View Full Version : math courses

israelecon

02-19-2008, 01:07 AM

i was wondering if anyone could tell me what the difference between calculus and analysis is in the US?

Also in general how rigorous are the math courses in the US?

pevdoki1

02-19-2008, 01:10 AM

Calculus in US: no proofs, concentrates on learning the mechanics (Taking derivatives, integrating, testing convergence of infinite series, Lagrange multipliers etc)

Real Analysis in US: Constructing the real numbers, compactness, Bolzano-Weierstrass theorem, rigorous convergence results (series and sequences), proving things like L'Hopital's rule (REAL proofs), integration, the Stieltjes integral, sometimes measure theory etc. The point is that you prove everything you use in calculus (AND precalculus) rigorously.

My courses in Algebra and Analysis have been quite rigorous. Basically, you can't write anything down unless you know how to prove it.

israelecon

02-19-2008, 01:18 AM

in israel nothing is taught without first being proven, very little stress on mechanics, they just assume you will pick that up by yourself. On tests and homework sets everything must be proven rigorously (i.e. you can't say "it is obvious that..." or "clearly...".

So i guess every math course in israel is rigorous like analysis and algebra in the US, which means we go through the material really slowly.

pevdoki1

02-19-2008, 01:24 AM

Most think on a calculus exam would be exercises (i.e. prove that d/dx(sin-1(x))=blah blah).. you still have to know to use the epsilon-delta definition of a limit (on a very basic level), but the fact is most people taking calculus are not interested in mathematics (just like some people taking differential equations ONLY want them to solve engineering problems)

It's funny how we are responding to each other in 2 threads at the same time.

israelecon

02-19-2008, 01:32 AM

so in israel there is no such thing as math courses for non math majors, no higher and lower level classes, so basically everyone there is interested in math and they start training you to be rigorous from the beginning.

in israel the average in most departments is about 80 in the math department its about 65 for the core courses, is this also the case in the US (much lower math grades than other grades)?

israelecon

02-19-2008, 01:33 AM

i started this thread cause i thought people would be more likely to answer here than in the nyu thread.

and yeah its kinda funny, but so is everything at 4 in the morning

pevdoki1

02-19-2008, 01:36 AM

I don't know if math grades are any lower than grades for other subjects. I don't think this is the case in my university (grades in my algebra class tended to be quite low, but I have no idea how the grades are in physics, chemistry, biology and so forth)

israelecon

02-19-2008, 01:44 AM

in israel all exact science grades are on average lower by 15 percent than grades in other faculties. it is not uncommon for the average in a course to be in the low 40's to high 30's. i once got a 65 in a course which looks really bad, but it was the third highest grade and the average was 43, and this was after they raised everyone's grades by 10 percent. (not surprisingly about half the first year math students drop out after the first year).

TruDog

02-19-2008, 01:48 AM

Just for what it's worth, isrealecon, you've got one of the strongest profiles I've ever seen. I'm glad that I'm in grad school now and not applying at the same time as you.

israelecon

02-19-2008, 01:52 AM

i hope you are right, trudog, but i don't really see whats so much stronger than others applying to top schools, except maybe the award i got from my university and my verbal gre, but they don't care about that so much anyway.

Ancalagon The Black

02-19-2008, 05:15 AM

At least at the undergraduate level (from experience and from having other people whom I know from my country talk about it) I know that most US mathematics courses are not that rigorous. Obviously, the system of education is much different.

Of course, you get special classes like Math 55 at Harvard which is supposed to be very very tough but I am not talking about specific cases but on the whole, in general.

israelecon, I agree with you, we are also taught to do proofs. Mechanics is secondary.

Again, the nature of mathematics (the exception is engineering mathematics) in India is very tough and grades awarded are very paltry. There is so much variation in grades that it is impossible to create any sort of model to assess them. For instance, in my introductory statistics class, I have a horrible looking grade of D (50% - 59%)

This looks really really bad. In reality, I have got 58 and the highest marks is 61 which makes it look really really bad (I am ranked 2nd in stats class) Even if we get the entire problem right, our professors do not give us full marks. Its just the way it is. And for more subjective courses, it depends on the professor - generally they are very stingy.

