Best Places to Recieve a Degree (Maths) From?

[matt grime]

a rigorous proof would be hard, but pretend you're a physicist answering it

 

[mathwonk]

then its "obvious"! my proof is rigorous of course. (to me anyway, since i know how to fill all the details.)

 

[James Jackson]

Indeed, that's what the relative was qualifying. Within the acceptance criteria there are those towards the weak end, those towards the strong end and all inbetween.

I completely agree with what you say about using syllabuses for comparative purposes, I was mearly trying to illustrate (badly, I accept) that Oxbridge isn't the be-all and end-all of a top class university education in the UK.

Anywho, back to the state of High Schools (I assume they're the US equivalent of Secondary Schools). My further maths A-Level covered group theory and discrete mathematics too - has this syllabus now changed (I was with OCR I think)?

I think it's difficult to compare A-Levels over the years, as the course content has broadend greatly. Perhaps now pupils are being taught more topics at a lower level, compared to being taught fewer topics at a higher level. This is in no means qualified with any evidence, it's just a suggestion. I'll ask my Dad what he covered in his Maths and Physics A-Levels way back when!

 

[James Jackson]

a rigorous proof would be hard, but pretend you're a physicist answering it

He, he. Approximate, expand, remove some small terms. The Physicist's way of answering anything...

 

[mathwonk]

lets see, what would it be like to be a physicist? ok, if the integral were zero, then for every e>0 the set of x such that f is greater than e, actually has content zero, so the whole interval would be a countable union of sets fo content zero, surely a contradiction to a physicist!

 

[NewScientist]

Physicists - we like this theory. Something agrees with this theory. It must be right - we don't know why it is right but it must be - we cannot prove it but we assert its validity.

Oops, something contradicts our theory. The theory must be wrong. Here is a different theory to describe the phenomena...we like this theory...ad nauseam.

NB, this takes place over 20/30 years :P!

-NS

 

[mathwonk]

or if the integral were zero, then the indefinite integral would be constant. but then its derivative, which is zero, would equal the original function which would then be zero.

but then it doesn't take very long. (and oops, its false.) so we add more assumptions,...

 

[mathwonk]

well i guess my proof in post 75 becomes rigorous if we use compactness. i.e. the interval is a countable union of sets of content zero, so for any d>0, the nth set has an open cover by a finite number of intervals of total length d/2^n. then the whole interval is covered by a finite collection of intervals (using compactness) of total length less than d.

this contradicts the assumption that the function was everywhere positive and the integral was zero. and this proof would take a whole for a student to write down, but is conceivable.

in fact this sort of proof occurs in hardy's pure mathematics.

is that the sort of book an applicant to cambridge would have already read?

 

[matt grime]

the question did just ask you to sketch (draw the graph rather than an otuline of a proof) the answer. remember the person taking this exam will not know anything abotu continuity or compactness or measure theory.

i was using high school to denote a pre 18 education (i'm english, my girlfriend american, i have lived in the US and we both now live in the UK. you get used to speaking in nongeographic specific terms sometimes and hope no one asks too closely what you mean).

i think it unfortunate that someone entering university doesn#t nkow how to solve 2nd order linear DE's, but i don't mind it being off the syllabus at high school. however, knowing how to sum a GP or 1+2+..+n is essential.

and for the last point in this post. yes in any university course there will be people obtaining 1sts and people failing. but there is no necessary guarantee of any absolute standard, and if the standard of a course is quite low then getting a 1st is unrewarding. many universities impose absolute standards, eg this exam is marked out of 100, anyoen getting over 90% will be marked down to 85, under 25 will be marked up to fit our preferred curves. it takes away a chance to shine. cambridge doesn#t do that - the exams aren;t percentage based in the same way, and it's almost physically impossible to do all the questions you are allowed to do. this was even more marked a century ago when people#s marks were on the scale of "so many thousands". i would rather see a system where it is a struggle to obtain marks, and score of 50% means you're a genius.

