Beweisen verstehen. Bildung durch Lehrkunst im Mathematikunterricht. Komposition, Inszenierung und Interpretation dreier Lehrstücke frei nach Wagenscheins Euklid-Exempeln: Entdeckung der Axiomatik am Sechsstern, Satz des Pythagoras, Nichtabbrechen der Primzahlfolge. Ein Beitrag zur Allgemeinen Didaktik aus fachdidaktischer Perspektive.

The possibility to prove statements once and for all is a unique feature reserved to mathematics. The theorems which Euclid of Alexandria (about 300 BC) proved more than 2000 years ago in his "Elements", are still valid today – and they will be so in 2000 years. Discovering and producing irrefutable truth is the characteristic feature of mathematics, and "proofs" are one of its central concepts. However, how to teach those mathematical concepts in the classroom is one of the crucial points of didactics. From the start, the abundance of formally deductive proofs prevents the pupils from discovering the proof process systematically since the ready-made proof results hide the fundamental ideas on which the proof process is based. Thus a paradoxical situation occurs: The characteristic feature of mathematics as a science hardly appears in school, and a way out does not seem to exist. The present work aspires to contribute decisively to the solution of this central problem by means of the “Art of Teaching” (Lehrkunst) according to Berg/Schulze/Wildhirt etc. The “Lehrkunst” didactics wages to introduce to the classroom aesthetically fascinating and philosophically profound cultural examples of achievements, breakthroughs and guidelines of European culture in a serious, revealing and downshifting way – whatever be the subject: staging lessons (Lehrstücke) is the name of the resulting teaching units. It is the educational and didactic topicality of the “Lehrkunst” didactics which makes it a promising partner for the solution of the problem: Already for some years, “Lehrkunst” has been implementing by the development of staging lessons what seems to become necessary by the change initiated by PISA in 2003 to output orientation: a new beginning of the input orientation. For, nevertheless, instead of the ruling alternative “either-or”, there should rather be the formula “both as well” – both input and output! In its first part, the thesis treats the question how proofs have developed starting with Euclid of Alexandria to the present day and to what extent this development is considered in the didactics of mathematics. In addition, starting with Martin Wagenschein's methodical trias of the genetic, the Socratic and the exemplary principles („Teaching to Understand“, 1968) and Wolfgang Klafki’s „Theory of the Categorial Education“ (1959) – in the meantime, both are recognized as classics of the educational theory – the concept of “Lehrkunst” didactics is developed historically and elaborated in detail. In the second part, three examples of Martin Wagenschein – the discovery of the axiomatic method in the hexagram, Pythagoras’ theorem, the non-cessation of the prime number progression – are developed into staging lessons, which were taught several times and are reflected, evaluated and interpreted. In the course of this process, the development of didactic works becomes especially clear as a process of cumulative optimization. An abbreviated version of the three staging lessons is found in the special issue "Lehrkunstdidaktik" of the journal MU (MU – der Mathematikunterricht, Friedrich-Verlag, issue 6/2013). In the third part, the results are recapitulated and evaluated. The results show that the three staging lessons allow the pupils to individually reenact genetic achievements of our culture – which is the essence of the educational process according to Klafki and Heymann („Allgemeinbildung und Mathematik“, 1996/2013). Moreover, it appears that formal-deductive proofs can always be only one aim of mathematics lessons at school, but we can reach them via preliminary stages of mathematical arguing using everyday language (cf. Brunner, 2013). Last but not least, the staging lessons, all of them taught about a dozen of times, make evident that proving as a process and proofs as its products cannot be separated from each other and that a profound, spiral treatment of the topic is possible in the lessons altogether. Proofs and deductive reasoning should be a guiding principle of mathematics lessons according to Heymann, which is why the educational standards of mathematics (of 2003 and 2012) have to be complemented in this regard.