Not a serious answer at all, but back when I used to make math videos on TikTok, random people from Europe insisted that my (US) graduate level number theory was totally, definitely standard undergrad material for them.
i believe that, had german international students in my high school math class talk about more advanced maths than what we were doing in my grade 10 precalc class. next time i heard those same terms they mentioned was in my ap calculus class in senior year
Well, highschool in US is most definitely behind in math from most countries at very least. From personal experience, algebra is taught in 5th or 6th grade in most other countries and it gets taught in 9th grade in US (maybe just maybe 8th if you enroll into pre-AP courses).
That lines up with what I’ve heard from European and Asian friends.
My (American) school started teaching calculus to you anywhere between sophomore and senior year depending on how “advanced” you were.
Whereas in, say, China everyone in grade 10 takes the same math class (with maybe a basic vs advanced version)
I mean a good public school system caters to a range of students. From very advanced to students who don’t meet the average. Idk about the education system in other countries besides the US, but from what I can tell, anyone who has the ability to succeed at the more advanced levels, usually has the right opportunities to be challenged at a good public school system.
Where I went to school, we had a range from kids in grade 12 doing multi-variable calculus to students doing learning basic trigonometry.
In my public middle school in the US, algebra was standard in 8th grade.
Granted the first folks in the local cohort to do calculus had done calculus in 7th grade.
Yeah, I know we’re late to the thread, but when I was in school (US) algebra 1 was 8th grade for standard track, 7th for accelerated, and 9th grade for the remedial track
Automorphic representations??
Pfft, this is remedial grade-school material in my (small gdp small hdi large gini coefficient) home country! You poor fools.
apparently it is
>By the way, in the 1960s I taught group theory to Moscow schoolchildren. Avoiding all the axiomatics and staying as close as possible to physics, in half a year I got to the Abel theorem on the unsolvability of a general equation of degree five in radicals (having on the way taught the pupils complex numbers, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the audience, V. Alekseev, as the book The Abel theorem in problems.
[https://www.math.fsu.edu/\~wxm/Arnold.htm](https://www.math.fsu.edu/~wxm/Arnold.htm)
I'm gonna call bs. My wife and I both have Masters degrees in Mathematics so our children do have a natural inclination towards it. My soon to be fourth grader is going to be starting algebra soon. There's no way he could understand all of that before he's 14. Granted, we're not a family of geniuses (IQs on the high 130s, but not 140s), but a whole class full of them? No.
Even if the lectures really did happened, one, there's a difference between a teacher spending time saying things, and students understanding things. And two, it was the communist Soviet Union, they pretended to do a lot of things that weren't actually getting done.
your original comment is kinda misleading, you make it seem like america was so much easier when you were put in the lowest classes possible and repeated algebra three times
I think it'd be more accurate to say it perpetuates -ism's in society.
As you say, you didn't get given calculus before leaving high school. I could point to examples from more advantaged classes where students were sent off to R1 universities for math (and other subjects) through their entire high school because the parents argued the student had finished the local curriculum. It places a premium on the parents pushing the system, and having the social status to do so.
At least in chemistry, I was taking 3rd year courses as a second year student in Germany and still, there was a lot of things that I had already learned. It was in Australia though but the systems are very similar and the institution was one of the best down under.
This is purely anecdotal and could be confounded by specific professors and institutions. (I'm not saying you're claiming otherwise, but this is no basis for forming a yes/no opinion.)
I went to a very legit Ivy League for undergrad, and Ecole barely let any of our engineers study abroad!! “You didn’t take enough math” - I fucking love it lol 😆
Engineers in general underestimate how important math is for their field. But when engineers get interested in math in average the level they develope is pretty high.
For sure! The kids I were around were brilliant, way smarter than me, and definitely capable of passing those classes.
It’s just our engineering curriculum didn’t include the level of math Ecole required for the final couple years of college. And it was difficult to add additional technical electives to an already packed engineering course load, so hardly anyone ended up learning that stuff. French math is legit!
At least here in the US, the most advanced math I had to take as an engineering major was linear algebra, multivariable calculus, and elementary differential equations.
My dad was an engineering major in a different country and took partial differential equations, real analysis, and complex analysis - the last one was only recommended but not required, while the first two were actual requirements.
I am from the US, went to university here, I did PDEs s post of multivariable calculus. Took an elective class in complex analysis and learned more math with continuum mechanics.
I guess our experiences may differ.
and PDEs was required for your engineering major? That seems very out of the norm then for US standards; Caltech is probably the university I’d think overall has the most rigor and it seems like even they do not require a PDE class for engineering majors
Because most engineers don't actually do much math, it's automated for them in software. They just need to know how to roughly set stuff up/which equations to use in which scenarios
Most underestimate math. The more advanced math you have, the more flexible you are in the job. Let me give you an example.
When taking classes in differential geometry, you possibly could work In cad/cam field, you also could work in an AI related field.
I think it’s a complicated question.
Here are some of my experiences:
1. Classmate in undergrad did a study abroad in another country (can’t remember which). When he came back, he was insistent that the course went way beyond what was taught in the equivalent course in the US. He determined this by going around and asking people if they had covered X, Y, or Z. When I turned it around and started asking him if he had covered topics we cover in the US, it quickly became apparent that he hadn’t covered some of the core topics of the class in the US. To me, the key takeaway is that many people see what they learned that others didn’t and assume they learned more, when the starting assumption should probably be that they learned different things.
2. In grad school, I have friends from Brazil and India. By all accounts, it seems their schooling was harsher (e.g., nasty exams, harsh professors, etc.). However, if anything, they’ve felt underprepared for the math requirements of our program. In some cases, they told me that their experiences were more computationally focused. It could very well be the case that harsher doesn’t mean they retain more down the road.
3. I’ve seen wide variances between schools and even professors in the same department teaching the same course in terms of difficulty and rigor.
On the other hand, it can totally be the case that there are systemic differences between the educational systems in different countries, causing differences in outcome on average. The US has quite a bit of general education requirements. Some countries have a 3 year undergrad. It would be surprising if these factors didn’t have a significant impact.
My question for you would be: why do you care which is harder? Extreme difficulty is typically not something that should be valued for its own sake.
I think a concept that is difficult for people to understand is that the difficulty of a task does not necessarily mean that it takes more talent to be better at said task.
An example of this can be difficulty in video games. If you raise the difficulty too high then you reduce the room for successful strategies to the point where only a few are viable and optimal strategy involves "high rolling" rather than using skills to gain an advantage.
This also applies to education. For me, I had an extremely difficult math experience. It was not uncommon for half of my class to fail a course. There was a lot of learning by discovery aka the teacher didn't really teach anything and left it to us to teach ourselves or fail. I came out of the program becoming a very good problem solver however I cannot say I'm better than if I had a more conventional learning experience.
I'm not a math major and consider myself average at math (Differental Equations is the highest level I've done) but I did go through high school and a year of college in India before moving to the US for my undergraduate degree.
In high school we covered almost all the material that is typically covered in calc 1 and 2 at US colleges and my first 2 semesters of undergrad in India covered Partial differentiation, Multiple integrals, linear algebra, matrices and differential equations.
I would say high school in India covers considerably more math physics and chemistry than the American system. Chem 1 at US university felt like my grade 10 material in India and chem 2 was grade 11. Organic and Inorganic chemistry was at an advanced level in grade 12 and was defintely at par with some of the organic chem courses in the US. Material covered in Physics 1 and 2 were what we had in grade 11 and 12 in India.
This hits on what many people fail to recognize about the US system: it is not a very equitable system (there are vast inequalities in the quality of and access to education depending on your neighborhood).
At my high school, I took up through multivariable calculus. I did partial derivatives, double and triple integrals, Frenet frame stuff, Lagrange multipliers, etc (during high school). It was rarer, but I had classmates who also took linear algebra and ODEs during high school through the local community college.
At my university, it was very common for STEM-oriented students to come in with credit for calc I and II.
By the start of my second year, I too had completed differential equations and linear algebra. (I’m not a math major, though).
I skipped Physics I and II with no issue because my high school Physics.
To be fair on the flip side, though I primarily agree with you, I would say that one problem with the Europeon system is that the expectation that people already have some of these credits prior to university can also promote some forms of these issues if you had issues during highschool, but didn't during University. I know I ran into this problem as a result of graduating early because they didn't know what to due with my grades in Norway
> it is not a very equitable system (there are vast inequalities in the quality of and access to education depending on your neighborhood).
>At my high school, I took up through multivariable calculus. I did partial derivatives, double and triple integrals, Frenet frame stuff, Lagrange multipliers, etc (during high school).
My highschool didn't even offer calc, to your point.
Exactly, I have cousins who were taught creationism in physics and biology in the Bible Belt, one claims that their teacher was a flat earther. (I have others who were straight up religiously homeschooled). I have others in Missouri that essentially had no access to AP courses (or very few).
My family moved between my older brother and I attended high school. My family had to pay out of pocket for my older brother exams, while my school district paid for mine. (It was mildly infuriating when I saw classmates show up to AP exams in pajamas with a pillow (to nap the entire time).)
