrate the characteristics

of these laboratories we summarize the contents and the aims of the GIL "Educate the sight" in

4.2, leaving to the following subsection the description of the general characteristics common

to all the GILs.

4.1 Design a GIL

In this subsection we discuss the common characteristics of the Globally

Interdisciplinary Laboratories that we shall exemplify in the next subsection with the

description of the GIL "educating the sight". We have maintained this planning also in the

design of the GILs “deciding how to choose in situations of uncertainty”; “knowing how to

organize the knowledge”; “knowing how to argue”, that will be described elsewhere.

The structure of a GIL should not be understood in a rigid manner. The nature of these

laboratories is to promote a creative interaction between the class and the group of teachers who

collaborate in carrying out the activities and therefore each realization is strongly characterized

by the specific interaction that is established between the groups and is substantially different

from the previous ones, although the process of consolidating the activities is in many cases

well underway.

The main phases of the construction process of these laboratories are summarized in the

diagram below

Fig. 2 Design model for a GIL

ISSN 1980-4415



We briefly comment on each of the different phases

Planning The choice of the topic does not follow (and must not follow) any rule. It

should be kept in mind, however, that it is not a matter of seeking a field in which to apply

mathematics, nor to look for mathematics in a particular field, but to deal with a transversal

theme, declined differently in different disciplinary fields and to compare the specific methods

and difficulties to approach it. For example, in the "Educate the sight" laboratory, the theme of

the transition from looking to seeing was chosen: looking at a geometric configuration, looking

for its properties and seeing its proof; look at a painting, look for its details and relationships,

understand its meaning.

Once the transversal theme has been chosen, an interdisciplinary work group of

researchers in education and teachers of different disciplines is formed which assesses its

interest, outlines its contours and discusses its feasibility in the form of a laboratory to be

brought to the classroom.

Teacher training During the planning phase the critical points of the project are

identified, both regarding to the knowledge needed by the teachers, and regarding the problems

connected with the didactic transposition in the classes. Training activities are therefore planned

for teachers in the form of laboratories in which, in addition to theoretical insights, the activities

that will then be brought, appropriately modified, are tested in the classes. The participation of

groups of teachers of different subjects, who will exchange the role of teacher with that of

student, is crucial in this phase.

The training phase ends with the preparation of activity sheets to be carried out in the

classroom, which are elaborated, tested and discussed by school teachers and university

researchers. The presence of professors of different subjects makes it possible to collect a rich

and articulated feedback on the activities sheets. The activity sheets, in Italian, for the GIL

“Educate the sight” can be downloaded at http://www.mat.uniroma1.it/people/rogora

Teaching At the end of the training phase, the groups of teachers involved bring the

activities to the classroom. It is crucial that all the teachers who worked fot the planning also

participate in each of the activities planned in the classroom. The presentation of the realizations

in class of the GIL “Educate the sight” is available online at the following web addresses:

ToKaLon, Perrotta, Possamai.

Debating At the end of the activities in the classroom, discussions are promoted on what

emerged from the laboratories, whether the didactic objectives were achieved in a satisfactory

way and, as often happened, if the activities highlighted other objectives not taken in

ISSN 1980-4415



consideration initially but worthy of development.

Updating The activities end with a rethinking of the laboratory, and with the decision

on modifications or the replacement of some activity sheets and the design of a new version of

the laboratory itself, which follows the same general outline but is profoundly renewed each

year.

4.2 The GIL “Educate the sight”

We think that improving the observational skills provide an example of transversal topic

that does not find its proper space in disciplinary based education models but is very important

in contemporary education. Among the many reasons for this, we have chosen the following

three:

1. observational skills are particularly suitable to develop and share a global

synthetic view on a complex problem;

2. it is a crucial step towards an “education to the beauty” that we consider a

necessary complement to labor market oriented training;

3. it helps to deal with specific disciplinary educational difficulties. In

mathematics, for example, developing observational skills may change students’

approach to problems: «it became a good habit for my students to take a good

look before trying to solve them, in order to let the problem speak to them and

not to fright them» (E. Possamai, reporting the outcome of the laboratory held in

her class).

