Best Places to Recieve a Degree (Maths) From?

[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 teh 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."

 

[gravenewworld]

AKG I go to Villanova University. Its is kind of well known school here on the east coast of the US. As far as the math requirements

-Calculus I-III
-Differential Equations w/ Linear algebra
-Foundations of Mathematics (which is like a intro course on proofs)
-Linear Algebra
-Advanced Calculus
-Modern Algebra
-Seminar in Math
-1 other upper level analysis class
-4 other upper level math elective classes

All in all I would say you need about 130 credits to graduate, so even with all the non-math courses I listed before and with these math required math courses you still have to take about 4 elective courses. The math major here leaves plenty of room for math majors here to get minors in comp sci., economics, physics, philosophy, or business which is typically what most of our math majors do.

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).

I asked the same question on the board before, and I got a bunch of flak from people. Why is that we have to study a bunch of things that aren't related to our major? Most US universities will say-"to give you a well rounded education." Believe it or not, most math majors that graduate from school don't pursue only mathematics for careers. We have had many of our math graduates work in all kinds of fields such as law, medicine, business, finance etc. Employers all say the same thing, they don't care what your degree is in, they want someone who can write and communicate well which is what the liberal arts studies here are supposed to help you improve on.

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".

Intro would be like taking basic calculus and advanced courses would be like taking real analysis only this would be for a liberal arts course. For example an intro course would be like world history while an advanced history course would be like Roman civilization ( a more specialized and indepth treatment of a specific subject.) The advanced courses usually have a lot more reading and writing required.

 

[mathwonk]

Zach, I always argue to my business students that claiming a tax exemption is like proving a theorem. the guidelines defining eligibility are the definitions, and what you write down is your proof that you deserve the exemption. So proofs are for everyone wanting to make his case successfully in any field.

the requirement for a math degree at harvard in those days was more or less:
" advanced calculus and any other 6 courses at or above (a certain comparable) level".

 

[matt grime]

if i ask you to explain what intermediate means to soemeone not familiar with the system then comparing it to another unfamiliar thing isn't gong to help! after all what you call basic calculus might be different from what i call basic calc (to me basic calc is stokes's green's theorem and harmonic analysis) and it further suppose some absolute standard - i studied the roman civilization at school when i was 10. does that make it compatible? this is supposed to highlight the spuriousness of the comparison.

 

[mathwonk]

i love this prerequisite blurb from a current course at harvard:

Mathematics Courses 2004/2005 213a. Functions of One Complex Variable
Catalog Number: 1621
Wilfried Schmid
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Review of basic complex analysis. Further topics will include series and product developments, uniformization, and special functions.
Prerequisite: Basic complex analysis or ability to learn quickly.

 

[mathwonk]

marlon, i do not know if you will readily believe this, but here are the core math requirements for a (non math) university student to graduate from my university. one could apparently not graduate from high school with only this in belgium.

"These are used to fulfill requirements in Areas A and D in the University's Core Curriculum .


MATH 1101 (Mathematical Modeling): can be used to satisfy a Core A requirement for majors outside science and business.

MATH 1111 (College Algebra, not taught here): except for students completing their entire Core Area A at another System Institution, College Algebra courses taken outside can not be used to satisfy Core Area A requiorements.

MATH 1060 (Mathematics of Decision Making): may be used to satisfy a Core D requirement.

MATH 1113 (Precalculus): may be used in Core Area A or Core Area D. It is required of all students in the College of Business and all Science majors in the College of Arts and Sciences. All sections of this course will carry 3 hours of credit. However, some of these sections will be designated as intensive . Intensive sections will meet 5 hours per week; they will include more review of algebra than regular sections.

There will be two occasions for referring students to intensive precalculus sections (subject to availability):

1. During orientation (see Placement Criteria ),
2. After the first computer test which will be given during the third week of MATH 1113.


MATH 2200 (Analytic Geometry and Calculus): this course, and the accompanying laboratory course, MATH 2200L, must be taken concurrently. They may be used in Core Area D. They are required of all students in the College of Business, and of science majors in the College of Arts and Sciences.
"

thats it.

