Best Places to Recieve a Degree (Maths) From?

[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?"

 

[mathwonk]

in regard to a point made above, the differences in difficulty between a top US school, and an average or below average one, are enormous.

In the US I suspect most people go to college, so there are colleges to accommodate everyone, at all different levels. At one extreme, people can even get PhD's on the internet from schools that are apparently little more than a website.

One such graduate was exposed last week here and fired from his job as a professor at a local college, where he apparently spent most of his classroom effort trying to be a "ladies man".

 

[mathwonk]

there are also US high schools on a higher level than some US colleges. Top high schools are called prep schools here, like Exeter and Andover, or Bronx high school of science, at least they were 40 years ago.

I do feel though that on the whole, the level of US college education is going down. I personally think it is a bad sign when Harvard has average grade inflation of 1 1/2 grades over the past 40 years, when according to a Harvard Alumni magazine article on the topic a few years ago, SAT scores (adjusted for SAT inflation!) were actually lower.

I do not see this sort of thing from Cambridge. Perhaps these pressures are inevitable marketplace issues in the US. Is it the case that a higher percentage of people attend college in the US than say in Belgium, or Britain? or does "everyone" go to college there too?

 

[gravenewworld]

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.

I looked at the cambridge website. Do math majors ever take a course that is not math? I couldn't even find a non-math course they were required to take.

These are the non-math courses that are required for the math degree at my university.

-1 year of foreign language at the intermediate level
-1 year of history- 1 semester @ the advanced level and 1 semester @ intro level
-1 year of philosophy- 1 semester at advanced 1 semester at intro
-1.5 years of social science- 1 semester at intro, 1 semester at advanced levels in same social science, 1 semester at intro level in another social science
-1 year of literature- 1 semester advanced, 1 semester intro
-1 year of science- 1 year of science and labs. Must be at science majors level.
-1 year theology- 1 semester advanced, 1 semester intro
-1 semester fine arts
-1 year of the core humanities (which is basically just classics)
-1 semester of college ethics
-1 semester computer science

Distribution of those courses must include: 4 writing intense and 4 writing enriched courses. You must also have to 2 different diversity courses 1 in women's studies, 1 in ethnic or minority experiences in the US, or 1 in courses which provide a focus on the culture, economics, politics or ecology of societies and nations other than those of Europe and the United States.


I am a "5th year sr." this year ( I will be graduating in Dec.) and over the course of my entire college career I unquestionably have written well over 750 pages (no exaggeration) in research reports, labs, essays, take home exams, etc.

If I didn't have to do all those other non-math requirements and all that writing, I'm sure I too would be able to get a more intensive study of math.


Highschools in America also reflect a similar system. When I went to high school the requirements were 4 years of science, 4 years of math, 4 years of history, 4 years of english, 3 years of language, 4 years religious studies (catholic school), other social sciences, and some electives.

 

[matt grime]

in the UK participation in higher education is around 40% when i last saw a statistic (abotu 4 years ago) i think with a plan to bring it up to 50%. a completely unworkable plan i may say and one that is complet BS when you look at the detail, and involves the particiapartion of around 60,000 more students per year., which would require the building of around 18 new universities. there is almost no funding to build these new universities and the onus is on existing universities to take more students. bristol has for instance taken on 50 moer undergraduate mathematicians than it did 3 years ago (approx 25% as required) and the department has insufficient space to teach them in and must spread its staff out over 3 (or more) buildings.

widening participation has essemtially meant that institutions offering vocational qualifactions now give degrees in them (leisure management, tourism, health care), it has not resulted in 25% more people reading chaucer and discussin post-enlightenment values.

 

[mathwonk]

oops, instruction at univ of ghent is in dutch. not too many american high schoolers need apply in that case. i forgot about our head in the sand language attitude here. British universities are almost the only ones where we could listen to lectures, and we are not that great at that language!

actually though my children attended dutch primary school in leiden for a week or so, once. if we had stayed, they could have become fluent enough in a few years i guess.

 

[matt grime]

I looked at the cambridge website. Do math majors ever take a course that is not math? I couldn't even find a non-math course they were required to take. .

they are there to study maths. it is a spurious exercise to compare but perhaps you should find out if the content of those "extra" classes you list is taught at high school in the UK? or perhaps you should justify why it is that we have to be forced to learn the classics (which are not humanties).

what does intermediate or advanced even mean in any of those contexts? for instance i am considered to posses a high school qualification that means i automatically pass the "ability to speak a foreign language" in many grad schools of mathematics in the US. i'd presume that is at least "intermediate".

 

[AKG]

These are the non-math courses that are required for the math degree at my university.

