Introduction

All the institutions set up training courses to prepare trainees teachers and teachers in service to teach abroad. All these training courses include different points of view: an intercultural, linguistic and mathematical education, therefore language and mathematical trainers were associated to prepare and assume the training courses in all the countries.
The intercultural education allows the trainees or teachers concerned to contact the general organization of teaching in the host country and with teaching practices.
As for the mathematical education, it permits to take interest in the various approaches of certain concepts, from the point of view of curricula, didactics and teaching in class.
However these training courses present important differences according to countries' as well in their organization as in their contents.
The first differences appear from an institutional point of view. Only Czech Republic already had a teaching in foreign language (English) included in its course. In the other countries, a new training course should have been created.
This new training course could be included like a part of the official training of student teachers in Italy, and in opposite, it was based on voluntary participation in others countries with an eventual reduction of the number of hours of teaching (Denmark) or without any institutional compensation or promotion (Austria, France). So these training courses have been more or less difficult to create according to the countries.
However it did not have a strong incidence on the duration of these new training courses, from 20 hours to 30 hours. It is on the other hand one of the reasons which can explain the important differences in term of total numbers, by example 1 or 2 student teachers in Austria and France and 40 in Italy.
These differences of total numbers can also be due to reasons of a cultural nature: the lack of strong tradition of teachers’ exchange with other countries is noted both by Austrian and Danish teams, while the Czech Republic had already joint a program in mathematics and foreign languages, and got a strong tradition of participation in exchange programs with other countries.
At last, the fact that the training courses took place, at least partly, in a foreign language had an influence on the recruitment of student teachers and teachers in service for this project. In France, the training course was done in Italian, the host country language, and it was difficult to find voluntary teachers speaking Italian. In others countries, English was the language used during the training courses and in spite of that, in Denmark, teachers have often considered their language level too low to participate to exchanges with foreign countries.
About contents, the conceptions of these training courses were very different according to the countries. The institutional rooting or not but also the exchange conditions, the chosen language for the training courses and the total numbers of participants were very important variables. In particular, respective shares devoted to a general training and to the specific training of visiting teachers to carry out the exchange are extremely different from one country to another.
In Italy, the training course was focused on a study of the different educational systems of the countries involved in the project, with comparison and discussion about their positive and negative aspects but the visiting teachers were in charge of the specific preparation for teaching in Austria and in German language. In the opposite, in Austria and France, training courses were focused on the host country and the preparation of visiting teachers to teach in this country about contents, language and teaching practices.
On one hand, the linguistic aspect, i. e. vocabulary (general vocabulary, specific vocabulary for teacher talk, mathematical vocabulary) and grammar, is interlinked with the rest of training in all the training courses. On the other hand, depending on whether the training course language was the host country language or not, the work remaining at the load of the visiting teachers was very different from one country to another.
Some methods were specific to certain countries: French participants had the opportunity to visit the «Liceo Leonardo da Vinci» in Paris where they were able to attend classes, and become acquainted with the culture of the Italian classes. The training course in Czech Republic has included a visit to a bilingual school and an observation of video recordings of mathematics lessons conducted in English.
In the same way, mathematical contents have been studied very widely like in Denmark and Italy or specifically focused on the chosen subjects (Pythagoras and fractions). So, if student teachers and teachers in service have prepared lessons during the training courses, these lessons were related to various mathematical subjects like in Czech Republic or they were focused on Pythagoras and fractions like in Austria and France. Everywhere, student teachers or teachers in service had the occasion to present teaching units during the training course. In some countries visiting teachers had even tested parts of the lessons they planned to conduct in the host country. Finally some of these presentations could be video-recorded like in Czech Republic, Denmark or France.