, Frankfurt ARGUMENTATION , PROOF AND THE UNDERSTANDING OF PROOF

Argumentation and proof play an important role in mathematics. In recent years several empirical studies have revealed deficits in students’ abilities in logical argumentation and in their understanding of mathematical proofs. In an empirical survey with upper secondary students we investigated different components of both mathematical, and general, competence like declarative knowledge, methodological knowledge, metacognition and spatial abilities in relation to students’ performance for geometrical proof items. The results indicate that these components explain a comparatively high proportion of inter-individual differences. 1 Theoretical framework 1.1 Argumentation and proof in the mathematics classroom Logical argumentation, reasoning and proof may be regarded as highly important topics in mathematics. Despite the fact that mathematics may even be regarded as a proving science the role of proof in the school curriculum did not always reflect that importance. In the last few years there was a significant change in the teaching of reasoning and proof in the mathematics classroom. During the lively discussions of the 70s and 80s as to whether proofs should be part of the mathematics curriculum in secondary schools, mathematics educators argued that proving in the classroom had developed into a topic that particularly emphasised formal aspects but disregarded mathematical understanding (Hanna, 1983). In the 90s the situation changed: proof is again regarded as an important topic in the mathematics curriculum (NCTM, 2000) and is an essential aspect of mathematical competence. However proof was not necessarily used as a synonym for formal proof. Several authors like Hanna and Jahnke (1993), Hersh (1993), Moore (1994), Hoyles, (1997), Harel and Sowder (1998) pointed out that proving spans a broad range of formal and informal arguments. Understanding or generating proofs is a significant component of mathematical competence. Moreover, mathematical argumentation has been identified as the essential element of higher order mathematical competence in the TIMSS study. Nonetheless not only the TIMSS study (cf. Klieme 2000) but also other empirical findings have revealed wide gaps in students’ understanding of argumentation and proofs (Senk, 1985; Usiskin, 1987; Healy and Hoyles, 1998). In a systematic investigation with high-attaining grade 10 students in Great Britain, Healy and Hoyles (1998) showed that there are deficits in their understanding of proofs, their ability to construct proofs, and their views on the role of proof. Even high-attaining students were far from proficient in constructing mathematical proofs, and were more likely to rely on empirical verification. However, most of them were well aware that once a statement has been proved it holds for all cases within its domain of validity. Moreover, they were frequently able to recognise a correct proof, though their choices were influenced by factors other than correctness, such as perceived teacher preference. 1.2 Geometrical thinking and geometrical competence Geometry is a good starting point to teach and learn mathematical argumentation, to explore mathematical concepts, to fill the gap between everyday life and mathematics, and to value mathematics as a part of human culture. Accordingly, geometrical competence can be re-