Matemáticas para viajar a las ciudades (in)visibles: eTwinning Project through Mathematics & Spanish as a second language.

Project Mathematics to travel to the (in)visible cities is aimed at a group of students between 14 and 15 years old and provides them with a multimedia scenario to discover through the exploration of a city the multiple optical from which it may be observed: mathematical, artistic, literary, social, emotional ...The initiative is an interdisciplinary collaborative work between two European secondary schools: Klasické a španělské gymnázium in Brno- Bystrc, one of six schools with bilingual Spanish- Czech section, located in the city of Brno in the Czech Republic and the school CPI Viaño Pequeno in Galicia (Spain), framed in a Comenius - eTwinning and developed during the academic year 2012-2013 .
For the development of this experience we selected twelve European cities: six Spanish (Barcelona , Granada, Madrid, Santiago de Compostela, Toledo and Valencia ) and six Czech (Brno, Cesky Krumlov, Lednice, Prague, Telc and Zd'ar nad Sazavou). The criteria for choosing these cities were to be UNESCO heritage towns, or that in them there is a monument listed by this Organization.
Combining the work in a real scenario as is the city itself with experience of working in a virtual environment, the project has encouraged contact among students from different geographical, social and cultural backgrounds, has enabled them to know our Common Heritage of UNESCO and has been a bridge for cultural exchange and to approach various academic disciplines from a multilingual context and even from an emotional standpoint .
The project title is indeed inspired by the work of the Italian writer Italo Calvino Invisible Cities (1972), who in turn interprets the travels of Marco Polo.
"Explore all costs and look for that city. Then returns to tell me if my dream responds to the truth" that's Kublai Khan's order to Marco Polo in Invisible Cities, a story that invites you to embark on the wonderful journey of imaginary worlds, which opens the door to the diversity of views of cities (visible city vs. invisible city) .
Every city speaks to us, just as a result of an open dialogue between itself and its citizens, who create a permanent cultural exchange with the ones that come as visitors or travelers .
Students, working in groups, explore and describe cities of Spain and the Czech Republic, linking urban aspects, architectural issues, scenics, history, tourism ... with mathematics, astronomy, film, literature, technology, fine and visual arts... , All integrated into a single project.
As a culminating activity each group creates its own imaginary city .
The work is developed through a collaboration platform: Twinspace, combining resources that are provided by this virtual environment (wiki, forum, chat ...) with some external others (webquest , video ...). Students use a variety of software tools (Prezi , Glogster, Geogebra, Skype ...) and hardware (electronic translators, iPad,...).
From the point of view of the study of Spanish as a Foreign Language (ELE ) it is a new way to understand and approach the Spanish language and culture, learning vocabulary about maths in Spanish.
From the point of view of Mathematics, it is a way to work realizing that they are part of the culture of civilizations and they are in many kinds of everyday situations, for example, just exploring a city. Topics developed in the Czech Republic: * Brno: The Villa Tugendhat in Brno. Bauhaus and Mies Van der Rohe. The construction of the golden rectangle. The golden rectangle in VillaTugendhat. The Greek cross and square. Escher mosaics. Final comments and conclusions.* Český Krumlov: circumference, ellipse, hyperbola and parabola. Apollonius of Perga. A cylinder section. The method of the gardener. Hypatia. The Cesky Krumlov´s auditorium. Acoustic and optical properties of an ellipse. Final comment.* Lednice: The University of Gregor Mendel and the gardens of Lednice. Garden design and geometric shapes Calculation with fractions in the gardens of Lednice. Conclusions.* Prague: numbering systems. The Babylonian civilization. The measurement of time on the clock in Prague. The Schwabacher numbers. The characters of the clock. The position of the Sun and the Earth. Tycho Brahe, Copernicus, Galileo and Kepler. Final Comment.* Telč: The plane motions: translations, rotations and symmetries. The square of Telc and its houses. The Spanish cards. More symmetries in functions and cards. Welcome to the world of numerical sequences. Final comments and conclusions.Žd'ár nad Sázavou: The regular decagon and pentagon. The golden measure. Dalí and the golden number. Zelena Hora 's Chapel in Zdar nad Sazavou. Geometric shapes and golden proportion at Zelena Hora. Conclusions. In Spain:* Barcelona: Paraboly or catenary: Do we know more than Galileo Galilei?. * Granada: The beautiful geometry of Alhambra. Mosaics and geometry.* Madrid: The Escorial: Art and Science. History of mathematics.* Santiago de Compostela: The baroque at Compostela. Geometrical shapes.* Toledo: Introduction of positional notation. Numbers. Arithmetics.* Valencia: The market of Valencia: its arches and columns. Geometry & Architecture. This project has been awarded a National Quality Label in Spain and a National Quality Label in the Czech Republic by the National Support Services Czech and Spanish respectively. It has also been awarded an European Quality Label by the Central Support Service for eTwinning in Europe.These awards are a great encouragement to all those who have worked on this experience, it would not be possible without the support and collaboration of many of our colleagues in our respective workplaces, the indisputable enthusiasm, effort and motivation from our students, who have participated in this initiative and the willingness and desire of the leadership of our centers. Our sincere thanks go to all of them.     
Note: All information and materials developed for this project are published with open access at the web address :  Olga Martínez Cancelas - Klasické a španělské Gymnázium Brno-Bystrc (Czech Republic)María del Carmen Buitrón Pérez - CPI Viaňo Pequeno (Spain)

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