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

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matt grime said:
it would thus be beneficial for the incoming student to know

complex numbers, 2 and 3d real vectors, matrices, determinants of 2x2 and 3x3 matrices, dot and cross product, 2nd order differential equations, all their trig identities, integrals via substition etc, hyperbolic trig
ohh come, these are just standard topics. If a student does not know these, what the hell is he/she gonna do at college ?

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

This is a clear example of the high school level being quite low in the UK. Here in Belgium, your list would make an self respecting future physics student laugh, really, that is the truth. Like i have stated before, the educational high school level in Belgium is the highest in Europe and about nr 5 in the world after all Asian countries and Finland i believe. Look for proof at the PISA survey if you do not believe me

regards
marlon

ps for proof look at the 'math aptitude internationally tested' entry at
https://www.physicsforums.com/journal.php?s=&action=view&journalid=13790&perpage=10&page=9 [Broken]
 
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marlon said:
ohh come, these are just standard topics. If a student does not know these, what the hell is he/she gonna do at college ?
I didn't do some of those topics, despite doing A Level Maths and AS Further Maths (AS is kind of a "half" A Level). At university, they went through most of it anyway, so it didn't really matter that much. However, they've recently (2004-05 being the first academic session, or it might have been 2003-4, I'm not sure :/) changed A Level Maths and made it easier, by taking out lots of stuff. It's kinda bad in a way, because I guess it increases the gap between A Level and university, but generally I think they go through everything anyway.
 

matt grime

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oh, a-level is incredibly poor these days ( i did group theory too at a-level), and i was merely indicating what would be considered a minimum amount of knwloedge, and i think if you asked a US high-school student that list i gave would be considered beyond their usual scope.

if you don't have that knowledge or ability to learn that stuff very quickly then you'll be lost as inside 2 weeks you'll hve gone from "C is the complex numbers" to "and the set of mobius transformations are the automorphisms of the extended complex plane".


i have taught maths students at bristol who cannot sum a geometric progression, or evaluate 1+2+...+n. However, any self respecting wannabe cambeidge mathematical student ought to think that a-level maths is easy. that is what STEP exists for. have a look at STEP III papers and see if they're something the average belgian high-school physics prospect would find easy.

in any case, it is the output of universities that was being considered here. would a 21 year old belgian doing mathematics have had a better education and possess a better degree than someone woh'd done a degree from Cambridge? what if it were extended to include the 4th year part3 course? at what age do you graduate from a belgian university? is it like germany, for instance?
 
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matt grime said:
oh, a-level is incredibly poor these days ( i did group theory too at a-level), and i was merely indicating what would be considered a minimum amount of knwloedge, and i think if you asked a US high-school student that list i gave would be considered beyond their usual scope. i have taught maths students at bristol who cannot sum a geometric progression, or evaluate 1+2+...+n. However, any self respecting wannabe cambeidge mathematical student ought to think that a-level maths is easy. that is what STEP exists for. have a look at STEP III papers and see if they're something the average belgian high-school physics prospect would find easy.
i am not gonna say every student will find it easy, but that is ofcourse the same in the UK. However these topics are quasi all seen in an 8 hour per week math-course. Ijn Belgium we have several math levels in the last two years of high school (4/6/8 hours of math per week). If you wanna study engineering or exact sciences at college, the 8 hour course is almost compulsory...

regards
marlon
 
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Marlon,

If you look at STEP then you will realise how hard it is. There 14 questions, 8 pure, 3 mechanics, and 3 statistical and you have to answer as manty as you can up to a maximum of 6. Most people might get 3 full solutions with errors in the three hours they have. This is a reflection of how hard the paper is.

-NS
 

matt grime

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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 highschools but of universities.

at what age do you graduate from uni in belgium?
 

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NewScientist said:
Marlon,

If you look at STEP then you will realise how hard it is. There 14 questions, 8 pure, 3 mechanics, and 3 statistical and you have to answer as manty as you can up to a maximum of 6. Most people might get 3 full solutions with errors in the three hours they have. This is a reflection of how hard the paper is.

-NS

very few people would get that score, almost no one in fact (assuming you are doing the relevant paper). 3 full solutions would mean you get a grade 1 out of 3 (1 being better than 3) and on step 3 this is obtained by the minority of people attempting the exam, and those attempting the exam constitute a very small fraction of the most able students in the country.
 

matt grime

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oh, and it isn't the topics that are hard in STEP, it's the questions that are hard.
 
But to be fair Matt, I'm sure you've also taught mathematicians at Bristol who do excel. There's people at all universities of all (relative) abilities. I know my course at Bristol (physics) had a wide range of abilities from just scraping through, find it very hard, hit their 'abstraction limit' quite early, through to those who breeze through it.

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. It would have to be this way to be certified by the Institute of Physics. However, I do know there are those who have struggled.

Anywho, I'm one of these mad ones staying on to do a PhD so I've clearly done alright...

I think the main point is that people shouldn't get hung up about what university out of the top ones to choose. Personally, I turned down Oxford over Bristol when all the offers were in - how you feel you'd integrate with a university is just as important as the reputation in the top centres.
 
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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
 

mathwonk

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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. thats 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.
 
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matt grime

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mathwonk said:
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. thats 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).
 
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did you guys have to complete a test to take 2nd year level courses in 1s year?
 

mathwonk

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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

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NewScientist said:
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 [Broken]

explains it
 
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matt grime

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James Jackson said:
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 highschool and unoversity, this part is about the poor state of highschools)

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.
 

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neurocomp2003 said:
did you guys have to complete a test to take 2nd year level courses in 1s year?
no, there w
 

mathwonk

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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.
 
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matt grime

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mathwonk said:
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 fucntion 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

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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)
 
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matt grime

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

mathwonk

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

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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!
 

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