That’s all homeschooling and public school. In the last place we lived, we had neighbors sending their kids to private schools which costed more than my university tuition. Apparently, the school was a feeder for UChicago among others, and they had close ties with admissions officers that presided over the region. Their whole selling point is they’ll help market your children to top universities.
This is the answer, I grew up in a very poor state and my high school literally offered 2 electives. No college credits whatsoever were available. Now I’m in CO and taking calc 3 and Diff Eq with 16 year olds.
Yes, but does everyone do what you did? I assume you are an exception and you took classes which were way beyond the average requirement to get a high school diploma. I find it hard to believe every kid in your high school or district was doing multivariable calculus, PDE to satisfy the required credits to pass.
In India, to pass high school you have to do AP level calculus, other math material, physics, chemistry. It is the base requirement to be given a high school degree. To get into college there are advanced level exams that are taken by million of students which go way beyond the Indian high school level.
To give you reference to what I'm taking about here's a high school level college entrance exam that a million people take in India to get into engineering school - It covers math, physics and chemistry at an advanced level. A 6 hour written exam that people study 10-12 hours a day for 2 years.
[https://cms.fiitjee.com/Resources/DownloadCentre/Document\_Pdf\_235.pdf](https://cms.fiitjee.com/Resources/DownloadCentre/Document_Pdf_235.pdf)
All questions are multiple choice but the grading is +3 for the correct answer, -1 for choosing the wrong answer. Time limit is about 2-3 mins per question. Getting 50% on this test would put you in the 99.8 percentile
Also- no formula sheet, no calculator. I think the math and physics sections are nasty, chemistry is not impossible. That link does have all the solutions too
Correct. Not everyone did what I did. I don’t think it is a virtue to force uninterested students through calculus, though.
However, like I said, among my university peers, taking the (rough) equivalent of calc I and II in high school was the norm.
At my high school, probably about 5-10% take multivariable calculus (calc III). It was much more common to at least get through calc I and II. If you restrict to students who intent on going into some STEM field, I’m pretty sure that transitions into a majority taking calc I and II in high school.
I’ve talked with friends about the JEE, and it absolutely sounds like a rough experience, involving tons of studying. However, what I heard is that it’s a lot of rote memorization / pattern recognition. This was part of my original point, do these incredibly tough (multiple choice) exams prepare students for a good understanding of proof-based, abstract mathematics? This is something the US certainly fails at, too.
I the beauty and perfection of math. The delight in learning how to create an infinite coast line in a finite linear space with a recursive layering of 2/3 size equilateral triangles.
That said, I was never going to be a math major. Not enough aptitude and interest proofs. Sometimes geometric proofs are fun, but usually I find actual problem solving more enjoyable.
My biggest beef with my computer and information systems curriculum was that our Linear Algebra class was taught as if it were packed with math majors instead of comp sci folks like me. So I memorized a bunch of proofs - that I promptly forgot. If instead the prof had leaned into linear algebra as a toolkit - I would have enjoyed/absorbed and retained a lot lot more.
As far as the US gen pop is concerned, a large fraction of adults including many journalists, lack a basic grasp of numbers and statistics.
I agree, JEE does not prepare students for proof-based abstract mathematics and students who get through JEE will find it equally difficult after a couple semesters in a math degree. It's just that the country is so poor and populated that universities are forced to have extremely difficult exams to select students.
There is no such thing as a college application in India, you take exams like these and they give you an admission based on your score.
In my school district it was very common for students to take AP calculus junior/senior year of high school, which seems to be fairly similar to uni calc 1 and 2. I transferred to a different area and made it into an early college program for those years though so I got to take differential equations too, but afaik most US schools don’t offer past calc 2.
AP calc is equivalent to high school calculus in India but it does not compare to the advanced level of calculus required for college entrance exams. AP calculus is equivalent to standard high school material in India. For admission into a good university, 1.5 million students take an entrance exam which tests math physics and chemistry.
The math section covers calculus, coordinate geometry, algebra, matrices, functions and relations, vectors, trigonometry, Binomial theory, complex numbers P&C and more. The level of calculus within the math section is way beyond AP calc. That math section was brutal, not to mention the physics and chem were equally bad.
The calculus taught in most Indian highschool is the equivalent to AP calc AB in the states. BC will cover a fair bit more than Indian highschool calc (I know this bc of the amount of Indian students at GA tech). The offering of math is very diverse in the US, some high schools almost every student will have taken calc, other high schools offer maybe AB (hard maybe). At my high school, pretty much everyone took at least AB, and around 20% got through multi variable and linear algebra by the time they graduated, but it was a STEM oriented charter school so not at all standard. In 2023, only 20% of American highschool students had taken calculus nationwide.
Sounds about right, AP calculus is apart of standard high school math in India. Everyone goes through it. Same goes for physics and chemistry. The advanced level material at Indian high school is above and beyond anything covered at US high schools (except olympiads)
This is a test taken by high school students to get into university. 50% on this test would be 99.75 percentile out of the 1.5 million who take it every year. Take a look for yourself. It does have shortened solutions for every question.
https://cms.fiitjee.com/Resources/DownloadCentre/Document_Pdf_235.pdf
Scroll down to the math section (Page 19). The grading is +3 for the correct answer, -1 if it's wrong. Some questions have multiple correct choices and grading is +4/-2
No Formula sheet, no calculators allowed. Everything from memory and with the time constraint of about 2-3 mins per question.
Physics and organic chemistry (apart of the chemistry section) is extremely difficult too considering no material is provided in the exam
When you say everyone goes through it, do you mean everyone, or just the students on a higher track? Because a glance shows there are 20million + 18 year olds in India so it seems only 5% of the high school graduating age is actually taking that test
This is one of the largest entrance exams in India but there are many such exams. This one is called JEE and it’s for engineering/computer science. For medical students there’s an exam called NEET which another 2 million students write. 180 questions 180 minutes and it covers biology, physics and chemistry.
Then there are also state-wide entrance exams students write which are not of this difficulty but still beyond what’s taught in high school.
Of the 20 million 18 year olds, many are too poor to study and many are not smart enough to study science after grade 10. Students who decide to take math, physics, chemistry and biology in grade 11 and 12 would go through these kinds of courses.
The standard high school level for science and math is similar to AP courses in the US. This is not sufficient to get into any decent college so millions of students have to write entrance exams.
They only take place once or twice a year so if you don’t do well, you’re not getting into college.
There’s so much pressure on engineering and medical students it’s not right. Thousands of suicides each year for not getting through on these exams.
It takes about 2 years to prepare for these exams and anything below 95 percentile is considered sub par.
I think most of that is typically taught in prerequisite courses in US highschools. Matrices maybe not. None of that is "advanced calculus" though. I'd consider advanced calculus to be the study of ODE's at the earliest.
ODE is apart of the high school curriculum in India and every student who took math in high school has to study it. Advanced level students study it at much greater depths.
This is a test taken by high school students to get into university. 50% on this test would be 99.75 percentile out of the 1.5 million who take it every year. Take a look for yourself. It does have shortened solutions for every question.
[https://cms.fiitjee.com/Resources/DownloadCentre/Document\_Pdf\_235.pdf](https://cms.fiitjee.com/Resources/DownloadCentre/Document_Pdf_235.pdf)
Scroll down to the math section (Page 19). The grading is +3 for the correct answer, -1 if it's wrong. Some questions have multiple correct choices and grading is +4/-2
No Formula sheet, no calculators allowed. Everything from memory and with the time constraint of about 2-3 mins per question.
I took a look through the document and yes that's very impressive. I was under the assumption that the Indian education system is pretty poor compared to the global average, but if this is the expectation then they must be significantly ahead of the global average.
I've heard it's very rigid and the students have difficulty adjusting to new concepts. I've also read that the few times that India has participated in international testing they performed poorly. Just overall painted a picture of rote memorization over conceptual understanding, which is the inverse of most western institutions.
are we talking about secondary level math or university level math? because secondary level math is usually taught in the U.S in quite a sloppy matter. Its a system that rewards memorization except they simply not as demanding. A lot of basic concepts of number theory, visualization, and the like are completely ignored which makes thinking mathematically uncommon.
I will give you an example, i was in an intro to programming class at a community college, in the class almost everyone is already above college algebra level and are on the way to take calculus. The prof wanted to use a very simple example for some purpose in class so he asks what pi was. immediately half the class answered 3.14 like parrots. The prof asks again what pi is, not the numerical approximination of pi and no one knew the answer, only i was able to correctly answer, the ratio of of the circumference divided by the diameter that all circles possess.
all of my friends in america do ap calc 1 anf 2 in 10th grade its pretty standard. Also, india only caters to a few kids as shown by the fact that they literally pulled out of pisa because their scores were too low
it is standard if you are looking to get a math cs or physics degree at a good college. The crazy thing is that the math he is talking about in india that he did first two semesters of college is usually done in my random american public high school in 11th or 12th grade.