 

[mathwonk]

having criticized the general level of my university, i want to say that we have some extremewly strong students here and some outstanding faculty. when those two get together, in the right courses, the result is an excellent experience for all concerned.

thus in regard to the general subject here of where to graduate from, the value of graduating from my university is as much dependent on the students ability as on the overall level of the university.

i once taught a high school student here a cousre of graduate algebra, decomposition of finitely generated modules over pid's, galois theory, commutative algebra, and so on.

he graduated from our modest university simultaneously with graduation from high school and entered berkeley graduate school the next year.

so really an educational experience can be what you make of it.

 

[marlon]

marlon, i do not know if you will readily believe this, but here are the core math requirements for a (non math) university student to graduate from my university.

ok i believe you

one could apparently not graduate from high school with only this in belgium.

Why Not ? What are you saying here ? It is not clear to me

marlon

 

[mathwonk]

i was saying that i gathered an average high school grad in belgium knew more math than we require of our university grads.

 

[marlon]

i was saying that i gathered an average high school grad in belgium knew more math than we require of our university grads.

yes that is very possible, especially those students that followed 8 hours of math per week, the last two years of high school. But that are not the average students ofcourse

marlon

 

[AKG]

AKG I go to Villanova University. Its is kind of well known school here on the east coast of the US. As far as the math requirements

-Calculus I-III
-Differential Equations w/ Linear algebra
-Foundations of Mathematics (which is like a intro course on proofs)
-Linear Algebra
-Advanced Calculus
-Modern Algebra
-Seminar in Math
-1 other upper level analysis class
-4 other upper level math elective classes

All in all I would say you need about 130 credits to graduate, so even with all the non-math courses I listed before and with these math required math courses you still have to take about 4 elective courses. The math major here leaves plenty of room for math majors here to get minors in comp sci., economics, physics, philosophy, or business which is typically what most of our math majors do.

My program is actually a mathematics specialist program (11.5 credits - our credit system is different as one full year course is 1 credit), which is more or less equivalent to a major (7 credits) and a minor (4 credits) in mathematics. Looking at your requirements, it is actually equivalent to 2 full years of extra requirements, whereas my extra requirements are only 2/5 of a year. Now that I think about it, though, I can see how those courses would fit in. A major and a minor would take up 11 credits, and doing all those requirements you have would take up 10 credits. A person can do 5 credits a year, so a person could do your program in 4 years and maybe take a full year course in summer school, or I guess most people count one of those requirements (like the computer science requirement) towards a minor (a comp. sci. minor). Are you allowed to count one of those requirements towards a minor?

I think the advantage at my school is that if you want to do a major and a minor, you can, just as you can at your school, but then you still have almost 2 years worth of courses which you are still free to play with, whereas it's kind of set in stone at your school. One has the freedom to double major at my school, I don't see how that would be possible in 4 years at Villanova. However, the courses you're "forced" to take look like an interesting variety anyways, but it seems that letting people choose if they want to take such a variety of courses or not would be advantageous. But if you're enjoying your education, that's all that matters.

 

[gravenewworld]

Here is a sample schedule a student would have to take to get a math degree w/ econ minor

http://www.math.villanova.edu/degrees/CurriculumSheetMathMajorEconMinor.doc

You really don't need to take summer school. The type of courses that we have to take are set in stone, but we are not limited to what course we can take in each subject. We can take the same classes say an english or a philosophy major would take. Yes, the classes that we have to take can be applied to minors, so a person who would want to get a philsophy minor would only have to take 4 more courses after taking the required 2.

 

[cronxeh]

Do they require you to take Numerical Analysis/Methods class as well?

 

[gravenewworld]

No, you can take that as an elective though.

These were all the math classes I took as an undergraduate.
-Calc III
-Diff eq. w/ linear algebra
-Foundations of math
-Linear algebra
-Advanced calculus
-game theory
-combinatorics
-complex analysis
-math seminar
-modern algebra
-topology
-independent study on topics in algebra
-independent study on math logic
-independent study on hilbert spaces
-I also took 3 graduate classes on linear algebra, geometry, and number theory

So it basically works out to only about 2 math classes every semester or 3 math classes if you decide to go "crazy".
-

 

[James Jackson]

In the UK, you go to university to read a subject and ONLY that subject. You might do a joint honours course, such as Physics and Philosophy, Physics with a Language, etc, but there is very little opportunity for study outside of that. I had one open unit in my first year, where I could've chosen just about anything from the university, but chose astrophysics as it kept my options open in terms of available second / third year courses.