The purpose of this GIL is to relate the activity of observing and interpreting a work of

art with that of facing and solving a mathematical problem, to take advantage of the parallel

observation of an art object and a mathematical problem. The laboratory compares the aesthetic

pleasure of viewing an artistic work with the intellectual pleasure of dealing with a

mathematical problem, underlining how these activities can be satisfying and of high

educational value, even without necessarily leading to correctly interpret the work of art or to

completely solve the mathematical problem, hence suggesting a possible way to help students

to overcome the "I don't know what to do" syndrome. Reasoning and debating on these

comparisons, suggests many not usual activities but of great educational value, both in teaching

mathematics and history of art. Some of the questions that have guided us in the design of this

laboratory, and which will also be considered in this paper, are: is there a method to stimulate

ISSN 1980-4415



intuition in facing a new problem.6 Is there a connection between the ability to interpret a work

of art and to solve a mathematical problem? Does highlighting the link between the two

processes make the analysis of a work of art more pleasant and more systematic? Does the

initial collection of details and observed relationships highlight structural analogies between

the two activities, which can help the development of both? Can the fear of making mistakes

be replaced by the pleasure of discovery by carrying out these activities in parallel? Is it possible

to effectively discipline the first reaction in front of a work of art or a mathematical problem,

in order to overcome the "I don't know what to say", "I don't know what to do" blocks?

The idea of the laboratory arises from the comparison with previous and still in progress

experiences of Valerio Vassallo and his collaborators at IREM in Lille, from reading of the

writings of Federigo Enriques, Emma Castelnuovo and Bruno de Finetti on “knowing how to

see” in mathematics and on the reflection on the value and specificity of geometric intuition, as

described in the works of Felix Klein and Enriques.

GILs on “Educate the sight”, have been organized for three years, starting in 2016 in

collaboration with art historians of the National Gallery of Palazzo Barberini and Palazzo

Corsini, and with the project “con la mente e con le mani” (With mind and hands) of the

Accademia Nazionale dei Lincei, which involved more than twenty schools in Rome.

Planning. The first planning of this GIL was made by Enrico Rogora (researcher in

didactics of mathematics), Valerio Vassallo (researcher in didactics of mathematics), Michele

Di Monte (art historian and responsible for educational activities for the National Gallery of

Palazzo Barberini and Palazzo Corsini), Silvia Pedone (art historian), Francesco Sorce (art

historian), Luigi Regoliosi (high school teacher of mathematics) and Maria Cristina Migliucci

(high school teacher of mathematics). In this phase a scheme for teacher training was designed,

which envisaged a visit at Galleria Corsini, where it was planned to let teachers observe and

discuss some paintings and a “visit” of some interesting geometrical configurations, during

which it was planned to let teachers observe and make conjectures on them.

Teacher training. The training phase of the Laboratory envisages the experimentation

and discussion of the training course designed by the proponents with the teachers interested to

take the GIL to their classes and the design of the classrooms activities, with thorough

discussion of their purposes, of possible answers to the questions we have posed at the

beginning of this paragraph and of the possible difficulties which may arise in classroom

transposition. The training phase takes four/five meetings. The simultaneous participation of

6 Make reference to Hadamard's book.

ISSN 1980-4415



the teachers of all the subjects that will be involved in the experimentation is required, in this

case mathematics and history of art teachers, but teachers of other disciplines are very

welcomed. Teachers of different disciplines working together on an interdisciplinary project

like this one allow the emergence of a rich perspective on educational problems raised by the

design of these activities, not only because of the sharing of their different expertise but also

of different point of views, that of the teacher and that of the student, since each teacher is

experiencing both perspective during this work: the teacher of mathematics experiences the

student sight when she confronts himself with a work of art and the same is true for the teacher

of art when he confronts himself with a problem of mathematics. The experience at the Corsini Gallery allowed us to understand how the students feel in front of something they do not know how to interpret (for example the geometric construction necessary to prove a theorem): What is important to understand the object I am considering? What are the essential details? And which ones are negligible? Am I following the right path or am I completely mistaken about the approach? (A. Perrotta, slides 2018)

Since the structure of the training process follows that which will then be brought into

their classrooms, we only present the latter, which is the result of the joint planning of the

teachers who participated in the design laboratory and who is adapted and enriched in each of

the classes. Further information (in Italian) about teacher training may be found at this link.

Teaching. The teachers trained in the previous phase, with the didactical material tested

and discussed with the other teachers and researchers and suitable adjusted within the group of

teachers of the same class in order to meet the specific educational needs of their classes begin

the GIL in their classis within few weeks from the last teacher training meeting. In classrooms

each activity takes place with the simultaneous participation of at least two teachers of different

subjects, who, for this GIL, are the mathematics and art history teachers.

Briefly, the proposed activities for this GIL, each lasting two hours, are the following:

● First Meeting: The first meeting involves two activities; the observation of Lloyd's star

and the observation of a painting.