 

[matt grime]

in fact this sort of proof occurs in hardy's pure mathematics.

is that the sort of book an applicant to cambridge would have already read?

no. absloutely not. the STEP idea is to test what you#ve been taught but with very difficult questions, not what you may have independently read ahead on. somequestions will be dependent on no backgorund, for instance the one showing that all the "fermat numbers" are relatively prime and hence there are infinitely many prime numbers is not a test of anything on a syllabus at A-level.

 

[James Jackson]

I believe one of my father's finals paper for biochem at Oxford consisted of the single question:

Discuss the properties of <some compound>

3 Hours.

Nice.

 

[juvenal]

As a US physics grad student, I've met my share of foreign grad students who've studied in other countries, and even some who've done their PhD's abroad. The Russians who've gone to MIPT in Moscow have definitely impressed me. I've also been impressed with some Italians. But I can't really say that I've been all that impressed, in general, with any of the Brits, some of whom I'm good friends with and have attended a number of different uni's over there, including Oxford and Imperial. Their knowledge/preparation seems to be on par with, if not inferior to, that of Americans. This is in physics, so it's possible that in mathematics, things are completely different.

I haven't met any Belgians, so I can't really support or refute anything Marlon has said.

 

[matt grime]

it is not unreasonable for a VIGRE funded US PhD student upon entering a graduate program to be ignorant of topics taught in the first term of a UK undergraduate course in mathematics. Ther are good, well prepared US students, just as there are underprepared British ones (please for the love of god stop calling us Brits, it's such an ugly word). there will always be extremes but most (all perhaps) graduating pure mathematicians who enter a grad program from an english university will know what simplicial homology, or a differential manifold, or a measure space is, and many will know all 3. if i think of the pure maths phds at bristol i know (about 8) then all will know at least 2 of them. if i think back to the US students at PSU only 2 of the 8 in my year knew of them upon arrival.

but then it's apples and oranges. i would, for instance, expect a harvard educated undergrad to know those things. but they are the minority in the US, and I am thinking about the general situation. of course it may well be me who experienced the minority, and of course one must factor in that there are many more PhD students in the US altogether than in the UK, perhaps that heavy restriction on numbers here skews the picture and only admits the well prepared.

 

[mathwonk]

well i think i have seldom met an undergrad freshman in calculus at my school who could get any of those level II STEP questions.

I agree it is a wonderful boon to education when the questions are much harder than anyone can do. this is the basic flaw in much US education, that everything should be so trivial that very few will miss anything.

that philosophy and orientation on the cambridge sitte reminded me of the talk at harvard when ai was an udergrad there in 1960. even at harvard it is different now, and "course evaluations" have succeeded in inflating the grades by more than an entire grade point, from a C+ to A-.

the frustration for those of us who did poorly was, that we knew that even a D- at harvard in those days was better than an A+ at some other schools, but no one else knew that.

so i expect that argument won out and they started giving higher grades.

the truth was, although i would not admit it then, that those of us admitted were capable of getting good grades even at harvard, and if we did not do so, it meant we had not tried hard.

so a low grade, even if we still knew more than someone else, meant we were not achieving to our full potential, and thus we deserved it.

 

[mathwonk]

if entering colege students at cambridge are not expected to know hardy, in what sense are they expected to "know calculus"?

and does the first year course there teach calculus at the level of hardy? (hardy was a recommended book, along with courant, for my first semester university course, i.e. my first course, in calculus.)

 

[juvenal]

Matt - I've never had a British person complain about the term "Brit" before. Maybe you guys are just too polite? Is it a fairly universally detested term? That's truly news to me.

 

[mathwonk]

as my friend said when I said my buddy at the meat market told me: "they calls me georgia, but my name is ted", my buddy says: " so you call him ted, right?".

 

[mathwonk]

if anyone is really reading this for advice, we have had people at my school with degrees from harvard, princeton, berkeley, etc etc etc, but two of the absolutely smartest guys there, and most valuable and respected, are ones who have degrees from grinnell in iowa and unc in north carolina.

i got a degree from harvard but i am still just me. nobody cares about that if i cannot answer their question.

so wherever you find yourself, do your best and you will rise to your natural level.

