I often wonder about Mathematics Education in China, particularly at the late high school and undergraduate level.

In the US, a typical undergraduate math degree will include lower division: 1-2 years of calculus (3 courses) different equations (1 course) linear albegra (1 course)

upper division 1 year of analysis (2 or three courses, real and complex) 1 year of algebra (2 courses) and a mix of: number theory, topology, more linear algebra, and so on.

I have heard from Chinese friends and seen on a few websites that Chinese students are often a few years ahead of our pace here in the US. For instance, they have often already started working on analysis and topology in late high school. It seems that they are doing US graduate level work by half way through their undergrad. What's more they are probably better at whatever subject they are learning too based on the sheer effectiveness of their education system.

I am wondering, for all that the romantic values of the US education are worth, why it seems that the Chinese system simply makes better, and frankly, more mathematically intelligent students. It seems that the best Chinese students read more, for longer hours, do more math problems etc. Whether or not this is by their own volition or because they are feeling pressure to do so (as is often case, so it's explained to me), the end result is the same regardless; They spend more time doing math, and they are better at it.

How can a student in the US honestly hope to learn enough math to be on an equal footing with a student who has worked 50% longer hours, and begun their mathematics education years earlier (if we're also not looking at particular cases of geniuses emerging at a young age in either country, but rather the average intelligent student)?

Please do share any knowledge, experience, or opinion you have related to these questions. I am a math student doing my undergraduate degree in the US, and I feel my education is not rigorous or thorough enough, despite being at a reputably challenging, well regarded institution. Furthermore, when I express my desire to work longer hours than other students, I am met with negative comments about how more work does not mean more knowledge. I think that's just plain wrong, so long as you're remaining healthy.

I have heard similar things about Russia, and even Japan and Korea. Please do share your thoughts.

Thank you for your input.

Edit and clarification: I agree with the sentiment that a broader education can help students deal with the real world, so to speak, better than a narrow one can in some respects. But first I'd like to suggest that we can't use the fact that the US was doing the best science in the last century to suggest this system is better. Many of the great discoveries and advancements I think of here in the last century are disqualified from this discussion by two factors. Firstly, the imperial history of the U.S. and its strategic footing during the World Wars allowed it to dominate in almost everything globally, from trade, to military power, to science. Secondly, many of the great advancements I think of in the past century were done by either exceptional geniuses who typically exhibited exceptional abilities at a young age, and furthermore many foreigners who came to the US for the reason above (I'm thinking Von Neumann, Einstein, etc). As the US loses its global dominance, I think that we will see less of this sentiment that the US education system really works exceptionally well in the ways that we imagine and discuss in this thread, and more recognition that other factors were at play. Furthermore, having first hand experience, it really does seem that my Chinese peers are better at math, their knowledge is not shallow in any respect. They work harder than most students here, and know the material better. They spend more time on homework and do more problems in their universities. It's not necessarily the breadth of our education that is responsible, but a combination of the breadth our lack of devotion to long hours and thorough understanding in exchange for serious labor.

Think, Chinese students will get perfect math GRE scores and be rejected from US math programs, while US students rarely if ever get such a score. Yet we still dismiss them as simply having memorized facts for the test. I think this is a big mistake, and will come back to bite the US, and its romantic liberal arts style education.

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    I am a tenured mathematician. I think the single undergraduate course that contributed the most to my professional development was a philosophy class (and it wasn't a logic or philosophy of math class). Mental flexibility is more important than technical knowledge, and the American system does a good job of encouraging mental flexibility. (For a famous example - Ed Witten was a History major as an undergrad!) – Alexander Woo Jul 22 '19 at 3:43
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    Thank you for your reply.But Edward Witten was a prodigy, raised by a physicist father, doing calculus in his preteens. I was hoping to ask in the name of the average intelligent student rather than exceptions. – Sam Jul 22 '19 at 3:47
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    I'm not specifically familiar with Chinese higher education, but you should note that the two factors you cite: study hours and early coverage of advanced topics, can be quite misleading, and in fact often suggest non-rigorous programs that provide useless degrees. For example I deal with lots of international students who think they know linear algebra, having taken it both in HS and UG, but actually didn't cover much beyond matrix algebra. They take many simultaneous classes with shallow depth, and all they do is cram for exams which test memorization. – A Simple Algorithm Jul 22 '19 at 11:01
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    China isn't the only example -- mainland European students are, at the end of their undergraduate career, also already at a level halfway through US grad school. It's really a romantic idea that the US education system is particularly good, but that's empirically just not true. (Though one might argue about what exactly "good" means.) – Wolfgang Bangerth Jul 22 '19 at 11:29
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    The StackExchange family of sites is a Q&A format, not a discussion forum. An "open discussion" such as what you're looking for doesn't fit well here. However, if you have a specific question about some aspect of Chinese or US mathematics education, you can ask it here or in the matheducators stack. – shoover Jul 22 '19 at 15:44

How can a student in the US honestly hope to learn enough math to be on an equal footing with a student who has worked 50% longer hours, and begun their mathematics education years earlier (if we're also not looking at particular cases of geniuses emerging at a young age in either country, but rather the average intelligent student)?