 

[mathwonk]

by the way when you exit school, in most cases I believe the question you should anticipate is not "what did you take?" or even "what do you know?", but "what can you do?"

 

[gravenewworld]

by the way when you exit school, in most cases I believe the question you should anticipate is not "what did you take?" or even "what do you know?", but "what can you do?"

Or basically "what do i remember?" To be honest with you I don't have a photographic memory. Most of the stuff I learned 2 or 3 years ago I have forgotten. If you mentioned a topic to me from a course I have taken I would tell you I am familiar with it or that I have heard of it before, but if you asked me to solve a problem I might have to read the text that I used for about 20 or 30 min to jog my memory. To tell you the truth I have no idea how in the hell i managed to fit so much information in my head the semesters I took 21 credits. It is a lot easier for a professional mathematician or a math professor to remember theorems and mechanics of solving problems since they do it almost everyday and get paid for it. And no, I didn't learn things just to pass the test and forget about it. When I took the math courses I took, I actually knew the material quite well. I learned in psychology that if you don't use or practice things you learned previously, the information stored in your mind deteriorates severely or you completely forget all together unless you have a photographic memory. That is the whole reason why I have been studying and will continue to study all summer long for the math subject GRE.

 

[mathwonk]

i'm not sure you understand me. being able to solve problems is quite different from remembering how to solve them.

people who are successful often "know" very little information, but are extremely capable at dealing with whatever situations they find themselves in.

the point is to learn how to think, not to memorize ways of doing things.

once you begin to understand what they were trying to tell you in those books it should be necessary to consult them again endlessly.

i.e. you should have a few well chosen tools that enable you to regenerate the main concepts and techniques.

try it, you probably actually do. i.e. what do you actually remember from school?

i only remember one thing from harvard calculus: "a derivative is a linear map".

in fact that is almost all you need to know from calculus, if you think about it.

i.e. the linear map is a local approximation to the original map, so properties of that linear map should translate into local properties of the original map, etc etc.
e.g. if the linear map is invertible, then the original map is locally invertible (inverse function theorem,...)

if thje lienar map is a surjective projection, then the original map is smoothly equivalent (locally) to a projection, (implicit function theorem,...)


what do you remember?

them point is to get where you do not need the books any more. not because you remember every stupid fact in them but because you have finally understood the key point they were making.

this summer while you are studying, take some time to think about what the ideas are. it will all become much easier when you do.

(listen for the grasshopper)

 

[cronxeh]

tell me something, because I'm pondering

if someone goes for a doctorate in mathematics vs applied mathematics - what is the advantage?

i mean from a real world perspective and from a perspective of understanding the mind of geniuses like erdos and euler. how does one get to a level where you can think in terms of functions and not use paper or computers - to only rely on your memory and imagination to plot everyting in your head, to sit in a card game and know what hand everyone has at the final moment, to have probabilities piling up with every action around you. how does one achieve the state of mind where you can formulate or even solve the riemann hypothesis? how many courses and books does it take?

 

[mathwonk]

well since no one can solve the riemann hypothesis, presumably none of us can answer your question.

in my opinion euler was a genius, erdos was an eccentric.

i would suggest however based on my own experience, that if you want to maximize your chances of doing some excellent work, you should read the works of those people you wish to emulate, like euler, gauss, riemann, erdos if you like.

oh, and the advantage of an applied math degree is probably a good job and good salary.

 

[cronxeh]

I'm curious what sort of research you are doing, and why you are doing it. I want to know what is the hardest problem in Math today in your opinion, why its significant, what would it mean for you personally to have it solved, and what do you think is the future of Mathematics, e.g. where are we going today in terms of the latest findings

Oh and also I'm curious what is the significance of Finsler manifolds and Collatz conjecture (in plain english)