-1 year of foreign language at the intermediate level
-1 year of history- 1 semester @ the advanced level and 1 semester @ intro level
-1 year of philosophy- 1 semester at advanced 1 semester at intro
-1.5 years of social science- 1 semester at intro, 1 semester at advanced levels in same social science, 1 semester at intro level in another social science
-1 year of literature- 1 semester advanced, 1 semester intro
-1 year of science- 1 year of science and labs. Must be at science majors level.
-1 year theology- 1 semester advanced, 1 semester intro
-1 semester fine arts
-1 year of the core humanities (which is basically just classics)
-1 semester of college ethics
-1 semester computer science

Wow! Where do you study? At my university (University of Toronto) all students in the Arts and Science faculty must have 1 year of social sciences, 1 year of humanities, and 1 year of science, plus whatever your program requires. As a math student, my program obviously requires math courses (which count as science) so I really only needed to take 2 full year courses outside of mathematics to get the distribution requirement. That's 2/5ths of a full year. Your requirements seem to require at least a full year of non-math. Our math requirements don't take up the rest of the time though, so with those extra courses in the year we are free to take anything: more math, science, philosophy, arts, etc. I would assume that our programs have similar math requirements, but your program seems to place more restrictions on what you do with your elective courses.

 

[mathwonk]

oops. i was prepared to argue that US college instruction is worse because "everyone" goes to college here. unfortunately for that argument, the census shows less than 35% of men in US aged 18-24 were in college in 1998, and less than 40% of women.

AHA! that's only 58% of all undergraduates in US. I.e. 42% are over 25 years old.

 

[mathwonk]

AKG, these things change over time. when i was at harvard in 1960 they had just instituted a "general education" program designed to thwart any more cases like the undergraduate they described in the guidebook for gen ed who had supposedly taken 4 years of study exclusively in "sanskrit and indian studies", as if that were a bad thing.

The distribution requirements at my current school are so complicated, it is a challenge to figure out when a student has actually satisfied graduation requirements, and i personally always need help doing so from more experienced faculty, having been there only 25 years or so myself.

 

[mathwonk]

Matt, what do these statements mean from the bristol website?

"Over half of our undergraduates achieve First Class or Upper Second Class Honours degrees - an indication of the high quality learning experience provided by this Faculty. This is further supported by the most recent HEFCE TQA Assessment results where almost all departments achieved scores of 23 or 24 points; no subject scored less than 22 points,"


are these objective nationwide measures of quality for student degrees and faculty performance?

here we just muddle along by our own set of standards, different at each school, except for the unscientific comparisons made by US news and world report magazine.

 

[matt grime]

given jon carlson's bemusement at cricket scoring how does he deal with UGA's policies?

 

[Zach_C]

The difference in learning mathematics more in depth really is the type of students taking the class. I had to know many proofs of theorems on limits, continuity and differentiation. Learning calculus without knowing the proofs seems alien to me. However, I went to a magnet school that specialized in science and mathematics. In a typical US high school it is probably flooded with future business majors. For the most part, it would be useless for them to know proofs. My philosophy is that if you take a course learn everything you can, which would include the proofs. This philosophy is not needed and is not used in the US public education system.

As far as Stanford not being on par with European universities, a close friend of mine went to Oxford for a semester. He said it was the hardest semester of his life. He also said that it was easier than he thought. That semester just required more study time than he was used to. And he did not attend an Ivy League school. From what I have seen, with students in the US and away, when you study abroad you do not receive the same academic education. You are in a different location primarily for a cultural education.

The quality of the education you receive is measured by what is achieved by its graduates, not its students. Look at that and the faculty when choosing a school.

 

[matt grime]

Matt, what do these statements mean from the bristol website?

"Over half of our undergraduates achieve First Class or Upper Second Class Honours degrees - an indication of the high quality learning experience provided by this Faculty. This is further supported by the most recent HEFCE TQA Assessment results where almost all departments achieved scores of 23 or 24 points; no subject scored less than 22 points,"


are these objective nationwide measures of quality for student degrees and faculty performance?

here we just muddle along by our own set of standards, different at each school, except for the unscientific comparisons made by US news and world report magazine.

this is an example of administrative BS.

1. TQA (teaching quality assessment) is nationwide on a unified scale. it is out of 24. i know of no university maths department (certainly none of the top 10) to score less than 23 out of 24. CAmbridge was denied 24 points since it failed to "provide sufficient information about examinations", mind you that was the second panel who examine them when i was there, the first was refused access by the university on the grounds the assessment panel was insufficiently qualified in mathematics to assess the lecturers' skill.

2. the department is free to give as many firsts or 2.1 as it sees fit (i know some places in the US don't have the concept of 1st etc, but you can imagine first, 2.1, 2.2 and third as being gpas of 4,3,2,1 resp.) there is no nationwide level of attainment this indicates.

 

[mathwonk]

jon just resigned recently, if that may be taken as a statement.

i just noticed as well on the cambridge website the mission statement includes:

" * the opportunities for broadening the experience of students and staff through participation in sport, music, drama, the visual arts, and other cultural activities
"


which suggests even maths students taking solely maths courses are exposed to many other things outside coursework.

 

[mathwonk]

here is an example of an exam from a harvard honors math course when i was there in 1965, but not the top honors course:

"assuming the reals form an archimedean ordered field, prove the implicit function theorem for maps from R^n to R^m."

(that was the whole 3 hour final exam.) It was taught by Joseph Kitchen, who failed to receive tenure and left the following year. the independent student evaluations on him taken by the student newspaper summarized them by saying " a large minority of Professor Kitchen's students think that he is God."