○ The activity related to Lloyd's star takes about fifteen minutes and has the

following purposes: to experience the difference between “seeing when looking”

and “seeing when searching”; to link intuition to language in order to become

aware of the transition from looking to seeing; experience the intensity of an

intuition, in a way to be used as a touchstone for subsequent activities.

○ The activity of observation and interpretation of a painting lasts about an hour

and a half and uses a form asking to: collect the significant details of the

ISSN 1980-4415



painting; search for relationships; tell the story of what the picture means; give

the picture a title. The activity has the purpose of experimenting and reflecting

on the passage from looking to seeing in the context of the observation of a work

of art, enhancing the phase of describing details and relationships, as an

exploration which is preliminary to the construction of stories that students

propose. At the end of the work, done in couples by the students, a teacher

coordinates the discussion on the different points and tries to sew the different

narratives to arrive at an interpretation, or a comparison between multiple

interpretations, of the painting. In the course of this activity the teachers will try

to make the most of partial or unorthodox responses, which seem to lead to dead

ends.

● Second Meeting: The second meeting involves the performance of an activity related

to the observation and description of a figure. Activities based on different geometric

configurations have been tested. We briefly describe the one in which we have chosen

to use the Tangram, which needs a minimum of prerequisites. The meeting is divided

into two parts.

○ First part, lasting about an hour. An activity sheet is distributed in which you

are asked to describe the figure and then have it drawn, without showing it, to a

companion or to the teacher. The activity sheet is structured in a very similar

way to the one used during the first meeting, which was used to guide the

observation of the picture, to stimulate the perception of the connection (and

differences) between the two activities. At the end, the teacher follows the

description of the students, trying to highlight the ambiguities and assumptions

implicit in the description of the children, to stimulate them to become aware of

the need for a precise description. We believe that too little time is given to

teaching mathematics to description and construction of a geometric figure, but

we believe that these are very formative activities that, if neglected, do not allow

students to understand the sense of more complicated activities, such as those

related to argumentation and demonstration.

○ Second part, lasting about an hour. The activity asks to count the figures that are

seen in the Tangram. By stimulating students to look for more of the seven

figures they find immediately, a discussion begins on the definition of the

concept of figure. The activity aims to highlight the component of freedom

present in the work of the mathematician who "creates", through definitions, the

ISSN 1980-4415



worlds he studies. The activity aims to bring out the hidden complexity behind

a definition and the importance of considering definitions critically.

● Third meeting. The third meeting plans to carry out an activity on the "first look" and

one on the "demonstration without words".

○ The activity on the first look lasts a few minutes and is limited to presenting a

problem that can be solved in several ways, and to ask that the students, after

reading and understanding it, to write the first idea that comes to their minds to

solve it, without worrying whether it is a good or a bad idea. The problems to be

proposed were previously selected by the group of teachers who set up the

laboratory. The teachers addressed the problems by filling in the same forms that

they will use, after having set them up, with their students.

○ The activity on the demonstration without words, which lasts about an hour,

guides the students to discover the meaning of a demonstration and to build it

starting from the clues that the class collects during the observation of a specially

prepared figure. Figures were also prepared with GeoGebra to be able to move

them, "putting ideas into motion" and making the class feel the importance of a

dynamic approach to a problem. This suggests a parallel activity in the

observation of a painting, where we can try to put ideas into motion by moving

"spatially" around the painting or moving "temporally" by comparing

interpretations and reactions that the work has aroused over the centuries. Even

the wordless demonstrations were selected and experimented previously by the

teachers.

● Fourth meeting. The fourth meeting plans to conclude the activity on the "first look".

The teachers, starting from the activity sheets collected during the activity on the "first

look" discuss where they can bring the first ideas of the boys, trying to consider all the

suggestions, both valuing those that indicate a feasible way, and discussing the reasons

why certain suggestions do not work, but may work in different situations. This way of

proceeding is inspired by the discussion of details, relationships and narratives,

proposed in the activity of interpreting a picture, to highlight how the collection of

"clues", the verification of hypotheses and the comparison of interpretations can

characterize also mathematical activities and show that curiosity and the fun of

understanding the meaning of a work of art can naturally accompany a mathematical

activity.

