MATHEMATICAL PROVING ON SECONDARY SCHOOL LEVEL I: SUPPORTING STUDENT UNDERSTANDING THROUGH DIFFERENT TYPES OF PROOF. A VIDEO ANALYSIS Esther Brunner , Kurt Reusser, Christine Pauli University of Teacher Education Thurgau, Kreuzlingen CH, University of Zurich, CH SUMMARY Within the framework of the Swiss-German study “Unterrichtsqualität, Lernverhalten und mathematisches Verständnis” [“Instructional Quality, Learning Behaviour and Mathematical Understanding] ∗ (Klieme, Pauli & Reusser, 2006, 2009) and by using the example of a purely mathematical problem, it was examined in 32 classes how teachers support the process of proving in classroom instruction from a subject- based and a communicative point of view. For this purpose, an analysis instrument was developed which describes content-related aspects of the problem-solving process as well as the students’ participation. The results clearly indicate that the individual teachers differ in terms of their choice and application of specific types of proof. A special group, however, is constituted by those teachers who prove in multiple ways. Keywords: Mathematical proving, Mathematics instruction, Secondary school level I, Support of the students, Video analysis INTRODUCTION In the context of educational standards (cf. NCTM, 2000; Blum et al., 2006; EDK, 2010), learning to prove and argue has gained new significance and experiences a real renaissance. This change was particularly furthered after the criticism of formal proof and its strictness could be neutralized with respect to public school instruction and was complemented by other concepts like, for example, pre-formal (or operative) proving (e.g. Krauthausen, 2001). Since mathematical proving is a demanding activity, it requires teachers to support their students in a way which is close to contents and understanding-oriented. And since a mathematical proof is accepted or rejected by the community, argumentation takes place within a discourse. For these reasons, two aspects are crucial to the support of proving: content-related support as well as participation in technical discourse. This paper is focused on content-related, technical support. Various empirical studies have shown that rather few students are able to give mathematical reasons for or to prove a given fact (cf. Healy & Hoyles, 1998; Reiss, Klieme & Heinze, 2001). As regards geometrical proofs, Reiss and collaborators (Reiss, Hellmich & Thomas, 2002) found that the varying capabilities of students in ∗ We thank the Swiss National Science Foundation (SNSF) for supporting the project.