 

[lyeinyoureye]

College choice matters a great deal. The teaching and extra-studial word (such as applications of math into physics, computing etc) is different at different institutions.

Saying college choice doesn't matter is like saying that going to the north sea is just the same as the carribean because they both have water!

-NS

Math is math. It's not like chemistry or physics where country "X" has a clear cut advantage in facilities... the basics are pretty common. and don't require anything more than text. Some schools may have bias towards certain fields, but for everything undergrad, students can pretty much learn it on their own if they so desire.
Because of this my suggestion was that it doesn't matter much where they go for undergrad so long as they figure out what they want to do research in. Then they can base their choice off of that.

 

[marlon]

of course the average student couldn't even start a STEP paper. they exist exactly because A-levels are poor determiners of ability at degree level. in any case the question wasn't about the standard oof high schools but of universities.

at what age do you graduate from uni in belgium?

you leave high school at 18 and college at 23 (most sciences and engineering take 5 years of college). The exact sciences like physics normally took 4 years of college (to obtain a masters degree) but this year the governement made it 5 years because of Bologna...

regards
marlon

 

[marlon]

my first year college math course then covered real numbers and complex numbers axiomatically with complete proofs from scratch, continuity, differentiation, integration, simple differential equations, infinite sequences and series, bolzano weierstrass, cauchy completeness, trigonometry via taylor series for e^z then sin, cos as functions of e^z, then vector spaces, inner products, prehilbert and hilbert space. that's about it.

really, but did you not see this in high school ? I mean stuff like the theorem of Bolzano , Weierstrass, Rolle, Cauchy, Heine Borel were all covered in high school. Ofcourse in the advanced math course but nevertheless i knew this when i went to college.

marlon

 

[rachmaninoff]

theorem of Bolzano , Weierstrass, Rolle, Cauchy, Heine Borel were all covered in high school.

Is the American system really that far behind everyone else? Where I am, these theorems aren't even mentioned until 3rd year of university in the standard progression...

 

[marlon]

Is the American system really that far behind everyone else? Where I am, these theorems aren't even mentioned until 3rd year of university in the standard progression...

normaly you should see these in any calculus course. They are used to formalism concepts like continuity and several function-type behaviours and properties (like if f(a) > 0 and f (b) < 0 and a > b then there must be at least one 'c' between a and b where f(c) = 0)

stuff like that

marlon

 

[juvenal]

Marlon - what textbooks do you use in high school? The typical American high school calculus sequence is not proof-based, and you'll never see Heine-Borel, for example.

As a benchmark, the most advanced intro freshman math class at Harvard uses books like Baby Rudin, i.e. Principles of Mathematical Analysis. At Caltech, the freshman use Apostol's Calculus book(s), and the sophomores taking real analysis use something like Strichartz.

 

[marlon]

Marlon - what textbooks do you use in high school? The typical American high school calculus sequence is not proof-based, and you'll never see Heine-Borel, for example.

I think that is one of the main differenes. Generally, quasi all theorems are proven here. It is how i was instructed. The theory is very important in the more advaced math courses. They are all Belgium books that are used throughout the country, like the DELTA or Jennekens series

marlon

 

[rachmaninoff]

That's different here. Most freshman calculus courses AFAIS are not considered 'advanced' maths - they're practical courses which teach just teach evaluating integrals and stuff. The theory behind it is left to 2nd- or 3rd- year Real Analysis classes - many non-math majors (including physics) never see Bolzano or Lebesgue. It might be different at places like MIT, where they have a freshman calculus track with theory (one of three tracks there):
http://student.mit.edu/@3336181.29109/catalog/m18a.html
Smaller departments like mine don't offer anything like that.

