Should I Become a Mathematician?

[mathwonk]

I recommend euclid, hartshorne, and archimedes.

 

[mathwonk]

heres apostol:

Calculus Vol. 1
Apostol
Bookseller: Larry Christian DBA metoyoubooks
(San Diego, CA, U.S.A.)
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Price: US$ 12.00
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Book Description: Blaisdell, 1962. Hard Back. Book Condition: Good. Dust Jacket Condition: No Dust Jacket. First Edition, 3rd Printing. 7x10. used but clean. and tight. Bookseller Inventory # 0017819

 

[bjgawp]

Thanks for your comments mathwonk. I've decided on either purchasing Apsotol's or Courant's textbooks as I want to see how the introduction of integral calculus FIRST plays out rather than what I've been taught in high school and even now in that differentiation is introduced and integration is simply the reversal. Do you have a recommendation on which of the two I should purchase first (I've decided that money won't be an issue as I really want to pursue this out of interest). I looked through the table of contents and it seems that the content are similar except for the linear algebra in apostol's and the last two chapters of courant's which seem interesting. I'm leaning towards apostol's seeing how I'm taking vector geometry & linear algebra this summer as an independent study course. Thanks again!

 

[Gib Z]

Though the last 2 chapters of Courant are interesting (Intro to multivariable functions and intro to differential equations) I would recommend Linear Algebra, Since both last chapters of Courant are covered in most multi-variable calculus books, and linear algebra is somewhat a prerequisite for multi-variable calculus.

 

[Gib Z]

I am shocked to say that the AP Calc exam you posted is easier to me than the geometry one...Just shows how stuffed up my learning has been :( I'll attempt the Calculus one first, but could i ask a favor of you mathwonk? I don't often have 3 hour blocks free, so can I do the exam in little blocks and perhaps you could give me a mark on it? Don't worry if you can't though, I'm sure you're very busy.

 

[mathwonk]

well probably i would go with apostol, especially since i noticed he does integral calculus thoroughly before differential, while courant does them roughly at the same time. but i own and recommend both books.

integration is the idea of defining areas and volumes and arclength and work, as limits of approximations. this is due to the ancient greeks.

calculus uses the fact that in the special case of continuously varying shapes, the derivative of the volume function is the area of the leading face, and uses this fact to compute the volume, or that the derivative of the area function is the length of the leading edge,...

the greeks already knew that the volume was determined by the area of these slices, but not precisely how to recover it from them, that's all. so its the introduction of differentiation and antidifferentiation as algebraic, or analytic processes, to augment and enhance the geometry of integration, that was added in modern times,

apostol shows as i recall, that after defining the integral as a limit of riemann sums, one can proceed to compute quite a lot of these sums and hence quite a lot of integrals, directly. usually today we just go right to the FTC using antidifferentiation, so quickly that we soon forget that riemann sums were ever involved.

as a result, students do not realize that some integrals are more easily computable by riemann sums than by antidifferentiation. that's why i put integrals like the one on the first page of my test. most of todays students simply say they do not know how to do it. they also claim that functions which are not continuous are not integrable.

It is entirely possible that a good high school student, like the ones who post here, can dispatch my calculus test with ease. In that case however, my point is that they still should NOT usually skip calc 1 or 2 in college and go on to non honors calc 2 or 3. Rather they should usually take a high level honors calc class like a spivak class, often from the beginning, to see the material done right, and to be in the company of the best students as well as the best teachers.

these AP courses are harming all the students, the good ones and the weak ones but in different ways. The weak students, and even the pretty good ones, are skipping my calc 1 and getting in over their heads in my calc 2. The strong ones are skipping my calc 1 or 2, getting into my calc 2 or 3, where they are able to do ok, but they are missing the more suitable spivak course that is designed for them, and they are missing having a top honors professor teach them.

well actually the AP course is functioning properly for students who use it to get into a good spivak course, but so few do that. we also have a mid range honors course, and students who take that lose AP credit. Since the students only have three choices, either the rather challenging spivak course, or the honors calc 1 but give up AP credit, or the non honors calc 2 or 3,, they usually choose unwisely the last option. of course that is the option they have been told AP courses are for, namely "ADVANCED PLACEMENT".