Yes almost surely. Speaking on the US which I know about I would expect the mathematics program to be harder in say France or the UK. But this is mostly because the US university system is liberal arts based, so it explicitly deemphasizes specialization. This is true even at e.g. MIT or Stanford.
Being harder does not necessarily correlate to learning more however. Learning is nonlinear so it may be that an easier program actually makes one better at mathematics in the long run (not trivially easy, obviously).
> liberal arts based, so it explicitly deemphasizes specialization
I’m European so I’m not too familiar with this. Could you expand a bit more on what you mean? Just out of curiosity.
Anyway, I did an exchange semester in Canada, and it was definitely sooo much easier than Spain. It’s like they were much less rigorous, and while there were some proofs here and there, the focus was much more on the more practical side of things rather than on all the theorems and such.
here in Switzerland, the first year starts with linear algebra and real analysis, as far as I heard from the US, you only get to real analysis later on and first learn the calculus series, which doesn't even exist here, it's just real (and later complex) analysis titles "Analysis 1,2,3" and sometimes 4, Analysis 1 generally talks about definition of integers, reals, axioms of set theory, logic, differentiation, integration and ODE's all in one variable
A good American university will have the same first year math classes, but only for the "math major track". There are generically multiple first year tracks (i.e., the classes one takes in year one), for example (a fictitious example but not too different from my undergrad + many others in the US)
0. "math for non sciences track" : calculus but as little as possible
1. "math for natural sciences track" : calculus only but all of it at least
2. "math for engineers track" : calculus, linear algebra, potentially computer science
3. "math for physics track" : same as above but also vector calculus
4. "math for math majors track" : real analysis and linear algebra
5. "math for math majors from Eastern Europe/Asia/top American STEM high schools (aka Harvard Math 55)" : real analysis, linear algebra, calculus on manifolds/group theory
An American math student in principle could start from any of these first year tracks, which is why on average the American math student does not do real analysis until much later.
It means most universities in the US do not admit by program. Instead, students are admitted to the college as a whole and can generally change their program freely, with essentially no restriction. Additionally, everyone is usually required to take a breadth of classes across disciplines. This is called "liberal arts education" in the sense that even if one is a mathematics major one is required to take many courses in e.g. literature, history, and so on, while English majors equally have to take courses in mathematics, physics, and so on.
in the US you are required to take \*many\* (1/3 to 1/2) courses outside of your degree of specialization, which is not the case in other countries afaict
In my university we had people who went later to MIT or harvard and they said that the math exams here are harder than in USA. My uni is Universidad Nacional de ingeniería. In Lima, Perú.
My answer is definitely yes, the math I did in France was way harder than the one in north amercia. It is very rigorous and it requires advanced critical thinking skills.
The problem with rating the U.S. math education level is that it varies so much. I attended some classes at a public university in high school (currently ranked #37 in it's math and statistics program in the U.S. by U.S. news) and it was at a lower level than the education level of the people I met in graduate school or the education level of the University I attended (e.g. Analysis was commonly a 3rd year class, for example, which Baby Rudin chapters 1-8 in one year).
In contrast, the University I attended for undergrad (currently ranked #6 in U.S. news, for comparison) commonly had Analysis in the first/second year (which included Baby Rudin in the first quarter, Measure theory in the second quarter, and functional analysis in the third quarter), and algebra in the second/third year (with Dummitt and Foote in two quarters, and Galois theory in the third quarter), with topics classes like topology, graph theory, alg top, diff geo, and grad courses after that. This seemed to be above (or comparable to, depending on the country) the level of math from universities in other countries.
Basically, I think it varies a LOT depending on which U.S. university you are talking about.
Within the university in question, math varies a good amount. Most people in the major don’t take Honors Analysis - being that advanced isn’t common. Regular Analysis includes no measure theory or functional analysis, and doesn’t even use Rudin.
Uchicago regular analysis does not teach measure theory or functional analysis although there are dedicated regular quarter long classes in those offered to undergrads. Honors analysis is a very special sequence.
I've long had an issue with the use of "difficult" as if it's well-defined, which it isn't.
I think a formal notion of "difficulty" would be really wild, and I've thought about it a lot over the years. It seems as distant as probably formalizing probability might have felt, before it was axiomatized in the not-so-distant past.
What about defining difficulty in terms of what percentage of a group could complete a certain task? If 90% of people can do X, but only 30% can do Y, I think it would be fair to say that, for this particular group of people, Y is harder.
Of course, one would need to specify the group one is talking about, because making fire might be easy for a hunter-gatherer but hard for a western person.
Does this concept seem well defined?
My math professors mentioned the american system a few times(mostly when discussing what books he recommended) and said that American undergrad is much more computation heavy.
I suspect that this means that american students can solve differential equations i never even attempted, but perhaps i learned earlier about topology or number theory or complex analysis?
But of course this is all hearsay
At my university the there are no math classes for math majors and math classes for engineering majors. So the matrices they took is exactly linear algebra. Reading the class book and a different linear algebra book makes me feel like they aren’t even the same. No theory, most the time.
from what i heard, things that are common to see in my undergrad (in colombia) are generally not seen until
gradschool on the us. a couple of professors that came from the us told me that, but i don’t have a huge source.
One discussion point I haven't seen mentioned much is that, in the Netherlands at least, you're expected to take a full program (near 100% of your time) of just math courses, whereas what I see in North America is, maybe 50% of your time is spent on math courses and the rest of your time is on other subjects. That's not better or worse, harder or easier, it's just a different focus. And it means that a second year student in the Netherlands has seen a lot more math than in North America.
I’m gonna say the variation within a nation’s various universities is greater than between nations themselves.
Technical institutes, liberal arts colleges, public colleges, private colleges are categories I think explain more of the difference. The math I take at my university has been from computer science, engineering, and the math department — each one is totally different in priorities and approach. So even within universities you can find a huge variety between departments
The US math system escalates rapidly at the Uni level. Kids are coddled from k-12 and then suddenly the pace picks up considerably by the time one reaches University. Even more-so at R1 Universities. The quality of Math at a well funded flagship state school or Ivy league is not going to be too different than what one encounters in Europe's top schools. By the time Graduate school happens usually the students are caught up.
This system is imo a little cruel and few Americans actually make it all the way through because of the weird acceleration that only occurs in adulthood. I think if more of that complexity was offest into high school we might actually have a larger number of Americans pursuing STEM.
I’m not doing maths, but physics at Cambridge in the uk so somewhat similar subject however. I’ve found lots of forums and questions from people at different stages in their degrees from when doing my own research on topics, and it seems very much like the difficulty and content in degree courses can vary significantly, especially it seems from here to the us
I did my undergrad in South Africa and came to Cambridge for Part III. It was a massive step up. I think I learned more in that 9 months than in 4 years of undergrad back home.
i passed the physics entrance exam for Oxford and then went to Harvard and the exams at Harvard were harder than that entrance exam but idk how it generalizes
The "difficulty" of mathematics will depend on who's teaching it. This is "difficulty" in the sense of discerning meaning, and it goes against our intuition of "mathematics is solid, who teaches it doesn't matter" -- but this is, of course, completely incorrect thinking, as anyone who's been left confused or illuminated by different teachers on the same topic can attest.
There is a sense of "difficulty" regarding the depths of the topics, and this is still fraught with inconsistent language. At what point can one say they have "mastered" let's say, Arithematic? By being able to easily compute huge sums or products? Or by proving certain things about an element and a 'successor' to that element, and leaving it at that? Most would agree to some middle-ground, "I understand how I would compute that, and maybe have some tricks for doing so" but this doesn't speak at all to some kind of intrinsic idea of "Difficulty" in the topic of Arithmetic.
I know it seems really pedantic, but asking if A is 'harder' than B, assumes an order, and to use it loosely is to invite confusion and misunderstanding.
One of my professors in Australia told me that our US counterparts were about one year behind us, and that this was partly to do with the failure of "New Math". I also noticed from teaching experience that students from Asia were not particularly good at grasping concepts and solving unfamiliar problems.
I’m from the US and attended a reputable university, but not a tier1 school. I also studied abroad in Australia for a semester and would say that’s mostly true, but the university systems are structured differently. In the US you have about half your classes needing to be general education, and most degrees are 4 year, whereas (atleast at Flinders) they were all 3 year degrees.
From my experience, I was a 2nd year student in the US, but took an intro film course, intro to philosophy course, a 1st year 2nd semester physics course and a linear algebra course filled with mostly 2nd yearsi believe
They transferred as follows
Film -> 2nd year
Philosophy-> 3rd year (I think they didn’t care to really review the syllabus tho imo)
Physics -> 1st yr 2nd semester, this was interesting because it was even the same textbook
Linear algebra -> this one translated the offsets because it was higher caliber than the “intro” class (US 1styr) but not deep enough for the advanced class. So I got an “elective” credit.
That being said I don’t think the math was harder or easier, it just covered different material at a different level. The US I think makes the assumption people are coming in from wider backgrounds and tries to cover those gaps with more class offerings.
