Axiomatic Thinking and Working in Teaching Mathematics

This article is motivated by the conviction that understanding the axiomatic method is a necessary component of an appropriate picture of mathematics. Therefore, it points out ways to investigate the role of axioms and proofs as well as the particular epistemological status of mathematical theorems as an additional topic at Sekundarstufe II. This includes recognising the necessity of axioms for a foundation of mathematical proofs, knowing the classical content-related and modern formalistic view on axioms as well as discussing general strategies for justifying mathematical theories. After sketching the historical development of the axiomatic method and of associated views in the philosophy of mathematics, we outline the most important pros and cons of dealing with axiomatics at school from a didactical perspective. We then delineate a concept of how the most important learning goals with respect to axiomatic aspects can be achieved by means of suitable topics. A teaching sequence from one of these topics, Kolmogoroff's axioms of probability, is presented in more detail