However, engineering schools award marks by the plenty - people get 70% - 80%+ quite easily. If you took a look at my transcript, you would be appalled to see so many Cs and Ds and really few Bs. However, I am ranked 4/150 in my department. Therein lies the dichotomy. :(

Karina 07

02-19-2008, 07:08 AM

Also in general how rigorous are the math courses in the US?

Ha ha ha!!!!! Not at all until analysis (imho). Upper-level courses are perhaps the same in rigour -- but it's hard for me to say since I don't know Israel's courses!

And I think math/sciences here do tend to have lower averages than other subjects, though there'll be some variation. To me, a typical median in say a humanities or social science will be a B+, while a typical median at least in the "core" courses of a math or science major will be more like a B-, at least in the intro courses before people are weeded out.

That being said, people who do straight up maths or sciences can have a better opportunity to get 100%s, too, or in letter-grade terms get all As and become valedictorian. I think the median in math/sciences is just pulled down by a bunch of people.

Just my own observations. Undoubtedly varies across schools. I'm personally in the middle of a vendetta against people who say that the *discipline* of math is harder than X other discipline, because it seems to me that's presuming a lot, and the preceding paragraph captures at least a bit (but only a bit) of that -- who's to say a non-math non-science major isn't trying for that 100%, in which case he's basically doomed, etc. It doesn't transfer so well to letter grades, and it's WAY off the point... maybe another day I'll flesh out my whole big argument.

israelecon

02-19-2008, 07:18 AM

i agree with you that math is not necessarily more difficult than other fields, just in my university more is expected of math majors than say, economics majors. economics can be made difficult, but its just not made as difficult for some reason. i guess it has to do with the culture of the department in any given university.

also, here there are definitely a few math majors who have 100% average, but the average is still low because in the math courses the grades are distributed in a U distribution, people have either good grades or bad grades, nearly nobody has a 70-80%, because in some sense in math you either know it or you don't.

but, math for me is not more difficult just in that the tests are harder, its mainly that to get 90 in a math course requires studying about 20 times as much as getting 90 in an economics course. and i mean that literally.

Ancalagon The Black

02-19-2008, 03:33 PM

I have a question. I have often heard of professors in the US awarding grades on a Bell shaped curve. Now, how does this work?

buckykatt

02-19-2008, 03:50 PM

also, here there are definitely a few math majors who have 100% average, but the average is still low because in the math courses the grades are distributed in a U distribution, people have either good grades or bad grades, nearly nobody has a 70-80%, because in some sense in math you either know it or you don't.

This phenomenon is widely-observed in computer science classes, especially at the introductory level. Some say that either your mind works the right way or it doesn't; some see it as evidence that we haven't really figured out how to teach programming yet.

buckykatt

02-19-2008, 04:07 PM

I have a question. I have often heard of professors in the US awarding grades on a Bell shaped curve. Now, how does this work?

Maybe not a true bell curve, but something like that. (Probably thinner at the lower end and fatter at the top, but that's my intuitive sense.) Here's a cynical view...

In my experience, students sort themselves out in three groups:

1. Those who truly master the material, with varying degrees of effort required.

2. Those who don't truly master the material, but who come to class, take good notes, show up at office hours, and study for the exams.

3. Those who couldn't care less and are only taking the class to satisfy a requirement (e.g. many of the business majors in an economics class).

Roughly-speaking, those are your A, B, and C students, respectively.

You could design a grading system that would place those students anywhere along the continuum of grades, but the students have certain expectations based on typical grades in comparable classes at the school. If you give them lower grades, they will give you bad evaluations and possibly they (or even their parents) will complain to your department head, the dean, etc. And anyway, that issue aside, no professor wants to penalize students for having the good judgment to take their class.

