It is established that the pass rates in Germany university courses are low. Typically people would say as explanation of how this happens, that the universities increase the difficulty of their courses or that there is not enough student support by teachers.
Disregarding support from teachers, in which way exactly are courses in German universities difficult for student? How do the teachers raise the course difficulty?
I am particularly interested in the case of bachelors in mathematics.
This can only be answered by comparison.
There are a number of differences in the teaching cultures of the USA and continental Europe. I have taught classes in both countries.
You pay virtually no tuition fees at German universities in comparison to US universities, even as a foreigner. Whereas in the US you are a paying customer and you essentially buy your degree, in Germany you are supposed to prove that you are worthy of getting the degree.
American universities focus on 'teaching' in the sense of the teacher feeding knowledge into students. Students have a rather passive attitude. At German universities the focus is on 'learning', meaning the students play the active role.
The general German attitude is that 'everyone gets a shot' at passing but they make it insanely difficult in order to filter out people.
At American colleges, students take a semester-long class where they practice mathematical reasoning. For example, they will talk one month about mathematical induction. At German universities, the teacher will talk about it for ten minutes, you get a few exercises, and then you are supposed to get it.
The focus at German universities is on axiomatic build up, even in classes for non-mathematicians, whereas American universities never do those foundations.
Typically, science classes are a few years ahead of their American counterparts.
German students finish high school much later. Much of what Americans do in their first few college years is considered high school material in Germany.
In addition to more material expected when entering university, the pace is much higher. What is graduate class material in the US will be undergraduate at many European universities. It is generally expected that students complement their studies with additional reading.
Depending on the place, 50% of students will fail at the exam, and that's the desired outcome.
If you want to get a direct impression in comparison to US universities, I'd recommend taking a look at, say, Koenigsberger's "Analysis I" book. On the other hand, the linear algebra books by Serge Lang have the reputation of being difficult (so Americans told me) but are considered accessible reading by most German students.
To add to Ambicion's and Maarten Buis' excellent answers, here are my two-cents based on getting a doctorate in Mathematics in Germany and then teaching in the US. From what I can tell, things in the US have not changed enough in Mathematics (I now teach in CS).
Beginning Mathematics in Germany was more abstract and used the axiomatic method, whereas for example Calculus in the US is aimed at natural scientists and especially engineers (at least at my school, the engineering school drives what is to be covered in Calculus). The equivalent class in Germany is Analysis. In the US, students are taught to calculate integrals and derivatives so that they can later do Fourier analysis and similar transforms, even though most of the students do never use them. In Germany, we were taught how to derive analysis from first principles and how to prove the important theorems. In the US, the quotient rule of differentiation is not often proven, whereas it is standard in Germany. In the US, students learn how to prove theorems maybe in the last semester of their sophomore (second) year and usually later. In my school, this is done with Discrete Mathematics and with Linear Algebra.
From what I can see what is happening in Germany right now, the engineering schools will teach their own Ingenieurmathematik classes as some did in my years, just after the Romans left the Rhineland.
So, even with the change in orientation towards the new B.S. degrees, I would still assume that beginners classes in Mathematics have a different goal than beginners classes in the US.
I am just supporting here what Maarten Buis and Ambicion wrote, even if I am not repeating their arguments. There are also great observations in the comments. This is a very interesting question that you are posing, worthy of a much deeper investigation.
You can raise the difficulty of a course by:
Comparing US and German courses in this respect is rather complex as the mathematics education in secondary education in the US is notoriously bad. So what is complex mathematics for a first year student in the US is something the German students learned in the 3rd or 4th year of Gymnasium.
I studied math in both the US and Germany (in the US at an expensive private university; i.e., not somewhere where the students need remedial math classes). In my experience, the programs are about equally rigorous. I would not say that the material in German courses is particularly hard, although it's possible that the pace is faster; in Germany not all the homework or test questions will be covered in class, and you have to supplement with your own reading.
The biggest difference is probably that in most (all?) German universities, your grade is based completely on a single final exam, whereas at US universities, a large proportion is based on homework. This means that in the US, a poor student can scrape by by just copying from a classmate. In my experience it was also much more common at German universities for students to show up unprepared for the final exam. There are multiple reasons for this; for example, Germany makes it fairly simple to repeat a failed exam, whereas at a US university, if you failed the final you would have to repeat the entire course. No doubt, however, this also contributes to the low pass rates.