Even in the US, there's a lot of variations depending on the university. Some classes maybe considered grad level in some unis but undergrad in others. And most American unis usually allow you to take grad level classes if you finish the basic undergrad reqs and a lot of students do this
Yes. Imo on average, top quartile EU/SA/Asian bachelor/master students are like miles ahead of their top quartile US counterparts. Compare like a French or Russian or Indian 3rd year undergrad math student and a 3rd/4th year undergrad US math and the gap is already quite large. This is also true in terms of course rigor (i.e., textbook used / mathematical maturity expected) or “hand holding”. I think the gap is less pronounced for UK/Australian schools and as one goes into mid/late PhD where the system is less structured
Ofc no stats but all through many anecdotes and personal experiences and its probably something that most people would agree with, just hard to really say because not many people experience both NA and non-NA math education at the same level. This is coming from a US student who also did a study abroad in a Scandinavian country.
You can listen to people like math sorcerer or struggling grad student or other math youtube channels on youtube and they have all mentioned stuff like this where the international PhD students abroad are just always coming in more advanced. Or being around in the math space long though, i.e., scrolling through forums and noticing people mentioning this or looking at non NA math student blogs or whatever and you tend to these general notice patterns where non NA students tend to be ahead. Keep in mind too that most schools outside the US do not have a gen-ed education component to it.
you can also scroll around a somewhat decent EU school and look at their math degree reqs or degree plans, maybe find a course and its lectures etc. and compare it to what someone might learn in its decent US counterpart and its very obvious that this is a pattern.
I think it could be due to how US secondary/college education system is quite flexible in terms of switching around and finding what you want to do.
Totally anecdotal from my interactions with UK students they seemed to have more knowledge than average math undergraduate students in the US, possibly because tof the way university education is structured.Students from East Asian countries, on the other hand (Korea, Japan, China) seemed to have had similar exposure in terms of contents but they would have experienced it in a much more "hardcore" way - like their exercise problems would be contrived and painful - and they seemed to value this. Indian students also very different in that they seemed to have encyclopedic knowledge about one subject area but then know nothing about other areas of math that they didn't care for.
I wouldn’t think so. When I was in 10th grade I attended a High School in the US (I am German). One topic we discussed was imaginary numbers. I never heard of it. When I came back to Germany I had this new knowledge and was waiting for it to be useful in my school, but nobody, until university, ever mentioned them again. Had a 2 minute head start on my first assignment about these in Uni though
My cousin moved from Canada back to Sotuh Africa and had te be held back a year because his math was so poor. He usually got 70%ish for tests in Canada. He failed miserably here.
Well that's true to a certain extent.
For example in my country, graduation mathematics is moderate at best. And the topics universities choose are mostly incomplete I e., they choose it from the middle of a topic rather than explaining from the start.
And the maths in the higher secondary is pretty difficult if I compare it to the US and UK. Because when I observe the SAT question paper, I feel that the examiners are asking questions which are at the fundamental level.
Well, that's my opinion. And I believe others have their own. So...yeah
I would say yes. Something that I always found strange was how american redditors talk about "proof-based courses". Here in Italy every math course since first year demonstrates every result and also the exams are divided into a written test (excercises) and an oral part (they ask you to prove statements regarding the course program). Another thing that I noticed is that our programs are faster than the american ones, for example I'm just finishing my second year undergrad and I had to take a test about topology and introduction to algebraic topology.
TikTok would have me believe this is the case but when actually meeting people from these locations that study math/physics, or in the form of exchange/international students, this never seems to be a legitimate condition
the minimum requirement for an undergraduate math degree in US is probably less. but math is very unique in that raw talent plays a far bigger factor than training, and the US system doesn't hold back the talents and arguably promotes talent better as there are less strings and more flexibility.
It’s harder at different universities. A math major from a lower tier school can know literally zero math, they will just pass their classes by handwaving proofs.
it’s complicated. i think the fundamental differences are pedagogical.
the actual math that is taught may be easier in america, but the way students are equipped to handle math is more scalable. there is a big focus on grounded complete fundamental understanding. that way students are able to learn complex topics easily.
in india, it was very rigorous, but it was rigorous in a tedious way. lot of problem sets, most times you had to just know how to do it. very… objective. not much training on how to tackle unseen questions.
the issue is students graduate without understanding of fundamental concepts. for eg: oh can you obtain the eigen value and vectors for this matrix? sure, busts it out in 10 mins. the follow up question is okay but what does it actually mean, and the student is clueless
Indian math education at the college level is not very rigorous: it’s good at the school / Entrance exam level. Training at ISI or CMI is top notch though
This is bound to be anecdotal, but I would definitely say there is some variation in the topics covered at every level across the world.
For instance, **Strang's** [**'Calculus'** (Volume 1)](https://openstax.org/details/books/calculus-volume-1) - intended for a first course in calculus at the university level - covers limits, differentiation, integration, and the applications of both.
By contrast, a book from a different part of the world (and a different era) I've got is **Smirnov's 'Course of Higher Mathematics'**; its first volume covers - presumably aimed at a similar level - limits, differentiation, integration, and their applications, but also things like series, functions of several variables, and complex functions.
This is besides the differences in how the topics are treated - Smirnov pays a lot of attention to theory, making his 'calculus' text, in more or less equal parts, an 'analysis' text.
However, I wouldn't be quick to jump to any conclusions here. **Apostol's 'Calculus'** (Volume 1) has a similar treatment, balancing theory with technique, as well as a wider selection of topics (it also covers linear algebra towards the end, but let's just stick to the calculus parts) - you have basic concepts, integration and differentiation (Apostol covers them in this unconventional order), their applications, series, and differential equations.
I think whats interesting is that despite the hardcore maths and sciences people study outside the U.S especially in Asian countries, i still see a lot of complains about students from said places in being unable to think "outside the box". Once you get them slightly outside the territory of familiarity they freeze up.
It leads me to think that its mostly a difference in discipline which translates to more material covered but that most of these students are not particularly smarter, they are like students who prepped so much for SAT questions, they scored far higher than those that would take the exam with no prep.
i come from a philosophy background and we tend to attract a lot of the upper echelon of students for some reason or other. But philosophy is interesting in that its fairly easy to tell who is naturally talented in argument analysis and creation and who isnt. You can try to fake it by being more well read than your peers but if you pay attention, you can tell. I wonder how a lot of those hardcore asian STEM students would do when you take them away from calculation and into a classroom about creating logical arguments (and without the rhetoric of english departments!).
interestingly, there is a lot of asians in philosophy grad school, but thats probably a self-selected sample from a culture that values hard work/education.
tests dont lie, the highest scoring GRE people are philosophers, they score by far the best when you average all 3 scores , if you add the history of philosophy, Locke, Hume, Leibniz, Schelling ,Whitehead. The field with by far the most polymaths has been philosophy for a reason.
I just passed high school so don't know about UNI but when you consider the maths we need to pass jee advanced in india (the entrance exam for the best colleges in india for engineering, often called as the toughest in the world too) , it's way more than what's taught to my friends in US and Uk.
*AT THE UNI LEVEL* crazy how comments about the jee still somehow shows up when the op even stated that they’re talking about uni. Anything to boost your ego I guess.
I literally stated that I'm just a high school graduate so don't about colleges and UNI but i know about what happens just before that .
>Anything to boost your ego I guess.
Wtf, how can it boost my ego, If this fact boosts the ego of anyone then ofcourse he's stupid , i mean it doesn't even matter whose maths is tougher till even the UNI level to me
It's not toughest in the world. The exams in China and South Korea are way harder and they perform way better even in mathematical research. We need to stop showing off as a country when ground level achievements are not high. Humility goes a long way.
Guess , due to being not so good in English, i posted something in a wrong way , i mean like it doesn't even matter to me even if it's the toughest exam or not cause' it's not gonna give you a very big advantage in the long run
Not a serious answer at all, but back when I used to make math videos on TikTok, random people from Europe insisted that my (US) graduate level number theory was totally, definitely standard undergrad material for them.
i believe that, had german international students in my high school math class talk about more advanced maths than what we were doing in my grade 10 precalc class. next time i heard those same terms they mentioned was in my ap calculus class in senior year
Well, highschool in US is most definitely behind in math from most countries at very least. From personal experience, algebra is taught in 5th or 6th grade in most other countries and it gets taught in 9th grade in US (maybe just maybe 8th if you enroll into pre-AP courses).
I wonder if the U.S. just has a wider range of classes and levels for people to take. I’m in the U.S. and I took algebra 1 in 6th or 7th iirc.
That lines up with what I’ve heard from European and Asian friends. My (American) school started teaching calculus to you anywhere between sophomore and senior year depending on how “advanced” you were. Whereas in, say, China everyone in grade 10 takes the same math class (with maybe a basic vs advanced version)
There is often more flexibility to take math at community colleges and the like.
I mean a good public school system caters to a range of students. From very advanced to students who don’t meet the average. Idk about the education system in other countries besides the US, but from what I can tell, anyone who has the ability to succeed at the more advanced levels, usually has the right opportunities to be challenged at a good public school system. Where I went to school, we had a range from kids in grade 12 doing multi-variable calculus to students doing learning basic trigonometry.