(Of course, if you're a tenured professor with a class that's overbooked, you can weed out students one of two ways: assign more work or grade more strictly. If you have a TA, the first strategy may be feasible; if not, you're probably going to go with the second. If you go with the second and your course is not required for anyone's major, you'll weed out the weak students and end up roughly back where you started in terms of the grade distribution.)

So, whatever the numerical scores you put at the top of each paper, when it comes times to assign letter grades for the course you look around at what other profs are giving for grades in similar classes (or, if you're a TA, you compare notes with the TAs for other sections of the class) and adjust the grades for your class until they are reasonably close.

A true cynic would note that this effect is likely to be more important for profs who haven't earned tenure yet, especially in a "teaching school". Thus, to maximize grades earned for effort expended, one should always take classes with junior faculty.

I have nothing beyond casual empiricism to back all of this up. ;)

I should add that no matter how much a professor "curves up" the grades, their self-respect may keep them from awarding an A to someone who clearly doesn't fall into group #1 above. I think that's why you sometimes see folks here claiming that there's a big difference between an A and an A-.

buckykatt

02-19-2008, 04:18 PM

Upper-level courses are perhaps the same in rigour -- but it's hard for me to say since I don't know Israel's courses!

Same here. The classes taken mainly by math majors (e.g. analysis) are taught at a high level, but in others (e.g. lower-level calculus and, to a lesser extent, upper-level calc and linear algebra) the non-math majors drag the level down. For example, profs still present proofs in class, but the exams require mostly problem-solving and mechanics. In a larger school, the math majors might be tracked into separate classes to solve this problem, though.

So, an A in, say, Calc II is a fairly weak signal compared to an A in Analysis or even Algebra.

Karina 07

02-19-2008, 04:36 PM

These are all really good posts.

One last thing to add: I know oftentimes when assigning final grades (fitting the exam etc. results to the letter grades) a professor will look at the distribution of the uncurved marks and identify "troughs" to help serve as the boundaries between marks. This is probably already implied by buckykatt's posts but I personally was excited by this detail when I first heard it so I thought to bring it out. Not only does that potentially help in identifying the kinds of students as described above, but it has an added perk: it minimizes the chance someone will come up and complain that they're 0.001% off a higher grade. So, sometimes it won't be a nice bell-shaped curve just because the uncurved grade distribution has troughs in slightly weird places.

Ancalagon The Black

02-19-2008, 04:42 PM

bucky:

Very well said. However, why would professors need to grade on a curve? That sort of puts grades into a quota no? Can they not judge every student individually without always putting them in context with the rest of the class or a predetermined quota?

buckykatt

02-19-2008, 04:43 PM

Yes, the students sort themselves out into groups pretty naturally. IMHO, it's often harder to figure out who gets, e.g., a B+ or B- instead of a B. Perhaps that's why it's so common to see profs making some explicit allowance for attendance, class participation, improvement over the course of the term, etc. If you have two students with identical "scores" and you want to smooth the distribution, it's easy to justify giving the B+ to the one who showed up to class and the B to the one who didn't.

israelecon

02-19-2008, 04:56 PM

the reason many professors put grades on curves is to avoid complaints by students that the grades are too low, because then the prof can say that there are x A's y B's etc. its also to avoid complaints by the dean that the grades are too high. i.e if the grades are bell-curved the prof insures himself from complaints in either direction.

Ancalagon The Black

02-19-2008, 04:59 PM

Thats very interesting. So students complain about receiving low grades? Thats like unheard of here.

buckykatt

02-19-2008, 05:05 PM

Very well said. However, why would professors need to grade on a curve? That sort of puts grades into a quota no? Can they not judge every student individually without always putting them in context with the rest of the class or a predetermined quota?

It's certainly easier to give all your students an A if you think they deserve one in some objective sense than it is to give them all a C. The top students might grumble a bit about the former case, but you'll have a riot on your hands in the latter case. Likewise, a tenured prof can deviate consistently from the norm with fewer repercussions than an adjunct.

