Knowledge and beliefs on problem solving of in-service and pre-service mathematics teachers

In many countries-as well as in Spain-primary teachers begin their careers on the basis of an instructive training and teach different subjects, if not all, to their class. The predominant teaching style in secondary education, in which each teacher focusses only on the subject in which they are an expert, is usually less pedagogical and more characterized by the epistemological emphasis on a specific subject, in our case mathematics. This article relates knowledge and beliefs on problem solving with teachers' training programs (and therefore the educational stage) and teaching experience. To this end, a study was conducted of cases with a pre-service primary teacher, an in-service primary teacher, a pre-service secondary teacher and an in-service secondary teacher, in which their mathematical knowledge for teaching (MKT) and their beliefs about problem solving are analyzed. The results show, on the one hand, the need for greater emphasis on mathematical content knowledge in the initial training of primary school teachers, and on the other, the need for ongoing training for both primary and especially secondary school teachers in the educational field.