● Fifth meeting. Visit to an art gallery. Our journey will end with a visit to some of the

ISSN 1980-4415



works exhibited at the National Gallery of Ancient Art in Palazzo Barberini. The works

were previously selected and are part of the initial path with the teachers, who observed

them filling in the same activity sheets and discussing with art historians who

collaborated in the preparation of the laboratory in the same way they used in the first

meeting. with students. The final visit closes the itinerary by repeating an activity

similar to the one proposed in the first meeting to discuss the links between the different

activities that have been taken by the students and to highlight the strengths and

weaknesses of the path.

We believe that, with this kind of activities it is possible, to a certain extent, to educate the sight,

stimulating curiosity to understand a painting and a mathematical problem through transversal

activities, of a laboratory nature, in situations that are not too characterized by the disciplinary

point of view, structured in such a way as to grasp and reflect on analogies and diversity and

on the importance of language to make the transition from looking to seeing effective. These

are activities that are not immediately aimed at developing skills but rather designed to stimulate

transversal knowledge. It is a type of teaching that is increasingly struggling to find space in

today Italian school because it does not place emphasis on know-how but on the 360-degree

formation of the person.

We reiterate that a fundamental characteristic of these globally interdisciplinary laboratories is

the presence in all phases of teachers of different subjects. This presence is an important added

value because it allows students to experience the profoundly interdisciplinary character of

culture. Another crucial feature is that teachers are personally involved in first experimenting

with the path they will take with their students. The presence of teachers of different subjects,

allows to observe the path from different points of view, returning students and observing

oneself with the eye of the teacher.

5 Interdisciplinarity and the teaching mathematics

In our opinion, the interdisciplinary approach used in the design of Globally

Interdisciplinary Laboratories helps effectively not only to address cross-cutting issues from a

correct interdisciplinary perspective, but also to address specific problems in the teaching of

the disciplines involved, in particular of mathematics. Let's see some examples, each of which

would deserve a specific in-depth study to fully illustrate the value of the interdisciplinary

ISSN 1980-4415



approach.

Language

It has been observed by several authors how some of the difficulties of learning

mathematics can be usefully addressed from the perspective of learning difficulties in a

specialized language Learning mathematics can be defined as an initiation into the specific discourse of mathematics, that is, initiation into a special form of communication known as mathematics. (Sfard 2001).

This point of view is close to one of the aims of the GIL “educate the sight”, which

highlights the importance of the communicative context and makes natural the comparison, the

relationships and the transition between different communication contexts and languages.

According to this perspective, the language of mathematics does not appear as unnatural but

deeply connected to others.

Fear of making mistakes and the importance of errors

There is no discipline in which the fear of error is so strong and rooted as in mathematics. It is evident that the demonization of error in mathematics has significant effects on an emotional level: the spread of the fear of making mistakes, turns out to be the emotion most associated with mathematics; the effects of the fear of making mistakes, ... on the block of thought processes. Di Martino Zan (2005, 2011).

The activities proposed in the "Educating the sight” GIL indicate a way to manage the

fear of making mistakes, removing the anxiety of having to say everything right and right away,

recognizing the fundamental value of errors in learning. In comparison with the activity of

interpreting a painting, the worry of immediately doing the right thing is less pressing. Looking

at a painting gives satisfaction even if you can't interpret it. On the other end, can you be

partially satisfied by facing a problem without being able to solve it? This satisfaction may be

obtained by realizing the importance of partial objectives towards the complete solution of a

problem, like, for example: describe, simplify, generalize, connect, modify.

Learning how to cope with errors is fundamental. avoiding mistakes is a petty ideal: if we dare not face problems that are so difficult to make the error almost inevitable, then there will be no development of knowledge. Indeed, it is from our most daring theories, including those that are erroneous, that we learn more. No one can avoid making mistakes; the greatest thing is to learn from them [Popper, 1972, tr. en. p. 242].

Arguing, interpreting and demonstrating

One of the critical points recognized in the teaching/learning of demonstrations concerns

the obscurity of its function The students' difficulties in the demonstration approach cannot be explained

ISSN 1980-4415



exclusively in cognitive terms of deficiencies in logical reasoning, but related to the difficulty of attributing a clear function to the demonstration. (de Villiers 1990).

In the GIL "Educate the sight”, and in an even more structured way in the GIL "Educate

to argumentation", the development of parallel interdisciplinary activities is of great help in

clarifying the function of the demonstration. For example, the activity of interpreting a work of

art compared to that of demonstrating a conjecture concerning a geometric figure is structured

in such a way as to bring out the connections between them and make the students participate

in the construction process of interpretations and demonstrations shared by the class. This

parallel structure naturally brings out the social component of the problems related to learning

to demonstrate (Stylianides, 2007).

