Über Vernetzungen im Mathematikunterricht - eine Untersuchung zu linearen Gleichungssystemen in der Sekundarstufe I
According to the TIMS-Study students in Germany show great deficits in problem solving abilities according to a lack of flexibility in thinking in mathematical networks. However, there exists hardly information about the exact lacks and the reasons of the deficits. The aim of the study in this dissertation thesis was to investigate how a mathematical network as it is presented in textbooks is transformed when carried over into students' minds during teaching and learning processes. The thesis subdivides into a theoretical part and the presentation of the empirical investigation. The theoretical part gives first an overview on the didactical discussion to the topic of connections in mathematics education. It follows a conceptual foundation of mathematical networks and their different aspects, pointing out that the underlying structure of a network is a graph, whose vertices represent mathematical objects and nonmathematical components related to these, and whose edges represent existing relations on them. According to the different sorts of relations, network categories with relevance for mathematics education in school are defined. Further, some theories and models describing aspects of genesis, memorizing and recalling connections are presented and information with regard to the different network categories defined above are brought out. The model of curricular frames is discussed as an approach to specify connections in teaching and learning processes; it serves, together with the modelling of connections as graphs, as aids for the empirical investigation. The empirical study particularly focuses on the topic "sets of two equations in straight form" in middle grade classes and restricts on the investigation of some network relations according to subject systematics and a special relation according to the application of mathematical objects, the model relation. The main research questions are: 1. Which relations according to subject systematics and which model relations to the topic focused on are part of curricular frames? 2. Which network transformations result in teaching and learning processes, on the way from the frame of the intended curriculum into the frame of the implemented curriculum and further in the frame of the achieved curriculum? The results show that connections as they are presented in text books are nearly unchanged translated into the frame of the implemented curriculum: teachers follow almost exactly the textbook in their lessons. On the further transfer of the implemented network into students' minds many connections are filtered out: the mainly learned connections by students are part of the links according to subject systematics (links between problems and solving algorithms, linkages according to a subconcept-superconcept relation), model links are hardly known. Students show great difficulties, when they have to use connections in problem solving processes. The study reveals the incompleteness of the transfer of implemented networks into students' minds. Moreover, the missing relations in the achieved networks are pointed out. The results of the study provide useful information for a possible improvement of teaching and learning processes with respect to the different sorts of connections.