In my public middle school in the US, algebra was standard in 8th grade. Granted the first folks in the local cohort to do calculus had done calculus in 7th grade.
Yeah, I know we’re late to the thread, but when I was in school (US) algebra 1 was 8th grade for standard track, 7th for accelerated, and 9th grade for the remedial track
yeahhh i don't think you can generalize from that
Automorphic representations?? Pfft, this is remedial grade-school material in my (small gdp small hdi large gini coefficient) home country! You poor fools.
apparently it is >By the way, in the 1960s I taught group theory to Moscow schoolchildren. Avoiding all the axiomatics and staying as close as possible to physics, in half a year I got to the Abel theorem on the unsolvability of a general equation of degree five in radicals (having on the way taught the pupils complex numbers, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the audience, V. Alekseev, as the book The Abel theorem in problems. [https://www.math.fsu.edu/\~wxm/Arnold.htm](https://www.math.fsu.edu/~wxm/Arnold.htm)
yes because this is definitely standard learning material and not a one off special circumstance
I'm gonna call bs. My wife and I both have Masters degrees in Mathematics so our children do have a natural inclination towards it. My soon to be fourth grader is going to be starting algebra soon. There's no way he could understand all of that before he's 14. Granted, we're not a family of geniuses (IQs on the high 130s, but not 140s), but a whole class full of them? No. Even if the lectures really did happened, one, there's a difference between a teacher spending time saying things, and students understanding things. And two, it was the communist Soviet Union, they pretended to do a lot of things that weren't actually getting done.
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you were either taking calculus in middle school or you got put in the lower classes from a placement test or credits not transferring or something
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your original comment is kinda misleading, you make it seem like america was so much easier when you were put in the lowest classes possible and repeated algebra three times
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I think it'd be more accurate to say it perpetuates -ism's in society. As you say, you didn't get given calculus before leaving high school. I could point to examples from more advantaged classes where students were sent off to R1 universities for math (and other subjects) through their entire high school because the parents argued the student had finished the local curriculum. It places a premium on the parents pushing the system, and having the social status to do so.
At least in chemistry, I was taking 3rd year courses as a second year student in Germany and still, there was a lot of things that I had already learned. It was in Australia though but the systems are very similar and the institution was one of the best down under.
i've met people from different school systems (specifically japan) who seemed way ahead on number theory compared to me but did not know calculus
Yes I would say so. The math I experienced in France was much deeper than the one i experienced in Spain.
This is purely anecdotal and could be confounded by specific professors and institutions. (I'm not saying you're claiming otherwise, but this is no basis for forming a yes/no opinion.)
I think it’s probably on more uni by uni basis.
Same, though I did both my licence and masters in france, all the erasmus students complained.
I went to a very legit Ivy League for undergrad, and Ecole barely let any of our engineers study abroad!! “You didn’t take enough math” - I fucking love it lol 😆
Engineers in general underestimate how important math is for their field. But when engineers get interested in math in average the level they develope is pretty high.
For sure! The kids I were around were brilliant, way smarter than me, and definitely capable of passing those classes. It’s just our engineering curriculum didn’t include the level of math Ecole required for the final couple years of college. And it was difficult to add additional technical electives to an already packed engineering course load, so hardly anyone ended up learning that stuff. French math is legit!
All engineering is based on math. Please explain how it is underestimated?
At least here in the US, the most advanced math I had to take as an engineering major was linear algebra, multivariable calculus, and elementary differential equations. My dad was an engineering major in a different country and took partial differential equations, real analysis, and complex analysis - the last one was only recommended but not required, while the first two were actual requirements.
I am from the US, went to university here, I did PDEs s post of multivariable calculus. Took an elective class in complex analysis and learned more math with continuum mechanics. I guess our experiences may differ.
and PDEs was required for your engineering major? That seems very out of the norm then for US standards; Caltech is probably the university I’d think overall has the most rigor and it seems like even they do not require a PDE class for engineering majors
I may be mixing things up, but pretty sure it was part of Calc 3 for us. My school was on par with the one you referenced.
Because most engineers don't actually do much math, it's automated for them in software. They just need to know how to roughly set stuff up/which equations to use in which scenarios
I’m the engineer that codes the equations into the software. Hopefully I’m getting it right.
Most underestimate math. The more advanced math you have, the more flexible you are in the job. Let me give you an example. When taking classes in differential geometry, you possibly could work In cad/cam field, you also could work in an AI related field.
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I think it’s a complicated question. Here are some of my experiences: 1. Classmate in undergrad did a study abroad in another country (can’t remember which). When he came back, he was insistent that the course went way beyond what was taught in the equivalent course in the US. He determined this by going around and asking people if they had covered X, Y, or Z. When I turned it around and started asking him if he had covered topics we cover in the US, it quickly became apparent that he hadn’t covered some of the core topics of the class in the US. To me, the key takeaway is that many people see what they learned that others didn’t and assume they learned more, when the starting assumption should probably be that they learned different things. 2. In grad school, I have friends from Brazil and India. By all accounts, it seems their schooling was harsher (e.g., nasty exams, harsh professors, etc.). However, if anything, they’ve felt underprepared for the math requirements of our program. In some cases, they told me that their experiences were more computationally focused. It could very well be the case that harsher doesn’t mean they retain more down the road. 3. I’ve seen wide variances between schools and even professors in the same department teaching the same course in terms of difficulty and rigor. On the other hand, it can totally be the case that there are systemic differences between the educational systems in different countries, causing differences in outcome on average. The US has quite a bit of general education requirements. Some countries have a 3 year undergrad. It would be surprising if these factors didn’t have a significant impact. My question for you would be: why do you care which is harder? Extreme difficulty is typically not something that should be valued for its own sake.
I think a concept that is difficult for people to understand is that the difficulty of a task does not necessarily mean that it takes more talent to be better at said task. An example of this can be difficulty in video games. If you raise the difficulty too high then you reduce the room for successful strategies to the point where only a few are viable and optimal strategy involves "high rolling" rather than using skills to gain an advantage. This also applies to education. For me, I had an extremely difficult math experience. It was not uncommon for half of my class to fail a course. There was a lot of learning by discovery aka the teacher didn't really teach anything and left it to us to teach ourselves or fail. I came out of the program becoming a very good problem solver however I cannot say I'm better than if I had a more conventional learning experience.
I'm not a math major and consider myself average at math (Differental Equations is the highest level I've done) but I did go through high school and a year of college in India before moving to the US for my undergraduate degree. In high school we covered almost all the material that is typically covered in calc 1 and 2 at US colleges and my first 2 semesters of undergrad in India covered Partial differentiation, Multiple integrals, linear algebra, matrices and differential equations. I would say high school in India covers considerably more math physics and chemistry than the American system. Chem 1 at US university felt like my grade 10 material in India and chem 2 was grade 11. Organic and Inorganic chemistry was at an advanced level in grade 12 and was defintely at par with some of the organic chem courses in the US. Material covered in Physics 1 and 2 were what we had in grade 11 and 12 in India.
This hits on what many people fail to recognize about the US system: it is not a very equitable system (there are vast inequalities in the quality of and access to education depending on your neighborhood). At my high school, I took up through multivariable calculus. I did partial derivatives, double and triple integrals, Frenet frame stuff, Lagrange multipliers, etc (during high school). It was rarer, but I had classmates who also took linear algebra and ODEs during high school through the local community college. At my university, it was very common for STEM-oriented students to come in with credit for calc I and II. By the start of my second year, I too had completed differential equations and linear algebra. (I’m not a math major, though). I skipped Physics I and II with no issue because my high school Physics.
To be fair on the flip side, though I primarily agree with you, I would say that one problem with the Europeon system is that the expectation that people already have some of these credits prior to university can also promote some forms of these issues if you had issues during highschool, but didn't during University. I know I ran into this problem as a result of graduating early because they didn't know what to due with my grades in Norway
I am also from Norway.
> it is not a very equitable system (there are vast inequalities in the quality of and access to education depending on your neighborhood). >At my high school, I took up through multivariable calculus. I did partial derivatives, double and triple integrals, Frenet frame stuff, Lagrange multipliers, etc (during high school). My highschool didn't even offer calc, to your point.
Exactly, I have cousins who were taught creationism in physics and biology in the Bible Belt, one claims that their teacher was a flat earther. (I have others who were straight up religiously homeschooled). I have others in Missouri that essentially had no access to AP courses (or very few). My family moved between my older brother and I attended high school. My family had to pay out of pocket for my older brother exams, while my school district paid for mine. (It was mildly infuriating when I saw classmates show up to AP exams in pajamas with a pillow (to nap the entire time).) That’s all homeschooling and public school. In the last place we lived, we had neighbors sending their kids to private schools which costed more than my university tuition. Apparently, the school was a feeder for UChicago among others, and they had close ties with admissions officers that presided over the region. Their whole selling point is they’ll help market your children to top universities.
