Contributions to mathematical knowledge and its acquisition

Abstract: This is a thesis in mathematics with didactical emphasis. The interest of didactics of mathematics is in mathematics in relation to how it is learned. The thesis comprises three parts, A, B, and C. Part A reports in three papers from research in mathematics within potential theory. Problems with electrical conductors form the historical background, where Gauss found mathematics interesting in its own right. He generalized the concepts and laid the ground of a rich mathematical area of research with many applications within other parts of mathematics. In Part A further generalizations are made. In seven papers in Part B reports are found from action research, or developmental research, in the classroom reality from primary to tertiary level. Fieldnotes are the source of data. In Part B one finds the basis of the ideas for Part C. Part C describes a project in a class of first year engineering students with the mathematics teaching completely void of exposition by the teacher. In the classroom the students worked in cooperative groups of four to the group with the teacher as adviser and discussion partner. One of the main questions was how this form of learning would affect the students' results from a cognitive as well as an affective perspective. The different methods of research used all point to the fact that the influences of the project have been positive. During the project a new theory was developed for the "fumbling" and not quite straightforward acquisition of mathematical knowledge may be described. The theory is useful for teachers as well as students and is in accordance with the way researchers in mathematics work. Thereby, in a natural way, this theory connects the three parts of the thesis