During an on-line course on Mathematics Didactics, carried out by n = 31 professors in service, it is analyzed how they study a question that could generate a Research and Study Path (RSP). The teachers investigated the question individually and in groups, and then they were asked to organize a possible teaching adapted to an institution known to them, based on the question. Written texts produced by teachers are analysed using two types of techniques: one qualitative and one based on lexicometric statistical methods. The results describe the difficulties of in-service teachers to organize a teaching according to the paradigm of research and the questioning of the world.