What is taught and what is learned. Professional insights gained and shared by teachers of mathematics

Abstract: The aim of the thesis is to contribute to knowledge about relationships between teaching and learning in school. The framework used in this research, variation theory, states that, to improve student learning, attention must be paid to what is being learned, the capability that is to be improved and the features (critical features) that it is necessary for the learner to discern. The studies reported here focus on the significance of critical features and are based on two ‘learning studies’ in mathematics, one of the density of rational numbers and one of the addition and subtraction of negative numbers. In a learning study, teachers work together with design, analysis and revision of their teaching of a single lesson with the aim of enhancing students’ learning by gaining insight about features that are assumed to be critical and enacting them in their teaching. The question answered in this research is whether the insight gained in the learning studies about critical features can be shared by other teachers and used to enhance other students’ learning. Two studies based on the previous studies were carried out together with a total of eight teachers and sixteen groups of students. Each teacher enacted two lessons with different conditions in terms of the critical features made use of. The lessons were video recorded and analysed with respect to which critical features were enacted in the lessons and what the students learned as indicated in pre and post tests. It is suggested that the critical features were transferable in two regards, in terms of student learning and in terms of a means of communication that could be shared among teachers. It was indicated that the critical features that were enacted in the teaching constrained what it was possible to experience in the classroom and what students learned. What was taught seemed to be reflected in what the students learned. Furthermore, the analysis indicates that it was not sufficient to simply name the critical features to the students; it seems that they must be discerned in order for learning to take place. It was found that the teachers made use of the critical features that were identified by other teachers in a learning study as a means to plan and teach lessons. This suggests that teachers can make use of the notion of critical features in their own teaching to enhance student learning.