 

[marlon]

That's different here. Most freshman calculus courses AFAIS are not considered 'advanced' maths - they're practical courses which teach just teach evaluating integrals and stuff. The theory behind it is left to 2nd- or 3rd- year Real Analysis classes - many non-math majors (including physics) never see Bolzano or Lebesgue. It might be different at places like MIT, where they have a freshman calculus track with theory (one of three tracks there):
http://student.mit.edu/@3336181.29109/catalog/m18a.html
Smaller departments like mine don't offer anything like that.

this MIT course indeed resembles the courses i had i my first year of college.

Ths is my whole point though. I am not denying that MIT and friends are top notch univesities in the US. However if you compare the level of difficulty with many Europea universities like the KUL or UGent in Belgium, it is not that big to say the least. I know a few people who have gotten their degrees at the universities and then went to Stanford, Caltech and Yale. Trust me, the difference is minimal. just think the level of the average US college is much lower then many European universities. Moreover, i even think that some US colleges have a lower level then some Belgian high schools.


marlon

 

[matt grime]

if entering colege students at cambridge are not expected to know hardy, in what sense are they expected to "know calculus"?

and does the first year course there teach calculus at the level of hardy? (hardy was a recommended book, along with courant, for my first semester university course, i.e. my first course, in calculus.)

entering they will know differentiation and integration and differential equations. though to what level these days i do not know.

at the end of year 1 they will know analysis proper (limits, sequences, etc) some complex analysis, differential forms as an applied mathematician would do it, stokes theorem green's theorem etc.

http://www.dpmms.cam.ac.uk/site2002/Teaching/IA/AnalysisI/2004ex1-4.pdf

here for example are the 4 examples sheets of the first year analysis course, these are the first half of the term, the second half they do vector caclulus.


juvenal. i have no idea if other people hate the word Brit too, but i;m trying to start a trend (if i did smilies now would be a good time to use em)


marlon, sounds like the belgian system is what i wish ours had been, and perhaps was 30 years ago. I've looked back at the first year exams from cambridge from the early 80's and it#s amaxing (in the sense that politicians are adamant that standards have noty dropped) how much more difficult they are.

from looking around finding thind out for this thread it appears that an approximate analogue for mathwonk would be "pick the hardest undergrad maths course in the US, and imagine a high school student jumping straight into the 3rd year, or certainly half way through the second, that is what it would be like to go to cambridge" it's not a fool proof analogy, admittedly, since i am attempting to digest the yale (etc) website's attempts to describe its courses and when one is expected to take them and they aren't very clear. i am basing it approximately upon when you start talking about algebra properly (groups, mainly)

one thing that i would like to know is why we in the UK aren#t strongly, openly and actively looking at europe to remodel our education system since it sounds (and is) far more admirable than ours. i was already aware that the university education was better both in provision and length, and that primary (elementatry, aged 5-10) schools were better (a certainly in a social sense), but i wasn't aware of such marked differences in th high schools. admittedly marlon did say these were "advanced classes", are these classes universally available?

looking back over the years at the changes in syllabus univeristy's here (and to some extent this covers cambridge too) are playing catch up for the first year compared to the situation 20 years ago. in some cases they never appear to catch up with the continental european levels.

i must admit though that my personal beliefs mean that i will always demand a higher standard in education, a standard that not all can attain. i found the syllabus at high school completely unchallenging and it wasn#t until i started practising for STEP that i really found motivation and failure came along. fortunately my teachers at school helped me learn how to do the papers and i ended up with a distinction in STEP 3 (but oddly a worse mark in an "eaiser" paper). i would suspect that many people didn#t have such a lucky experience (state schools like mine with this extra help would'nt be common place) and i wonder how many talented individuals are put off from applying to cambridge because of it. but this way leads to an even more off topic ramble about misinformation and applications. sufficed to say how many other countires would have a system where it is casually accepted (against the evidence) that oxbridge is biased against state school applicants and where teachers in schools even tell students not to bother applying because they won#t fit in rather than because they aren#t clever enough?

 

[matt grime]

just think the level of the average US college is much lower then many European universities.

a good point that hasn't been made enough is that the differences are large only on a large scale.

 

[misogynisticfeminist]

ohh come, these are just standard topics. If a student does not know these, what the hell is he/she going to do at college ?