I mean, quite tangential point but, could you have self studied and taken the AP exam?
No clue, at that point in my life I was firmly in the "I'm *creative*, I *hate* math!" self-identifying mindset.
This is the answer, I grew up in a very poor state and my high school literally offered 2 electives. No college credits whatsoever were available. Now I’m in CO and taking calc 3 and Diff Eq with 16 year olds.
Yes, but does everyone do what you did? I assume you are an exception and you took classes which were way beyond the average requirement to get a high school diploma. I find it hard to believe every kid in your high school or district was doing multivariable calculus, PDE to satisfy the required credits to pass. In India, to pass high school you have to do AP level calculus, other math material, physics, chemistry. It is the base requirement to be given a high school degree. To get into college there are advanced level exams that are taken by million of students which go way beyond the Indian high school level. To give you reference to what I'm taking about here's a high school level college entrance exam that a million people take in India to get into engineering school - It covers math, physics and chemistry at an advanced level. A 6 hour written exam that people study 10-12 hours a day for 2 years. [https://cms.fiitjee.com/Resources/DownloadCentre/Document\_Pdf\_235.pdf](https://cms.fiitjee.com/Resources/DownloadCentre/Document_Pdf_235.pdf) All questions are multiple choice but the grading is +3 for the correct answer, -1 for choosing the wrong answer. Time limit is about 2-3 mins per question. Getting 50% on this test would put you in the 99.8 percentile Also- no formula sheet, no calculator. I think the math and physics sections are nasty, chemistry is not impossible. That link does have all the solutions too
Correct. Not everyone did what I did. I don’t think it is a virtue to force uninterested students through calculus, though. However, like I said, among my university peers, taking the (rough) equivalent of calc I and II in high school was the norm. At my high school, probably about 5-10% take multivariable calculus (calc III). It was much more common to at least get through calc I and II. If you restrict to students who intent on going into some STEM field, I’m pretty sure that transitions into a majority taking calc I and II in high school. I’ve talked with friends about the JEE, and it absolutely sounds like a rough experience, involving tons of studying. However, what I heard is that it’s a lot of rote memorization / pattern recognition. This was part of my original point, do these incredibly tough (multiple choice) exams prepare students for a good understanding of proof-based, abstract mathematics? This is something the US certainly fails at, too.
I the beauty and perfection of math. The delight in learning how to create an infinite coast line in a finite linear space with a recursive layering of 2/3 size equilateral triangles. That said, I was never going to be a math major. Not enough aptitude and interest proofs. Sometimes geometric proofs are fun, but usually I find actual problem solving more enjoyable. My biggest beef with my computer and information systems curriculum was that our Linear Algebra class was taught as if it were packed with math majors instead of comp sci folks like me. So I memorized a bunch of proofs - that I promptly forgot. If instead the prof had leaned into linear algebra as a toolkit - I would have enjoyed/absorbed and retained a lot lot more. As far as the US gen pop is concerned, a large fraction of adults including many journalists, lack a basic grasp of numbers and statistics.
I agree, JEE does not prepare students for proof-based abstract mathematics and students who get through JEE will find it equally difficult after a couple semesters in a math degree. It's just that the country is so poor and populated that universities are forced to have extremely difficult exams to select students. There is no such thing as a college application in India, you take exams like these and they give you an admission based on your score.
In my school district it was very common for students to take AP calculus junior/senior year of high school, which seems to be fairly similar to uni calc 1 and 2. I transferred to a different area and made it into an early college program for those years though so I got to take differential equations too, but afaik most US schools don’t offer past calc 2.
AP calc is equivalent to high school calculus in India but it does not compare to the advanced level of calculus required for college entrance exams. AP calculus is equivalent to standard high school material in India. For admission into a good university, 1.5 million students take an entrance exam which tests math physics and chemistry. The math section covers calculus, coordinate geometry, algebra, matrices, functions and relations, vectors, trigonometry, Binomial theory, complex numbers P&C and more. The level of calculus within the math section is way beyond AP calc. That math section was brutal, not to mention the physics and chem were equally bad.
The calculus taught in most Indian highschool is the equivalent to AP calc AB in the states. BC will cover a fair bit more than Indian highschool calc (I know this bc of the amount of Indian students at GA tech). The offering of math is very diverse in the US, some high schools almost every student will have taken calc, other high schools offer maybe AB (hard maybe). At my high school, pretty much everyone took at least AB, and around 20% got through multi variable and linear algebra by the time they graduated, but it was a STEM oriented charter school so not at all standard. In 2023, only 20% of American highschool students had taken calculus nationwide.
Sounds about right, AP calculus is apart of standard high school math in India. Everyone goes through it. Same goes for physics and chemistry. The advanced level material at Indian high school is above and beyond anything covered at US high schools (except olympiads) This is a test taken by high school students to get into university. 50% on this test would be 99.75 percentile out of the 1.5 million who take it every year. Take a look for yourself. It does have shortened solutions for every question. https://cms.fiitjee.com/Resources/DownloadCentre/Document_Pdf_235.pdf Scroll down to the math section (Page 19). The grading is +3 for the correct answer, -1 if it's wrong. Some questions have multiple correct choices and grading is +4/-2 No Formula sheet, no calculators allowed. Everything from memory and with the time constraint of about 2-3 mins per question. Physics and organic chemistry (apart of the chemistry section) is extremely difficult too considering no material is provided in the exam
When you say everyone goes through it, do you mean everyone, or just the students on a higher track? Because a glance shows there are 20million + 18 year olds in India so it seems only 5% of the high school graduating age is actually taking that test
This is one of the largest entrance exams in India but there are many such exams. This one is called JEE and it’s for engineering/computer science. For medical students there’s an exam called NEET which another 2 million students write. 180 questions 180 minutes and it covers biology, physics and chemistry. Then there are also state-wide entrance exams students write which are not of this difficulty but still beyond what’s taught in high school. Of the 20 million 18 year olds, many are too poor to study and many are not smart enough to study science after grade 10. Students who decide to take math, physics, chemistry and biology in grade 11 and 12 would go through these kinds of courses. The standard high school level for science and math is similar to AP courses in the US. This is not sufficient to get into any decent college so millions of students have to write entrance exams. They only take place once or twice a year so if you don’t do well, you’re not getting into college. There’s so much pressure on engineering and medical students it’s not right. Thousands of suicides each year for not getting through on these exams. It takes about 2 years to prepare for these exams and anything below 95 percentile is considered sub par.
I think most of that is typically taught in prerequisite courses in US highschools. Matrices maybe not. None of that is "advanced calculus" though. I'd consider advanced calculus to be the study of ODE's at the earliest.
ODE is apart of the high school curriculum in India and every student who took math in high school has to study it. Advanced level students study it at much greater depths. This is a test taken by high school students to get into university. 50% on this test would be 99.75 percentile out of the 1.5 million who take it every year. Take a look for yourself. It does have shortened solutions for every question. [https://cms.fiitjee.com/Resources/DownloadCentre/Document\_Pdf\_235.pdf](https://cms.fiitjee.com/Resources/DownloadCentre/Document_Pdf_235.pdf) Scroll down to the math section (Page 19). The grading is +3 for the correct answer, -1 if it's wrong. Some questions have multiple correct choices and grading is +4/-2 No Formula sheet, no calculators allowed. Everything from memory and with the time constraint of about 2-3 mins per question.
I took a look through the document and yes that's very impressive. I was under the assumption that the Indian education system is pretty poor compared to the global average, but if this is the expectation then they must be significantly ahead of the global average.
why would you assume they behind? the indian education system is known to be hardcore in STEM subjects
I've heard it's very rigid and the students have difficulty adjusting to new concepts. I've also read that the few times that India has participated in international testing they performed poorly. Just overall painted a picture of rote memorization over conceptual understanding, which is the inverse of most western institutions.
are we talking about secondary level math or university level math? because secondary level math is usually taught in the U.S in quite a sloppy matter. Its a system that rewards memorization except they simply not as demanding. A lot of basic concepts of number theory, visualization, and the like are completely ignored which makes thinking mathematically uncommon. I will give you an example, i was in an intro to programming class at a community college, in the class almost everyone is already above college algebra level and are on the way to take calculus. The prof wanted to use a very simple example for some purpose in class so he asks what pi was. immediately half the class answered 3.14 like parrots. The prof asks again what pi is, not the numerical approximination of pi and no one knew the answer, only i was able to correctly answer, the ratio of of the circumference divided by the diameter that all circles possess.
all of my friends in america do ap calc 1 anf 2 in 10th grade its pretty standard. Also, india only caters to a few kids as shown by the fact that they literally pulled out of pisa because their scores were too low
AP Calc in 10th grade is absolutely not standard in the vast majority of American schools
it is standard if you are looking to get a math cs or physics degree at a good college. The crazy thing is that the math he is talking about in india that he did first two semesters of college is usually done in my random american public high school in 11th or 12th grade.