Your question contains a flawed premise, which is that in order to succeed in your studies and/or career you need to be (or that it even makes sense to ask if you are or are not) “on equal footing” with some arbitrary group of students.

Empirically, every year a couple of thousand students in the US graduate with a PhD in the mathematical sciences, about half of them US citizens (see here). It is an empirical fact that many of those US mathematicians go on to extremely successful careers in academia and elsewhere. We can argue from now until next week about the philosophical differences between US, Chinese and other nations’ education systems, but it seems pointless to argue with facts. The evidence simply suggests that American-educated mathematicians compete just fine with those educated outside the US. So the answer to your “How can a student in the US hope ...” is: they can certainly hope it, because that is what the reality on the ground is telling them.

Second, let me address the “on equal footing” issue. What comes to mind here is the notion of comparative advantage from economics - the idea that people in different places are better at producing different goods - I will apply it in the case when the “goods” are mathematical results rather than economic foods. Let’s assume for the sake of discussion that Chinese mathematics students indeed study and work harder than their US counterparts. I don’t know if this is true, but I have heard some similar things in other contexts (for example that classical music prodigies in China work themselves half to death from a young age, and as a result achieve levels of virtuosity that western musicians find essentially impossible to match) so it wouldn’t surprise me if it were true. Now, as it happens I agree with you that a person who works harder will end up knowing more and knowing the material better than someone who doesn’t work as hard. That’s absolutely true, and if you want to be successful, being a hard worker is a terrific advantage.

But now, guess what? Knowing more does not necessarily translate to being more successful. It turns out that American students enjoy their own set of comparative advantages over those from many other nations, very probably including China. For example: the environment in which American students are raised and educated is more economically prosperous, safe, healthy, and (to some extent) psychologically supportive than those in many other countries. Their country is one that famously encourages freedom of thought and of speech, creativity, innovation, risk-taking, and many other values that are positively correlated with personal and national success. Even in the limited context of mathematics, I feel reasonably confident in estimating that all of those circumstances can add up to quite a significant comparative advantage, that enables those students to produce certain kinds of high quality mathematics that their Chinese and other peers are not able to produce.

The bottom line is: hard work is important; cramming your head full of facts and knowledge is important; but they are not the only important things. There is much new work in mathematics that can be better done (or in some cases can only be done) by someone who is very creative and has a flexible and original mind than by someone who works extremely hard and knows a lot but isn’t as imaginative or creative or willing to take risks. (And conversely, someone who works hard and knows a lot can do things that a “lazy” but super-creative person cannot do. The principle of comparative advantage works both ways.)

Focus on your advantages, and you will be fine.

  • I appreciate your input, especially with respect to the advantages of each individual. I am skeptical of whether the advantage that an American or similar culture has (sounds like American exceptionalism to some extent) is really that of flexibility, creativity, etc, or just as you allude to, favorably tilted starting conditions in life and a less competitive job market, but I very well may just not understand what the reality of the situation is for working mathematician. Regardless, you answered the question .The bit about creativity seems important. Thank you for the thoughtful answer, Dan! – Sam Jul 23 '19 at 5:12
  • "that enable those students to produce certain kinds of high quality mathematics that their Chinese and other peers are not able to produce." This seems important. I am curious to know more about this idea of different qualities of mathematics, so to speak. I can imagine what you're talking about, but please do message me (or post here) if you can suggest further reading about this. – Sam Jul 23 '19 at 5:23
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    @Sam you’re welcome. To be clear, I’m by no means an exceptionalist of any sort. There are many great nations in the world, each of which can inspire us to be better in some ways that are unique to it. And some countries do regard mathematics more highly in their culture and traditions than the United States, and as a result produce more than their fair share of excellent mathematicians. Again, this isn’t necessarily correlated with working long hours, so I wouldn’t say that’s the thing one should obsess about. – Dan Romik Jul 23 '19 at 5:23
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    @Sam I don’t have any further reading - these are intuitive and entirely speculative thoughts based on my personal experience. – Dan Romik Jul 23 '19 at 5:29

Hmmm. Are apples better or are oranges better? Hmmm.

