PROBLEM SOLVING AND OPEN PROBLEM IN TEACHERS’ TRAINING IN THE FRENCH AND MEXICAN MODES Alain Kuzniak*, Bernard Parzysz*, Manuel Santos-Trigo **, Laurent Vivier* * Laboratoire de Didactique André Revuz, Paris-Diderot University, France ** Center for Research and Advanced Studies, Cinvestav-Mexico Mathematics education may differ in traditions and theoretical approaches throughout countries, but it is generally acknowledged that problems and tasks play an important role in learning. Two research teams, one in Mexico and the other in France, decided to work together in giving mathematics teachers in both countries a same task to solve during a training session. This paper summarizes the results of this study. It shows that in the two countries the reference to different theoretical frameworks results in putting emphasis on different but complementary aspects of teaching. INTRODUCTION In different countries, mathematics teachers’ education and research programs in math education exhibit differences both in principles and ways to implement them. There are also different research traditions or paradigms to frame mathematics education projects, but all of them recognize that mathematical problems or tasks play an important role in fostering the development of teachers and students’ mathematical knowledge. Contrasting different systems and approaches, in this study French and Mexican, may be of interest to identify features, common goals, and differences associated with their traditions in education. Our research interest is to analyze and document both the processes put into play by pre-service teachers while working directly with problems and later the way they use these problems in their teaching practices. Specifically, the purpose of our common project is to investigate ways in which pre- service and in-service teachers work on a series of problems in order to prepare, organize, schedule and implement their lesson plans. Moreover, the development of technologies in class is quickly changing teaching practices and it appears necessary to take it into account. During their interaction with the tasks, pre-service teachers were encouraged to use computational tools. In the present contribution, we will focus on the initial step of the project in which a common problem was used as a teachers’ training tool in both systems of education. Our main interest was to see the implementation in contexts in which “solving a problem” has not necessarily the same meaning. Introduced by Polya (1945), problem solving constitutes the traditional framework for teachers’ training in Mexico. In France, the traditional framework for teachers’ training relies much more on Brousseau’s Theory of Didactical Situation (TDS, 1998).