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This question is inspired by what Steven Krantz wrote in his book "A mathematicians survival guide."

If you are an American student, educated in the USA, then you may find that the international students in your graduate program appear to be way ahead of you. First of all, students in Italy, France, and other European countries often specialize much earlier than we do in the United States. They have had many more advanced math courses than you will have had. [I was once told that a typical Italian math student graduating from college will have taken 22 year-long math courses. I have not had 22 year-long math courses in my entire life.]

If this is not true, then in your experience, from which countries/universities do you get the most advanced graduate student?

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    I have to admit that the introduction of the bachelor degree, following the Bologna process, spoiled a lot the quality of the mathematical preparation in Italy, but having taught to many students coming from different countries (but not the US) both at the bachelor and master's levels, I recognize that Italian students have generally a better mathematical background (but in many cases I consider it still too low).
    – Massimo Ortolano ♦
    Mar 11, 2017 at 20:12
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    Yes, the Scuola Normale Superiore di Pisa (SNS) is excellent: I met several graduates from the SNS, and they showed really excellent (and I mean, excellent) preparation, both in mathematics and physics. But SNS is really not for the faint of heart, or at least, it used to be: I heard that they had to lower a bit the standards too.
    – Massimo Ortolano ♦
    Mar 11, 2017 at 20:19
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    Western european students are bloody amateur mathematicians when you have seen people from the good (high) schools in Russia or Persia. ;-)
    – Karl
    Mar 11, 2017 at 20:48
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    Depends on the county and university. At my university measure theory is a 3rd semester compulsory course. In the US it's not even required at most universities.
    – TryingToGraduate
    Mar 11, 2017 at 21:01
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    My impression is that the German system and the US system don't match up year for year, in the sense that a German STEM student of the same age will have done more math than the US counterpart.
    – aparente001
    Mar 12, 2017 at 3:40

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First and foremost, it is a important to note that there is a vast amount of variation between institutions and individuals, so generalizations are not terribly helpful here.

That said, historically there have indeed been differences in the philosophy and organization of European and American approaches to education. European education has tended to put more emphasis on knowledge, where American education has tended to put more emphasis on creativity and communication. As a result, many European students are already specializing in high school, while many American students are required to remain generalists until university and may in fact still be taking classes outside their "focus" all the way through the end of their undergraduate studies.

This does not, however, provide any clear advantage to one group or the other when it comes to doing a Ph.D. At the doctoral level, one needs both knowledge and also creativity and the ability to communicate well with colleagues. Thus, while students educated in one tradition are more likely to enter graduate school being able to prove Urysohn's Lemma, students educated in the other tradition are more likely to be able to effectively communicate why that proof is interesting.

In short: the word "appear" is in the original quote for a reason.

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  • American education has tended to put more emphasis on creativity and communication [citation needed]
    – Dan Romik
    Jan 14, 2018 at 15:23
  • @DanRomik jeff schmidht, book disciplined mind.
    – SSimon
    Jan 14, 2018 at 15:30
  • @DanRomik Consider the prevalence of core curricula and distribution requirements at US institutions vs. "focusing" requirements such as the A-levels in the UK
    – jakebeal
    Jan 14, 2018 at 16:24

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