Analyses ZDM 2004 Vol. 36 (3) 98 The teaching of proof at the lower secondary level – a video study 1 Aiso Heinze, Kristina Reiss Augsburg (Germany) Abstract: Teaching mathematical proof is one of the most chal- lenging topics for teachers. Several empirical studies revealed repeatedly different kinds of students’ problems in this area. The results give support that students’ abilities in proving are signifi- cantly influenced by their specific mathematics classrooms. In this paper we will present a method for evaluating proof instruction and some results of a video study that describe proving processes in mathematics classrooms at the lower secondary level from a mathematical perspective. Kurzreferat: Der Beweis im Mathematikunterricht ist eine der größten Herausforderungen für Mathematiklehrer. Empirische Studien haben wiederholt verschiedene Schülerprobleme in diesem Bereich aufgezeigt und lassen annehmen, dass die Schü- lerfähigkeiten im Beweisen signifikant durch den spezifischen Unterricht beeinflusst werden. In diesem Beitrag präsentieren wir eine Methode zur Evaluation von Unterrichtsbeweisen so- wie Ergebnisse einer Videostudie zum Beweisen im Mathematikunterricht der Sekundarstufe I. ZDM-Classification: C73, D43, D53, E53 1. Introduction Teachers’ experiences in the mathematics classroom suggest that it is difficult for students to learn how to proof. Moreover, there are several studies on proof and argumentation providing empirical evidence of students’ difficulties with proofs. Our own research indicates that students’ views on proofs and their abilities in proving are significantly influenced by the specific form of mathematics instruction. However, the reasons for these differences in the students’ performance remain unclear. Accordingly, we conducted a video study in order to analyse proof instruction in Germany. In this article we will present a method for evaluating instruction and will give some results that describe proving processes in the mathematics classroom at the lower secondary level from a mathematical perspective. Our research is integrated in the context of the international studies TIMSS and PISA. It aims at identifying cognitive and noncognitive factors playing a role in the development of proof competencies of students 2. The role of proofs in mathematics and in mathematics education Concerning the role of proof in mathematics and in the mathematics classroom we would like to emphasize three aspects of mathematical proofs, namely proof as a social construct, the different functions of a proof, and the 1 This research was funded by the Deutsche Forschungs- gemeinschaft in the priority program „Bildungsqualität von Schule (Educational Quality of School)“ (RE 1247/4). distinction between the process and the result of proving. Since a detailed discussion of these three topics would go beyond the scope of this article, we will restrict ourselves to the main ideas outlining the framework of our research. There is an extensive discussion about the nature of proof in mathematics (e.g. Hanna and Jahnke, 1996). Though mathematics is regarded as a strict and exact scientific discipline there are no precise definitions for basic notions like “proof”. Mathematicians will probably be able to give some necessary conditions for a mathe- matical proof, but a proof is accepted by more or less implicit rules of the mathematic community: “A proof becomes a proof after the social act of ‘accepting it as a proof’” (Manin, 1977, p. 48). This acceptance of a proof by the mathematics community depends on various factors. First of all, the proof has to validate whether a conjecture is true, though a proof has to satisfy various other functions. As stressed by Hanna and Jahnke (1996), de Villiers (1990) and others, proving in mathematics is more than validation. Hanna and Jahnke (1996) describe eight different functions of a proof (like explanation, systematization etc.). Mathematicians know through their own work, that they must distinguish between the proving process and the proof as an outcome of this process. Sometimes the process of proving a theorem may take years and may include various approaches which may (or may not) lead to a success. In general, none of these efforts can be seen in the final product, the formal written proof. Conse- quently, the teaching and learning of proof should not be restricted to presenting a correct proof. It is more important to stress the process of proving rather than to give the outcome of this process. In order to differentiate between the process and the outcome of proving, Boero (1999) described an expert model of the process. It is divided into different stages and gives insight into the combination of explorative empirical-inductive and hypo- thetical-deductive steps during the generation of a proof. We refer to Boero (1999) for the original description of this model; an adapted version for the analysis of proving processes in the mathematics classroom is given in section 4.3. 3. Students’ proof competencies – empirical results Several empirical studies on proof and argumentation provide an overview about students’ mathematical competencies in different countries. These studies give evidence that comparatively few students are able to deal with proof items or to reason mathematically (e.g. Healy & Hoyles, 1998; Lin, 2000; Reiss, Klieme & Heinze, 2001). In a study with 669 students in grade 7 and 8 of the German Gymnasium (high attaining students) we focus on the question how cognitive and noncognitive factors influence the students’ abilities to perform geometrical proofs (cf. Reiss, Hellmich & Reiss, 2002). We were able to identify three levels of competency: (I) basic compe- tency, (II) argumentative competency (one-step-argumen- tation) and (III) argumentative competency (combining several steps of argumentation). Low-achieving students