Programs Best Places to Recieve a Degree (Maths) From?

[mathwonk]

I did not know any of that "standard" stuff when i went to college, and i went to a good college. all i knew was euclidean plane geometry and algebra up through quadratic equations, and a little logic and elementary probability (dice, cards), no trig, no calculus, no linear algebra.

However I knew that material well, and could use it.

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.

i also sometimes failed to hand in any hw, or take the midterm, so effectively it was all on the final.

so it sounds similar to cambridge. the only prerequisite was a willingness and ability to hang in there.

(I am not saying I had that ability.) it has changed now though i believe, and no one is likely to get in as ignorant as i was. I also knew what a group was, and could prove the reals uncountable, so sort of snuck my way into the course, over the objections of the prof.

Even though I did not succeed under that accelerated program, I liked it because it showed me what level I was supposed to be at, and allowed me to aspire to be there.

the point was to set the goals high enough to be useful, not low enough to be achievable.

fortunately it turned out later i did have the ability, i just needed the work ethic. Or perhaps i did not have enough ability for the work ethic i started with. so i just needed to elevate my work ethic until it was enough to compensate for my lack of ability.

There is nothing wrong with failing, if you are at least attempting something worthwhile, a concept that seems completely lost in our system today.

At the school where I teach now almost no one knows any of that material you listed coming in. Unfortunately that includes the ones who have been "taught" it high school. so I personally would prefer they come in really understanding even the tiny amount that I myself had on entering, rather than not understanding anything as it often seems now.

i also expect hard work, much harder than most are used to. that expectation is what really sets the best schools apart i think.

 

[matt grime]

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.

so it sounds similar to cambridge. the only prerequisite was a willingness and ability to hang in there. .

add in discrete probability, continuous r.v.'s multivairiate normals, branching processes and discrete maths, partial orders, combinatorics, generating functions, and group theory, geometry (of the complex plane), mechanics, more DE's (ones requiring series solutions, and coupled ones). subtract hilbert spaces and completeness, but then add in stokes theorem green#'s theorem etc. subtract general vector spaces but add in summation convention and more 3-d stuff that is useful in applied maths.

the good students then "pull-forward" (take a secodn year class early) linear maths (jordan normal form stuff).

 

[neurocomp2003]

did you guys have to complete a test to take 2nd year level courses in 1s year?

 

[mathwonk]

i did not mean it covered the same material as at cambridge. I meant the expectation of moving you well beyond where you were before. my course was actually off limits to anyone having had calculus. but maybe it still was not as hard. it was hard enough for me.


wow! what a pleasure reading the description of grades and expectations on the STEP webpage.

and the faculty of maths at cambridge look terrific. there is alan baker, and j.h. coates, and hey I know him! Nick Sheperd Barron.

boy it would be fun to be young again and go back to school at a place like that, in fact either one would do.

 

[matt grime]

Matt,

I've never seen the grade boundaries/descriptors for S, I, II, III but I thought that 3 semi correct solutions was quite good - but not especially so.


-NS

I was slightly misremembering but it's almost correct.

If you have 4 full predominantly correct answers out of the 6 attempted you have a 1, if you have 3 almost entirely correct then that would be a 1 on step 3 which is what i was thinking. that link i gave

http://www.maths.cam.ac.uk/undergrad/admissionsinfo/admissionsguide/text/node6.html

explains it

 

[matt grime]

But to be fair Matt, I'm sure you've also taught mathematicians at Bristol who do excel.

some, yes, but ability is different from knowledge. (this thread has two distinct flavours high school and unoversity, this part is about the poor state of high schools)

There's people at all universities of all (relative) abilities.

no there aren't, that is why universities have selection criteria, unless that is what the (relative) is supposed to mean.

Personally I found the course easy; looking at the course descriptions at Oxford, the syllabus at Bristol is very near identical, perhaps with Bristol offering slightly more bredth in the final year.

certainly i can believe that oxford and bristol have about equal reputations, but, a syllabus isn't worth the paper it's written on for comparative purposes. find me a syllabus that states it wishes to teach half arsed easy rubbish that won't stretch its students' intellectual capabilites, by all means, and prove me wrong. my students will be expected to "understand number theory to include finding HCF's and sing euclid#s algorithm as well as being introduced to group theory" to paraphrase, however that doesn't state what is basic and so on. certainly there are good students at bristol, and i don#t think that the first year number theory and group theory course will have remotely tested them or made them want to investigate the subject more because the material isn't very testing. whereas the mechanics course is demanding of them.

 

[matt grime]

did you guys have to complete a test to take 2nd year level courses in 1s year?

no, there w

 

[mathwonk]

well after looking at a sample STEP test level II, or something, it looks extremely different in sprit from the sort of question we were asked in first year college. Instead of computing some gruesome looking integral we were asked to prove say that every odd degree polynomial had a real root.

judging by hardy's problems, i suppose specific integrals have a long tradition on tripos.

of course the question that a positive function has a positive integral looks interesting. are you suppose to assume to function is riemann integrable, lebesgue integrable? i guess i could look at the syllabus, but it doesn't say.

 

[matt grime]

well afgter looking at a sample STEP test level II, or something, it looks extremely different in sprit from the sort of question we were asked in first year college. Instead of computing some gruesome looking integral we were asked to prove say that every odd degree polynomial had a real root.

of cousre the question that a positive function has a positive integral looks interesting. are you suppose to assume to function is riemann integrable, lebesgue integrable? i guess i could look at the syllabus.

STEP is predicated from the idea that the examinee will have some core set of knowledge (the A-level syllabus) and then asking as hard questions as they can from there. there are also questions that are essentially combinatorics too and are content free, often these are things about difference equations. they also want to see sustained reasoning and hence the tediously long integrals (which probably have a trick solutioon too)

 

[mathwonk]

ok here's my attempt at showing a positive riemann integrable function has positive integral. since f is riemann integrable, it is continuous almost everywhere, hence has a lipschitz continuous indefinite integral G which is differentiable almost everywhere with G'(x) = f(x) for any x where f is continuous. Moreover f>0 implies G is at least weakly increasing on [a,b]. But since the integral equals G(b)-G(a), and G has positive derivative somewhere, G(b) > G(a), so the integral is positive.

But I would be surprized, i.e. amazed, if an applicant is supposed to be able to do that sort of thing out of high school!

i am going to guess they were allowed to assume continuity of f. or maybe just a more elementary proof would be in order direct from the definition.


by the way i do not advise applying to any of these schools, and asking to "recieve a degree"! (just kidding)

 

[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