Ausbildung algebraischen Struktursinns im Klassenunterricht : Lernbezogene Neudeutung eines mathematikdidaktischen Begriffs

Abstract

When confronted with algebraic expressions, many see little more than a random combination of letters, numbers, and other signs. However, others easily approach the algebraic structures formed by the symbols; they manipulate expressions, evaluate them, get solutions. They have algebraic structure sense. This work investigates the processes of the development of algebraic structure sense in classroom instruction, drawing on data from two teaching units (one on linear equations, one on linear funtions) that were planned in cooperation with the mathematics teacher of the class. Two networked theoretical views - the GCSt model by Bikner-Ahsbahs and the Theory of Objectification by Radford - allow for the identification and interpretative description of situations in which students see structures and under the right circumstances develop algebraic structure sense. Using the documentary method, the long-term observations of classroom action are first understood in their complexity and then analyzed for their meaning with regard to the research questions. As a result, the development of algebraic structure sense can be described as a process of becoming familiar with the activity connected to the respective algebraic structure. With regard to linear equations a detailed pathway for this process is laid out. It is shown how the new view on the algebraic structure manifests itself in verbal descriptions and inscriptions. Furthermore, the role of prior knowledge is discussed. However, the development of algebraic structure sense is not a purely mental achievement: Situational interest and established roles influence the learning process. A social form of interaction is described that integrates all these aspects: Students tune their views on the algebraic structure with that of the teacher. In the closing discussion the empirical results are bound to the theories that were used. Furthermore, connections are made to the wider research landscape and it is shown how the results from this work are of relevance beyond its concrete topic.