Overcoming the blocks “I don't know what to say", "I don't know what to do"

Too often a student's first reaction to a math problem is dichotomous: «I can do it

because I have already seen a similar one» (and therefore the problem is actually a exercise) or

«I can't do it because I've never seen anything like it before».

In the GIL “Educate the sight” we search, reflecting on the analogies with the

observation of a work of art, tools for facing the difficult moment of the first approach to a new

problem in a more conscious and creative way.

The activity of making explicit the "first glance" to a problem, which consists in writing

precisely the first idea that comes to mind to tackle a problem, without developing it until the

solution of the same and the subsequent activity required of teachers to discuss and with the

whole class the various suggestions, helps to appreciate the mathematical work regardless of

the result that is immediately achieved and to realize the existence of different possible general

strategies to face a problem that can be effectively employed before throwing in the towel

(studying special cases , replace the problem with a simpler problem, generalize the problem,

link it to a problem already seen, highlighting the common and specific elements, transforming

a geometric problem into an arithmetic problem or vice versa, etc.)

These discussions highlight different strategies to address a problem, how to distill

strategies and how strategies can be reused. Look for those aspects of the problem in question that can be useful for future problems - try to put In evidence the general scheme behind the present concrete situation. (Polya, 1971).

5.1 Conclusions

The complexity of the problems of contemporary world makes it necessary to develop

ISSN 1980-4415



interdisciplinary and collaborative approaches and the School should develop new teaching

methods to cope with them. In the Globally Interdisciplinary Laboratories proposed in this work

the teachers are involved in the interdisciplinary planning of an activity and in the co-

participated realization in every phase of it. The unit to which a GIL is addressed is no longer

the class with its teacher but the class with its teachers. The educational problems that arise in

the design and implementation of a GIL are at an intermediate level between those faced by

general educational theories and those addressed by disciplinary didactics. This intermediate

position makes it possible to effectively examine some specific disciplinary issues related to

teaching/learning by looking for a link with those specific to other disciplines.

References PAMPLONA, A. S. A formação estatística e pedagógica do professor de matemática em comunidades de prática. 2009. 269f. Tese (Doutorado em Educação: Educação Matemática) – Faculdade de Educação, Universidade Estadual de Campinas, Campinas, 2009. PONTE, J. P.; CHAPMAN, O. Mathematics teachers' knowledge and practices. In: GUTIERREZ, A.; BOERO, P. (Eds.). Past, present and future. Rotterdam: Sense Publisher, 2006. p. 461 - 494. SHULMAN, L. Those who understand: the knowledge growth in teaching. Educational Researcher, Washington, v. 15, n. 2, p. 4-14, feb. 1986. AAVV Licei Matematici Retrived from Licei Matematici, June 21st, 2019, https://www.liceomatematico.it. APOSTEL L. Interdisciplinarity; problems of teaching and reserach in universities. Paris: Organization for Economic Co-operation and Development, 1972. CAPONE R.; ROGORA E.; TORTORIELLO F. S. La matematica come collante culturale nell'insegnamento. Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana Serie 1 (2), 2017, p. 293-304. DE MARCHIS M.; ROGORA, E. Attualità delle riflessioni di Guido Castelnuovo sulla formazione dell'insegnante di matematica. Periodico di Matematiche, 2017, p. 71-79. ENRIQUES, F. Insegnamento dinamico. Periodico di Matematiche, 1921, p. 6-16. LAKATOS, I. Proofs and refutations. Cambridge, Cambridge university press, 1976. ODIFREDDI, P. G. Il computer di Dio. Pensieri di un matematico impertinente. Milano: Raffaello Cortina, 2000. ORTEGA Y GASSET, J. The revolt of the masses. New York, New American Library, 1932. POLYA, G. La scoperta matematica. Capire, imparare e insegnare a risolvere i problemi, vol. II. Milano: Feltrinelli, 1971. ROGORA, E. I test INVALSI. ROARS Review, 2014.

ISSN 1980-4415



ROGORA, E. Ancora su INVALSI, test di apprendimento e modello di Rasch. ROARS Review., 2014 ROGORA, E. Evaluating and choosing: the role of mathematics. Lettera Matematica PRISTEM, p. 161-164. Stylianides, A. J. Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38 (3), 20017, p. 289-321.

Documents

Title in Portuguese/Spanish (secondary language)programmi.wdfiles.com/local--files/umi/ManifestoUMI.pdf · worlds, but scientific in methods, since it uses logic (Odifreddi, 2000).5