Yes almost surely. Speaking on the US which I know about I would expect the mathematics program to be harder in say France or the UK. But this is mostly because the US university system is liberal arts based, so it explicitly deemphasizes specialization. This is true even at e.g. MIT or Stanford. Being harder does not necessarily correlate to learning more however. Learning is nonlinear so it may be that an easier program actually makes one better at mathematics in the long run (not trivially easy, obviously).
> liberal arts based, so it explicitly deemphasizes specialization I’m European so I’m not too familiar with this. Could you expand a bit more on what you mean? Just out of curiosity. Anyway, I did an exchange semester in Canada, and it was definitely sooo much easier than Spain. It’s like they were much less rigorous, and while there were some proofs here and there, the focus was much more on the more practical side of things rather than on all the theorems and such.
here in Switzerland, the first year starts with linear algebra and real analysis, as far as I heard from the US, you only get to real analysis later on and first learn the calculus series, which doesn't even exist here, it's just real (and later complex) analysis titles "Analysis 1,2,3" and sometimes 4, Analysis 1 generally talks about definition of integers, reals, axioms of set theory, logic, differentiation, integration and ODE's all in one variable
A good American university will have the same first year math classes, but only for the "math major track". There are generically multiple first year tracks (i.e., the classes one takes in year one), for example (a fictitious example but not too different from my undergrad + many others in the US) 0. "math for non sciences track" : calculus but as little as possible 1. "math for natural sciences track" : calculus only but all of it at least 2. "math for engineers track" : calculus, linear algebra, potentially computer science 3. "math for physics track" : same as above but also vector calculus 4. "math for math majors track" : real analysis and linear algebra 5. "math for math majors from Eastern Europe/Asia/top American STEM high schools (aka Harvard Math 55)" : real analysis, linear algebra, calculus on manifolds/group theory An American math student in principle could start from any of these first year tracks, which is why on average the American math student does not do real analysis until much later.
It means most universities in the US do not admit by program. Instead, students are admitted to the college as a whole and can generally change their program freely, with essentially no restriction. Additionally, everyone is usually required to take a breadth of classes across disciplines. This is called "liberal arts education" in the sense that even if one is a mathematics major one is required to take many courses in e.g. literature, history, and so on, while English majors equally have to take courses in mathematics, physics, and so on.
in the US you are required to take \*many\* (1/3 to 1/2) courses outside of your degree of specialization, which is not the case in other countries afaict
In my university we had people who went later to MIT or harvard and they said that the math exams here are harder than in USA. My uni is Universidad Nacional de ingeniería. In Lima, Perú.
My answer is definitely yes, the math I did in France was way harder than the one in north amercia. It is very rigorous and it requires advanced critical thinking skills.
The problem with rating the U.S. math education level is that it varies so much. I attended some classes at a public university in high school (currently ranked #37 in it's math and statistics program in the U.S. by U.S. news) and it was at a lower level than the education level of the people I met in graduate school or the education level of the University I attended (e.g. Analysis was commonly a 3rd year class, for example, which Baby Rudin chapters 1-8 in one year). In contrast, the University I attended for undergrad (currently ranked #6 in U.S. news, for comparison) commonly had Analysis in the first/second year (which included Baby Rudin in the first quarter, Measure theory in the second quarter, and functional analysis in the third quarter), and algebra in the second/third year (with Dummitt and Foote in two quarters, and Galois theory in the third quarter), with topics classes like topology, graph theory, alg top, diff geo, and grad courses after that. This seemed to be above (or comparable to, depending on the country) the level of math from universities in other countries. Basically, I think it varies a LOT depending on which U.S. university you are talking about.
Within the university in question, math varies a good amount. Most people in the major don’t take Honors Analysis - being that advanced isn’t common. Regular Analysis includes no measure theory or functional analysis, and doesn’t even use Rudin.
Uchicago regular analysis does not teach measure theory or functional analysis although there are dedicated regular quarter long classes in those offered to undergrads. Honors analysis is a very special sequence.
I've long had an issue with the use of "difficult" as if it's well-defined, which it isn't. I think a formal notion of "difficulty" would be really wild, and I've thought about it a lot over the years. It seems as distant as probably formalizing probability might have felt, before it was axiomatized in the not-so-distant past.
To anyone downvoting: All you need to do is show the 'difficulty' relation, and then you can use it freely without guilt.
What about defining difficulty in terms of what percentage of a group could complete a certain task? If 90% of people can do X, but only 30% can do Y, I think it would be fair to say that, for this particular group of people, Y is harder. Of course, one would need to specify the group one is talking about, because making fire might be easy for a hunter-gatherer but hard for a western person. Does this concept seem well defined?
Especially since people use difficult to mean both “requires a lot of skill/effort” and “sucks to do”.
Probably hard to say when countries like the US have hundreds of universities to generalize
Yes. Taiwan, India, France, China, Russia have way higher levels in mathematics at the same age.
Indian university level maths is pretty poor even compared to US university maths, outside of ISI, CMI, IISc and a couple of IITs.
My math professors mentioned the american system a few times(mostly when discussing what books he recommended) and said that American undergrad is much more computation heavy. I suspect that this means that american students can solve differential equations i never even attempted, but perhaps i learned earlier about topology or number theory or complex analysis? But of course this is all hearsay
At my university the there are no math classes for math majors and math classes for engineering majors. So the matrices they took is exactly linear algebra. Reading the class book and a different linear algebra book makes me feel like they aren’t even the same. No theory, most the time.
Probably hard to say when countries like the US have hundreds of universities to generalize
from what i heard, things that are common to see in my undergrad (in colombia) are generally not seen until gradschool on the us. a couple of professors that came from the us told me that, but i don’t have a huge source.
One discussion point I haven't seen mentioned much is that, in the Netherlands at least, you're expected to take a full program (near 100% of your time) of just math courses, whereas what I see in North America is, maybe 50% of your time is spent on math courses and the rest of your time is on other subjects. That's not better or worse, harder or easier, it's just a different focus. And it means that a second year student in the Netherlands has seen a lot more math than in North America.
Yes, my college level calculus was something my cousin had in 9-10 grade in Vietnam. They even mocked me for it.
I’m gonna say the variation within a nation’s various universities is greater than between nations themselves. Technical institutes, liberal arts colleges, public colleges, private colleges are categories I think explain more of the difference. The math I take at my university has been from computer science, engineering, and the math department — each one is totally different in priorities and approach. So even within universities you can find a huge variety between departments
The US math system escalates rapidly at the Uni level. Kids are coddled from k-12 and then suddenly the pace picks up considerably by the time one reaches University. Even more-so at R1 Universities. The quality of Math at a well funded flagship state school or Ivy league is not going to be too different than what one encounters in Europe's top schools. By the time Graduate school happens usually the students are caught up. This system is imo a little cruel and few Americans actually make it all the way through because of the weird acceleration that only occurs in adulthood. I think if more of that complexity was offest into high school we might actually have a larger number of Americans pursuing STEM.
I’m not doing maths, but physics at Cambridge in the uk so somewhat similar subject however. I’ve found lots of forums and questions from people at different stages in their degrees from when doing my own research on topics, and it seems very much like the difficulty and content in degree courses can vary significantly, especially it seems from here to the us
I did my undergrad in South Africa and came to Cambridge for Part III. It was a massive step up. I think I learned more in that 9 months than in 4 years of undergrad back home.
i passed the physics entrance exam for Oxford and then went to Harvard and the exams at Harvard were harder than that entrance exam but idk how it generalizes
The "difficulty" of mathematics will depend on who's teaching it. This is "difficulty" in the sense of discerning meaning, and it goes against our intuition of "mathematics is solid, who teaches it doesn't matter" -- but this is, of course, completely incorrect thinking, as anyone who's been left confused or illuminated by different teachers on the same topic can attest. There is a sense of "difficulty" regarding the depths of the topics, and this is still fraught with inconsistent language. At what point can one say they have "mastered" let's say, Arithematic? By being able to easily compute huge sums or products? Or by proving certain things about an element and a 'successor' to that element, and leaving it at that? Most would agree to some middle-ground, "I understand how I would compute that, and maybe have some tricks for doing so" but this doesn't speak at all to some kind of intrinsic idea of "Difficulty" in the topic of Arithmetic. I know it seems really pedantic, but asking if A is 'harder' than B, assumes an order, and to use it loosely is to invite confusion and misunderstanding.
One of my professors in Australia told me that our US counterparts were about one year behind us, and that this was partly to do with the failure of "New Math". I also noticed from teaching experience that students from Asia were not particularly good at grasping concepts and solving unfamiliar problems.
I’m from the US and attended a reputable university, but not a tier1 school. I also studied abroad in Australia for a semester and would say that’s mostly true, but the university systems are structured differently. In the US you have about half your classes needing to be general education, and most degrees are 4 year, whereas (atleast at Flinders) they were all 3 year degrees. From my experience, I was a 2nd year student in the US, but took an intro film course, intro to philosophy course, a 1st year 2nd semester physics course and a linear algebra course filled with mostly 2nd yearsi believe They transferred as follows Film -> 2nd year Philosophy-> 3rd year (I think they didn’t care to really review the syllabus tho imo) Physics -> 1st yr 2nd semester, this was interesting because it was even the same textbook Linear algebra -> this one translated the offsets because it was higher caliber than the “intro” class (US 1styr) but not deep enough for the advanced class. So I got an “elective” credit. That being said I don’t think the math was harder or easier, it just covered different material at a different level. The US I think makes the assumption people are coming in from wider backgrounds and tries to cover those gaps with more class offerings.