But I don't want to oversell the external incentives as the reason. It's also very difficult to be objective about what constitutes an "acceptable level of progress", or to know what's fair when you have a student who has arrived in your class ill-prepared because they've been failed by the rest of the educational system. The first time I graded student papers I was extremely harsh, because I had unrealistic (though, I contend, not unreasonable) expectations of what college-level writing should be. You know--things like complete sentences, verbs that agree as to tense, and words used according to their accepted definitions.

I think all of us want to be fair when we assign grades. Unless we believe that we truly ended up with a less capable or hardworking group of students than the instructor across the hall (which sometimes happens), it's natural to want to make our grade distribution similar. Whatever we may think about the poor state of education, it doesn't seem fair for a student to get a C instead of a B (or vice versa) simply because they drew us in the scheduling lottery.

buckykatt

02-19-2008, 05:10 PM

Thats very interesting. So students complain about receiving low grades? Thats like unheard of here.

I hear that it's even worse than when I was teaching in the early/mid-90s. Some of my friends tell me about these so-called "helicopter parents" who will complain on their students' behalf and even--get this!--show up to job interviews with them! I keep hoping that they're exaggerating, but deep down I know they're not... :(

Ancalagon The Black

02-19-2008, 05:22 PM

Thats ridiculous !! Surely, students cannot be allowed to complain just for the heck of it and certainly not parents !! And the university listens to them?

israelecon

02-19-2008, 05:28 PM

my father is a prof here in israel and he says that students begin complaining before they even get the grades back. when he comes home after a test his e-mail is already full of letters complaining that the questions weren't clear enough. basically what does a student care to complain, it doesn't cost anything to complain and sometimes it actually raises your grade so why not complain?

AstralTraveller

02-19-2008, 07:27 PM

Back when I lectured Statistical Inference, I received a complain from a parent who argued that I should pass her daughter because she was under a medical condition (stress), despite she not having taken any of my exams! I didn't even know the kid (she never showed up in class).

I passed on the highly inappropriate pressure to the highest authorities of the department. I mean, maybe she could get an exemption for the semester instead of failing, but getting a passing grade without any evidence for me to pass the kid, it was unheard of! Of course she did not pass. The department decided ultimately to fail the kid.

Ancalagon The Black

02-19-2008, 07:37 PM

Heh !! I reckon if you try to complain in my university they will flunk you in the next test or just refuse to entertain any such requests.

econorama

02-19-2008, 08:23 PM

In my econometrics class, after the first test some students complained to the professor that he hadn't adequately prepared us for the test. We hadn't gotten our grades back, but they were assuming (correctly) that they'd done badly.

One student got a score in the 90s, I got an 87, and everyone else got scores in the 60s or below. But the test had been entirely made up of homework problems we'd been assigned. For most of the questions he hadn't even changed the numbers. People will complain about anything if they think it'll help.

Ancalagon The Black

02-19-2008, 08:25 PM

Excellent point econorama. I remember that I once complained to the department head about the teaching (or lack of it) of a particular teacher. Nothing happened. He still teaches us. Ugh !!

Karina 07

02-19-2008, 09:41 PM

Undergrads have the right incentives to complain, I guess: they don't usually think their profs' opinions of them matter, at least not the vast majority of their profs because even if they need one or two profs to write them recommendations later they can still blow off entire departments.... It's maybe like a lesser version of people posting fake things on thegradcafe.com, where it's basically costless to complain and you might get an improved mark back, even if the chances are slim. I had a prof once who, noticing this, said that you were free to complain to him about any mark and try to raise it by X, but if he disagreed with you he'd reserve the right to take X *off* your original mark.

Incidentally, grad students should never complain about marks. A prof told us this quite frankly at the beginning of the year. He said even if you're right, even if you did deserve more, the fact that you're complaining will be more of a black mark against you than any miniscule benefit you could derive from it, and tales of your complaining would spread throughout the faculty.

israelecon

02-19-2008, 09:45 PM

i know a prof that gives everyone 5 extra points on every test, and you forfeit those points if you complain and he doesn't accept your complaint. this creates a disincentive for dumb complaints.