It is difficult to compare educational systems with such a narrow focus as is done in the question. Where do we study Linear Algebra? Ultimately there are more important questions. These questions are attempted to be answered through a somewhat ill formed national educational set of objectives that may be driven centrally (China) or widely distributed (US).

In the US, there are problems with the pre-college education system due to a lack of resources and an unwillingness of politicians to raise taxes or to think creatively about how to provide those resources. More could be done, if more money could be provided and if it wasn't seen as a political strategy to attack teachers, partly because they have traditionally had strong unions.

At the undergraduate level in the US, the philosophy is that the education should be very broad. One doesn't only study mathematics if your major is math. You also study history, philosophy, language, sociology, art and music (possibly), literature, and other things.

At the masters level in the US, the study narrows, but not so much as in other places. One typically studies the field intensively, but mathematics is, itself, a broad field. Insight in algebra is quite different from insight into analysis. Some insights carry over, but not all.

At the doctoral level in the US, as is true elsewhere, one studies a small part of a field very intensively and focuses on research in that field and extending what is known there. Some MS programs start on this narrowing, but not all.

In the US, each university, generally speaking, defines its own curriculum. Within fields, the faculty realizes a curriculum and there is generally fairly wide acceptance about what is possible. Pre college education is normally defined at the State level, with Michigan and New Jersey, perhaps, having quite different standards.

In some other countries the educational system is very hierarchical with, in theory, every student studying the same things at the same age. But even there, differences of implementation can cause differences of outcome.

But what is better?

That depends on what you are trying to achieve. If your goal is to create narrowly educated "technocrats" who don't know or care much about life then a narrow education is what would be preferred. But if you want, instead, to educate the "whole person" then a broad education is better overall, even if it takes longer to develop technical skills.

But, as a CS professor, I always considered it much more important to understand what should be built rather than how to build a given thing. If you don't understand the should you can do great damage to the world (take note Facebook, Twitter, ...).

Traditionally this broad educational system has served the US pretty well. In the 20th and early 21st centuries, many (most?) of the world's most important scientific discoveries were done here, though often enough by immigrants who studied elsewhere. I note that that is now changing, but I also note that US education is receiving even less support - witness too many colleges depending far too much on loan supported student tuitions, rather than grants or straightforward, tax supported, funding.

I used to warn people in highly technical fields who focused too much on details, that, eventually, the history and philosophy majors who knew nothing of what they, themselves, did, would be their employers. The history and philosophy majors took a broader view and could better judge what was important. Even in CS courses, I predicted that the students that asked why would have a better future than those who could only ask how.

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    For all the traditional (and non-traditional) American focus on breadth in undergraduate studies, I'm seeing little of it manifest when I have a conversation with mathematicians who studied in the US. Are they sitting in all those history classes and not learning anything? Does it all get forgotten when the sheepskin arrives? Am I only picking up European cultural references and are they only learning non-European culture? I know you're supposed to broaden horizons rather than hear answers in those kinds of classes, but the average European academic seems to have broader horizons, ... – darij grinberg Jul 22 '19 at 14:52
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    ... at least to me, than the average American one. And as far as independent thinking is concerned, my impression is that it cannot be learned from any class, but European academia, with its sink-or-swim structure, is at least good at filtering out the non-independent thinkers. And in mathematics, American universities are definitely not teaching independent thinking when they only start with proofs in the 2nd or 3rd year. – darij grinberg Jul 22 '19 at 14:54
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    @darijgrinberg What does "independent thinker" mean? As far as sink-or-swim academic culture goes, it's mostly an effective filter for students from educationally privileged families. American higher ed culture (at least nominally) recognizes that this is a bad thing. – Elizabeth Henning Jul 22 '19 at 21:07
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    "Independent thinker" means a whole cluster of things; on the level of (say) maths graduate students it includes an ability to figure out when to stop reading and start writing; what to learn and what to re-do; how to spot connections between different subjects and what to make out of them; more generally, the ability to take the driver's instead of the passenger's seat. These are hard to learn from any structured class-like activities -- stacking extra breadth requirements in particular is unlikely to be useless; but at least in mathematics, there is one obvious thing that you can do to ... – darij grinberg Jul 22 '19 at 21:27
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    ... set this independence process in motion, and that's starting proof-based classes early. I'm not saying the European system is generally very good (the cost-benefit ratio is unbeatable, but I'm not going to claim that freshmen classes with 1000 students, half of whom are only seeing the lecturer through a video screen, are superior to US offerings), but this is something it is clearly good at, while US universities for some reason aren't even trying. Linear algebra is another thing (undergrads get 2 classes of LA in Germany and 1 in the US, and it shows). – darij grinberg Jul 22 '19 at 21:31

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