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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.

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    What you you mean by "mathematics"? Formal studies in math (e.g., leading to a B.Sc. in math), or "math for X", like "math for business majors"? If the former, my understanding is that in the US, courses will rarely have proofs and much more rote application of differentiation, integration or linear algebra formulas - whereas math in Germany is almost nothing but proofs. Your very first session might be devoted to the construction of the rationals from the integers through equivalence relations. The effect of a "salutary shock" is probably entirely intended. Commented May 13, 2023 at 6:47
  • I understand that this is the case @StephanKolassa
    – Babu
    Commented May 13, 2023 at 6:52
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    Possibly mathematics has a natural level of difficulty, or as I sometimes call it, "brutal honesty". Arguably the real challenge is how the U.S. system makes it so easy to pass. The mount-Everest amount of institutional effort at my college that's gone into trying to simplify tests, merge classes, remove requirements, provide corequisite supports, etc., is really jaw-dropping. At the high schools I'm told it's come down to flat-out administrative fiat that you must pass students even if they do literally nothing. Commented May 13, 2023 at 9:38
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    Because they are in German!
    – EarlGrey
    Commented May 17, 2023 at 8:29

4 Answers 4

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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.

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    Valid points, but, I feel this still doesn't answer exactly "how the difficulty is raised". It answers to me, more of "what is the reasoning behind the raised difficulty"
    – Babu
    Commented May 13, 2023 at 7:37
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    I have added a few more points. However, I think it's very important to understand the broader economic/cultural/social context rather than a few examples. If you understand the former, then everything else follows.
    – Ambicion
    Commented May 13, 2023 at 8:33
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    @TrystwithFreedom: Just a small note, but I think the wording "how the difficulty is raised" is a bit misleading, since it seems to presume that there were somekind of natural "ground state" with low difficulty, from which German universities or teachers chose to deviate for some reason by "raising the difficulty". From the opposite perspective one could just as well ask (and I would just as well object this wording) "how is the difficulty in the US lowered?" Commented May 13, 2023 at 9:40
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    The general German attitude is that 'everyone gets a shot' at passing but they make it insanely difficult in order to filter out people. – I find “insanely difficult” exaggerated. If everybody gets a shot, those with bad aim will miss that shot, and be filtered out. Of course, individual courses may be insanely difficult due to bad teaching or exam design, but in general, this is not required for filtering and also not done in my experience.
    – Wrzlprmft
    Commented May 13, 2023 at 10:33
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    his Algebra book still a good and accessible book in comparison -- I find that Lang tends to use shorter and less complex sentences than many authors (in this sense, he is somewhat like how Asimov compares to most other popular science writers), which makes for more straightforward reading and probably works especially well for non-native English speakers. On the other hand, students who like a lot of motivational exposition and a stylistic/poetic literary style probably find his texts to be a bit terse and bland. (continued) Commented May 15, 2023 at 9:42
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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.

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You can raise the difficulty of a course by:

  • covering more material
  • covering more complex material
  • covering the same material but accepting fewer failures
  • any combination of the above

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.

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    Don't underestimate what students find hard, some have trouble with fractions or that multiplication comes before addition... Commented May 13, 2023 at 10:25
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    @JochenGlueck: hechingerreport.org/… Commented May 13, 2023 at 14:38
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    @DanielR.Collins and MaartinBuis: After reading Maarten's response and the first paragraph of the article linked by Daniel, my first impulse was to point out some nitpicky reasons why we might be comparing somewhat different things here. But then I read the rest of the article and now I have no idea what to reply. Maybe you're just right that I'm overly optimistic (I guess there's a considerable bias in Germany to always think about those few fancy elite universities when tertiary education in the US is mentioned). Shiver. Commented May 14, 2023 at 11:00
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    Universities in Germany generally do not select either, other than having finished gymnasium. Commented May 14, 2023 at 18:27
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    @RegressForward Historically the Abitur has been a real barrier. Now not so much. In the cohort of 25-30 year olds in 2019 (the most recent data I could find on quick notice) 54% had Abitur. On top of that there are various other ways of attaining a "Hochschulberechtigung" Commented May 15, 2023 at 7:09
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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.

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