Discourse Understanding is hard. This seems to be especially true for mathematical discourse, that is proofs. Restricting discourse to mathematical discourse allow us, however, to study the subject matter in its purest form. This domain of discourse is rich and welldefined, highly structured, offers a well-defined set of discourse relations and forces/allows us to apply mathematical reasoning. We give a brief discussion on selected linguistic phenomena of mathematical discourse, and an analysis from the mathematician’s point of view. Requirements for a theory of discourse representation are given, followed by a discussion of proofs plans that provide necessary context and structure. A large part of semantics construction is defined in terms of proof plan recognition and instantiation by matching and attaching. 1

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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