Doing Mathematics – authentically and discrete. A perspective for teacher training

School has to qualify students to use mathematical knowledge and competences in a flexible way in different situations. For that they have to understand fundamental concepts and techniques of mathematics. They have to be able to use standard-procedures to solve problems in well-known situations as well as to use flexible different strategies to solve non-routine problems. Learning the basics lies in the past, learning critically thinking, solving problems and making decisions using mathematical reasoning is one of the requirements of today’s world. Therefore teachers are necessary, who be capable to engage their students in building up these competences. But doing this – what seems like a banality – is only possible, if they get such competences by themselves and had been fully award of such learning processes.