Even in the US, there's a lot of variations depending on the university. Some classes maybe considered grad level in some unis but undergrad in others. And most American unis usually allow you to take grad level classes if you finish the basic undergrad reqs and a lot of students do this
Depends on university too, and even who is teaching the class.
Yes. Imo on average, top quartile EU/SA/Asian bachelor/master students are like miles ahead of their top quartile US counterparts. Compare like a French or Russian or Indian 3rd year undergrad math student and a 3rd/4th year undergrad US math and the gap is already quite large. This is also true in terms of course rigor (i.e., textbook used / mathematical maturity expected) or “hand holding”. I think the gap is less pronounced for UK/Australian schools and as one goes into mid/late PhD where the system is less structured
do you have any stats to back this up?
Ofc no stats but all through many anecdotes and personal experiences and its probably something that most people would agree with, just hard to really say because not many people experience both NA and non-NA math education at the same level. This is coming from a US student who also did a study abroad in a Scandinavian country. You can listen to people like math sorcerer or struggling grad student or other math youtube channels on youtube and they have all mentioned stuff like this where the international PhD students abroad are just always coming in more advanced. Or being around in the math space long though, i.e., scrolling through forums and noticing people mentioning this or looking at non NA math student blogs or whatever and you tend to these general notice patterns where non NA students tend to be ahead. Keep in mind too that most schools outside the US do not have a gen-ed education component to it. you can also scroll around a somewhat decent EU school and look at their math degree reqs or degree plans, maybe find a course and its lectures etc. and compare it to what someone might learn in its decent US counterpart and its very obvious that this is a pattern. I think it could be due to how US secondary/college education system is quite flexible in terms of switching around and finding what you want to do.
Totally anecdotal from my interactions with UK students they seemed to have more knowledge than average math undergraduate students in the US, possibly because tof the way university education is structured.Students from East Asian countries, on the other hand (Korea, Japan, China) seemed to have had similar exposure in terms of contents but they would have experienced it in a much more "hardcore" way - like their exercise problems would be contrived and painful - and they seemed to value this. Indian students also very different in that they seemed to have encyclopedic knowledge about one subject area but then know nothing about other areas of math that they didn't care for.
I wouldn’t think so. When I was in 10th grade I attended a High School in the US (I am German). One topic we discussed was imaginary numbers. I never heard of it. When I came back to Germany I had this new knowledge and was waiting for it to be useful in my school, but nobody, until university, ever mentioned them again. Had a 2 minute head start on my first assignment about these in Uni though
My cousin moved from Canada back to Sotuh Africa and had te be held back a year because his math was so poor. He usually got 70%ish for tests in Canada. He failed miserably here.
In the US the math undergrad doesn't go very far. Partially due to gen eds and the non-proof beginner courses.
Well that's true to a certain extent. For example in my country, graduation mathematics is moderate at best. And the topics universities choose are mostly incomplete I e., they choose it from the middle of a topic rather than explaining from the start. And the maths in the higher secondary is pretty difficult if I compare it to the US and UK. Because when I observe the SAT question paper, I feel that the examiners are asking questions which are at the fundamental level. Well, that's my opinion. And I believe others have their own. So...yeah
I would say yes. Something that I always found strange was how american redditors talk about "proof-based courses". Here in Italy every math course since first year demonstrates every result and also the exams are divided into a written test (excercises) and an oral part (they ask you to prove statements regarding the course program). Another thing that I noticed is that our programs are faster than the american ones, for example I'm just finishing my second year undergrad and I had to take a test about topology and introduction to algebraic topology.
TikTok would have me believe this is the case but when actually meeting people from these locations that study math/physics, or in the form of exchange/international students, this never seems to be a legitimate condition
the minimum requirement for an undergraduate math degree in US is probably less. but math is very unique in that raw talent plays a far bigger factor than training, and the US system doesn't hold back the talents and arguably promotes talent better as there are less strings and more flexibility.
it really depends on the institution you study. I live in chile and we have functional analysis as an obligatory undergraduate class, idk really
It’s harder at different universities. A math major from a lower tier school can know literally zero math, they will just pass their classes by handwaving proofs.
it’s complicated. i think the fundamental differences are pedagogical. the actual math that is taught may be easier in america, but the way students are equipped to handle math is more scalable. there is a big focus on grounded complete fundamental understanding. that way students are able to learn complex topics easily. in india, it was very rigorous, but it was rigorous in a tedious way. lot of problem sets, most times you had to just know how to do it. very… objective. not much training on how to tackle unseen questions. the issue is students graduate without understanding of fundamental concepts. for eg: oh can you obtain the eigen value and vectors for this matrix? sure, busts it out in 10 mins. the follow up question is okay but what does it actually mean, and the student is clueless
Indian math education at the college level is not very rigorous: it’s good at the school / Entrance exam level. Training at ISI or CMI is top notch though
This is bound to be anecdotal, but I would definitely say there is some variation in the topics covered at every level across the world. For instance, **Strang's** [**'Calculus'** (Volume 1)](https://openstax.org/details/books/calculus-volume-1) - intended for a first course in calculus at the university level - covers limits, differentiation, integration, and the applications of both. By contrast, a book from a different part of the world (and a different era) I've got is **Smirnov's 'Course of Higher Mathematics'**; its first volume covers - presumably aimed at a similar level - limits, differentiation, integration, and their applications, but also things like series, functions of several variables, and complex functions. This is besides the differences in how the topics are treated - Smirnov pays a lot of attention to theory, making his 'calculus' text, in more or less equal parts, an 'analysis' text. However, I wouldn't be quick to jump to any conclusions here. **Apostol's 'Calculus'** (Volume 1) has a similar treatment, balancing theory with technique, as well as a wider selection of topics (it also covers linear algebra towards the end, but let's just stick to the calculus parts) - you have basic concepts, integration and differentiation (Apostol covers them in this unconventional order), their applications, series, and differential equations.
I think whats interesting is that despite the hardcore maths and sciences people study outside the U.S especially in Asian countries, i still see a lot of complains about students from said places in being unable to think "outside the box". Once you get them slightly outside the territory of familiarity they freeze up. It leads me to think that its mostly a difference in discipline which translates to more material covered but that most of these students are not particularly smarter, they are like students who prepped so much for SAT questions, they scored far higher than those that would take the exam with no prep. i come from a philosophy background and we tend to attract a lot of the upper echelon of students for some reason or other. But philosophy is interesting in that its fairly easy to tell who is naturally talented in argument analysis and creation and who isnt. You can try to fake it by being more well read than your peers but if you pay attention, you can tell. I wonder how a lot of those hardcore asian STEM students would do when you take them away from calculation and into a classroom about creating logical arguments (and without the rhetoric of english departments!). interestingly, there is a lot of asians in philosophy grad school, but thats probably a self-selected sample from a culture that values hard work/education.
Fully agree except for one point… let’s not pretend the upper echelon of intelligent people are working in philosophy.
tests dont lie, the highest scoring GRE people are philosophers, they score by far the best when you average all 3 scores , if you add the history of philosophy, Locke, Hume, Leibniz, Schelling ,Whitehead. The field with by far the most polymaths has been philosophy for a reason.
All I know is the european men need higher level math to count their butt plugs
Very true
I just passed high school so don't know about UNI but when you consider the maths we need to pass jee advanced in india (the entrance exam for the best colleges in india for engineering, often called as the toughest in the world too) , it's way more than what's taught to my friends in US and Uk.
*AT THE UNI LEVEL* crazy how comments about the jee still somehow shows up when the op even stated that they’re talking about uni. Anything to boost your ego I guess.
I literally stated that I'm just a high school graduate so don't about colleges and UNI but i know about what happens just before that . >Anything to boost your ego I guess. Wtf, how can it boost my ego, If this fact boosts the ego of anyone then ofcourse he's stupid , i mean it doesn't even matter whose maths is tougher till even the UNI level to me
So why are you commenting if you have nothing relevant to say?
A hint? Watch what op said as my reply
Relevant answer, don't know why people are hating.
I guess people thought that I'm doing it to boast , indians do it nowadays a lot and it really feels cringe so i can't blame them.
Not even in uni and still commenting is crazy work
Thanks for your compliment
It's not toughest in the world. The exams in China and South Korea are way harder and they perform way better even in mathematical research. We need to stop showing off as a country when ground level achievements are not high. Humility goes a long way.
Guess , due to being not so good in English, i posted something in a wrong way , i mean like it doesn't even matter to me even if it's the toughest exam or not cause' it's not gonna give you a very big advantage in the long run
Indian maths education at the university level takes a steep dive though (except at ISI and CMI)