How about adding the concepts linear algebra (base vectors, linear transformations, groups, ...)



regards
marlon

in Singapore the norm is that, in pre-university. 2 kinds of mathematics, standard math and further math is offered. Only a small number take further math. Standard math doesn't even talk about ODEs higher than 1st order, linear algebra is totally out, things such as hyperbolic functions and polar coordinates are totally not covered. the F maths people do a little bit on linear algebra but most is centered on matrices, and they hardly touch on vector spaces.

the amount of material covered is quite bad actually.

 

[gravenewworld]

I would like to compare the breadth of an American education with the breadth of a European university education. I think the reason that american universities don't go into as much depth as European univisities is because American universities stress breadth over depth at the undergraduate level. Most American universities stress a liberal arts education even if you are in the sciences rather than just specializing only in your major at the undergraduate level.

 

[mathwonk]

Marlon,

As I said earlier, all I knew from math upon entering university was euclidean plane geometry and algebra up through quadratic equations, plus a little logic and simple combinations and permutations. no trig and no calculus, and no linear algebra.

nonetheless, i was much better prepared than students i have today who have taken calculus in high school, as i understood and could use the topics i had taken, whereas most of today's entering college students here not only do not understand calculus, they also do not understand algebra or geometry or trig, much less logic.

It is also misleading just to list topics covered in a course without any idea of the depth to which they are covered. in my high school, there was no depth at all, and in college the depth was as great as I use now as a professional mathematician, at least in the math courses I took, but not in all courses for all students.

there was wide variation in level of math courses at the same college. as a freshman i took the honors course and encountered questions worth only 1/4 of a point out of 10 on homework, that were worth 25 points out of 100 on a midterm in a non honors course i took as a sophomore. The difference was laughable.

 

[mathwonk]

Many people here seem obsessed with compoaring the lists of topics covered in their cousres as if that were a good thing, and better in proportional to how many there are. In my opinion, the increased emphasis in the US at least on high school courses that cover a lengthy specific array of topics has harmed college preparation greatly.

Instead of a student who has read a specific list of books e.g. I would prefer students who know how to read critically, and generate and argue their own position well.

It is not what material they know, but how well they know it, and what intellectual skills they have acquired.

the same holds in math.

rather than having studied calculus shallowly, i would rather an entering student have a good grasp of algebra and geometry, and some acquaintance with logic and proof. It would be nice if they have some imagination as well, and computational strenbgth, such as is measured by the STEP questions.

But what I especially like is the philosophy expressed on the Cambridge website toward excellence and the high expectations, and I am tempted to copy these guidelines for my colleagues' consideration.

The Belgian system also sounds very impressive. If you will suggest some websites where i could learn more I will enjoy them.

It is hard to learn anything from lists of courses as Matt has remarked, but in the old days the catalog said things like: "we attempt to place every student in the most advanced course for which he/she is prepared."

the honors course also carried warnings like:" this course requires not only roughly twice as much time as the regular course, but also a high level of possibly undefinable 'mathematical ability'."

 

[matt grime]

I would like to compare the breadth of an American education with the breadth of a European university education. I think the reason that american universities don't go into as much depth as European univisities is because American universities stress breadth over depth at the undergraduate level. Most American universities stress a liberal arts education even if you are in the sciences rather than just specializing only in your major at the undergraduate level.

that would be a good point, though you would have to allow for the fact that the liberal arts you study there at a US university may have been taught at high school in Europe.; if the level of scientific education entering an average state university is two years behind that of a european university in the sciences then why not in the arts too? in mainland europe, though sadly not the UK foreign languages are taught to students before the age of 11. and there is natural breadth in the baccalaureate system as well. the UK is (and the isn#t and then is again etc) in the process of thinking about (we don't like to hurry these things) adopting a broader education system between 16-18 to refelct the baccaluareate system.

 

[mathwonk]

at my childrens school, 15 or 20 years ago when I tried to lobby for foreign languages before high school, one intelligent middle class parent asked me "why would anyone want to know French?"