Theory and Practice: a Case of Fruitful Relationship for the Renewal of the Teaching and Learning of Algebra

The paper consists of three parts. After presenting the viewpoint of different authors on the feedback of theoretical studies onto the teaching praxis, in the first part we portray the Italian model of research for innovation and explain its genesis together with its latest evolution, particularly the role of 'teacher-researcher'. The second part is specifically devoted to one of our innovative long-term projects, about the teaching/learning of algebra, and it focuses the attention on the teachers' behaviour. We specify the theoretical frame in which this problem is located, the problems concerning its starting up with reference to the teachers' preconceptions and beliefs, what kind of interventions and how they were carried out with and for the teachers, so that they could see and get in tune with the main objectives to be pursued in such a research. We produce examples of their contribution to the development of the project and of the feedback of the activity in the pupils. The third part contains a reflection on the problems that arise on enacting researches according to this model, both from the point of view of the teachers involved and from that of the director of the research. In particular, we first concentrate on some problems that the teachers have to face when they approach the double role of teacher-and-researcher, on what influence the development of the research can undergo owing to their beliefs or to the environment in which they work. Then we analyse the more general problem of the social incidence of such a research, also with reference to the present status of the Italian system for teacher training.