Matemáticas para viajar a las ciudades (in)visibles
Mathematics – to Travel to the (In)Visible Cities is an imaginative Project that allows us to approach six cities and to discover them through various aspects of the Mathematics Curriculum.
The search for Mathematics in real life, far from the classroom, and the connections between other apparently humanistic areas of interest is the original and inventive focus of this Project. The two project founders inform us in the following article of their experience during the Project and of the benefits of the Project to the educational community:
This Project takes its name from Italo Calvino's book, “The Invisible Cities”, a work of fantasy inspired by Marco Polo's travel logs. In “The Invisible Cities”, Calvino allows us to approach his cities through his polyhedral mirror full of hidden places, memories, hidden desires, deals and signs... He shows us many diverse points of view from which to approach cities.
This idea was fascinating to us since we thought that it was a way of representing other European cities which would be different and exciting to our pupils. The pupils would become familiar with the many diverse faces of the new cities. We thought that these ideas would fit into our proposal for an eTwinning project in which it would be important to know and share experiences with other European countries.
Marco Polo and other travellers like him who through their journeys across the Silk Trade Route (for example), have contributed to our knowledge of other cultures and made Europe what it is today. We want Europe to continue to be what it is, a place where many different cultures can find a place and mix together side by side.
As Doris Lessing said during a speech at the Prince of Asturias Prize Giving Ceremony:
“When I feel pessimistic about the situation of the world, I often think about that time, here in Spain, at the beginning of the Middle Ages, in Cordoba, Granada, Toledo and the cities of the south when Christians, Muslims and Jews lived together in harmony. This period lasted for three centuries. This marvelous culture lasted for three centuries. Have we ever seen anything similar? What has been can continue to be.”
On these firm foundations we have based our Project. The work lasted for six intensive months and involved 44 pupils. Both Czechoslovakian and Spanish pupils participated and the communication took place in Spanish.
We wanted to work with a multidisciplinary focus on cities in both countries. The first step was the selection of cities which should feature in the Project. We decided to select cities which featured in the Heritage List of UNESCO. The cities that we selected were: Brno, Český Krumlov, Lednice, Praga, Telč, Zd'ar nad Sazávou in the Czechoslovakian Republic y Santiago de Compostela, Toledo, Granada, Madrid, Barcelona y Valencia in Spain.
In this selection, our intention was that the pupils knew some of the invisible and diverse aspects – not only mathematical but also artistic, historical, literary etc. We threw ourselves into the this adventure of creating a series of didactic units that help the pupils to discover these aspects, that are, “less visible aspects” of mathematics, such as: Compostela´s Baroque; the golden section and Mies van der Rohe's Villa Tugendhat in Brno; Salvador Dalí and the golden number; the geometry of the Žďár nad Sázavou; the mosaics of the Alhambra in Granada; the Silk Exchange in Valencia; and the catenary of the Sagrada Familia in Barcelona.
The TwinSpace workspace has provided useful tools to promote teamwork which have helped to resolve some of the intrinsic problems caused by a project of this type. We have made good use of the communication tools offered by the system, such as, chat rooms, forums, wiki editor and web editor although for some tasks it has been necessary to use communication tools available outside the TwinSpace platform. Webquest and Skype are an example of two such tools which have been frequently used.
The first phase of the Project was to get to know each other. We wanted the pupils to get to know each other and to talk about their countries, about what they most liked; about anything and everything that could bring them together socially.
We created a forum in the TwinSpace workspace that consisted of a virtual café called, “Café Doña Urbanita”, where the pupils could share their ideas about books, music, TV series and even carnival fancy dress outfits and traditional celebrations.
In this forum, the pupils completed tasks and discussed a fragment of Italo Calvino's book where Marco Polo describes a bridge stone by stone:
In this forum, the pupils completed tasks and discussed a fragment of Italo Calvino's book where Marco Polo describes a bridge stone by stone:
- Which is the stone that holds up the bridge? asks Kublai Kan.
Marco Polo responds that, the bridge is not held up by this or that individual stone but by the line of the arch that the stones form.
Kublai Kan remained in silence and later asked the following question: why do you talk to me about stones if it's only the arch that is important?
Marco Polo replies that without the stones there would be no arch.
Besides these activities in the forum, in order to break the ice and get to know one another and start to work as a team, each country (working in groups) prepared small documents about cities to upload on to the TwinSpace workspace. Some pupils presented this information to others via video conference.
https://picasaweb.google.com/lh/photoEj9l9IHpgdfKIT6RCqINQtMTjNZETYmyPJy0liipFm0?feat=directlink
The rest of the working process and methodology used in each of the parts of the Project can be seen in the link shown above.
In addition, you can access more information about eTwinautas and the invisible cities by clicking on the following links:
As a conclusion, we would like to say that we are pleased with the results of this Project although the Project hasn't been without its difficulties. We would like to share some of our stories about these difficulties in the hope that it might be useful for others to see how we managed to resolve these difficulties.
The Viaño Pequeño Primary and Infants School is a rural centre and there were many difficulties related to Internet access in this centre. In order to overcome these difficulties we reduced the requirement for activities which required access to the Internet and opted for the use of communication tools, such as, Webquest. Webquest offers the opportunity to work offline and the results of this work can be uploaded later. The pupils from the Czech Republic used used the wiki space on the TwinSpace site as their platform.
Owing to the timetable differences between Spain and the Czech Republic, we had to reduce the video conference sessions that we had initially planned. In order to communicate effectively, the pupils increased their use of blogs and forums.
We believe that one particular factor that complicated the work process was our enormous ambition. We created a project which involved a great deal of material and work in a short space of time. However, we were delighted by the pupils' response to the Project. The main benefit has been that the pupils have been very highly motivated and are keen to continue with this Project. This year we have noticed an increased interest in this Project with many other countries and eTwinautas wishing to participate in the Project. This obviously reinforces the Project since new cities will be included and new centres added in the future.
This Project has made us appreciate and value the eTwinning Projects a great deal since young people in Europe are having meaningful contact through these projects. These young generations are the future of Europe and, we believe, that this Project in particular has made a valuable contribution to their appreciation of the cultural richness of this union of nations.
Mª del Carmen Buitrón Pérez
CPI Viaño Pequeno, Viaño Pequeno, España
Olga Martínez Cancelas
Klasické a španělské gymnázium, Brno-Bystrc, Brno, República Checa
Source of images: Project TwinSpace.