That name is almost completely a misnomer, as although high school AP courses do resemble many college courses today, precisely because those have watered down to accommodate AP students, they do not compare in difficulty, especially for students who only got a 3 or a 4, to courses like mine. and my course is not hard, but it is not entirely computation.

mathematics is not just about computation, but also and primarily about reasoning. in my experience there is little or no reasoning taught in an AP course, nor tested on an AP test. Many of my AP students did quite well on the part of my test where all they had to do was compute some antiderivatives. but when they were asked to compute an integral by a geometric series they did not even know what to do.

many of them did not even realize it was an infinite series, and just gave the sum of the first 4 terms.

they also did extremely well on the vector algebra part where all they had to do was arithmetic. but earlier when i asked them to figure out what arithmetic to do, they did poorly. so i too am dumbing down my calculus class to accommodate weak student preparation. for example very few would have succeeded in using vector algebra to show the median of an arbitrary triangle meet 2/3 of the way up each median.

so i just used the trivial question i asked as a means of reminding them of that geometric fact, and to verify they knew the basic vector operations. as soon as i asked something harder, like an angle, or a projection, most missed it.

 

[mathwonk]

Gibz one reason the geometry exam looks harder than the calc exam is that archimedes could use riemann sums and cavalieri's principle to do harder volume and area problems than many of us can do using the FTC, even though the FTC supposedly makes them "easier".

if you think about it, the greeks had the idea of computing areas and volumes by taking limits of approximations from within and without, i.e. upper and lower riemann sums.

then the cavalieri principle follows from this, i.e. solids with the same cross sections have the same approximating riemann sums of cylinders or shells. hence they knew that solids with the same cross sectional areas have the same volumes.

then they began with the observation that a cube can be decomposed into three congruent right pyramids, to see that a right pyramid whose height equals the edge of its base has volume 1/3 that of the circumcscibing cube.

then cavalieri let's them see that changing the angle of the sides does not change the volume, and the approximation concept also let's them see that scaling the height changes the volume by the scale factor.

so they knew the volume of all pyramids. then approximating a sphere with them the way we approximate a circle by triangles, they get that the volume of a sphere is 1/3 the product of its surface area by its radius.

then archimedes crowning achievement was to notice that the cross sectional area of a sphere, plus that of a (double) cone, equals the cross sectional area of a cylinder. hence he obtained the volume of a sphere by subtracting two known quantities. (see the very clear pictures on Ted Shifrin's website at UGA math dept, from his AMA talk.)

all this without the FTC.

so this is why integration, as archimedes did it, should precede learning the FTC and differentiation. of course in a geometric sense his relation between surface area of a sphere and volume of that sphere IS differentiation, wrt radius.

his final result whose proof was erased and still lost, was to relate the volume of a bicylinder to that of its circumscribing cube. what object do you think should be used here to replace his previous use of a cone? it helps to know the cross sections of a bicyinder are squares.

(yes! you are right,.. its a ...oops my answer got erased for a greek orthodox prayer book. you do see it though don't you?)

 

[Tickitata]

I have a pdf version of Apostol's Calculus book (both volumes) that I am working through

Obviously it's not the same as owning a physical copy, but I am finding it very useful. Would it be against the rules of the forum to post a link to it?

/edit: I also have Spivak's Calculus on Manifolds

 

[Gib Z]

Actually, The riemann sums of the volumes etc was the only parts of the exam I could do =] I learned off Courant, where he didn't introduce the FTC until after he used Riemann sums to find the integrals of x^n where n is any rational number (he later proves after introducing FTC for an irrational n, but it doesn't use FTC either), and sin x/cos x.

 

[bjgawp]

Alright thanks for the suggestion guys. I'll get myself a copy of Apostol.

Wow didn't think the AP Calculus course was so flawed. What brought about this change in the first place if integral calculus was initially meant to be taught?

 

[Gib Z]

Because if we teach integral calculus first we must learn how to evaluate integrals with riemann sums, and then later move onto differentiation and connect the two. This flawed way goes the other way around but skips out on Riemann sums pretty much. It's *easier*.

 

[Tickitata]

you skipped Riemann sums in your AP program?

We sure didn't...

we also didn't skip epsilon-delta. Although we took the standard modern route (differentiation first, then integration), I strongly believe my AP Calc BC course was fantastic.

 

[Gib Z]

Well actually, we skip Riemann sums and epsilon-delta proofs in the Australian Equivalent of the AP program, neither of which I have taken. I thought they would have been similar, never mind.

 

[Tickitata]

Well, Riemann sums are often on the AP test, so I'm sure that most programs include it. However, epsilon-delta isn't on the test, so I'm not sure how many US programs teach it

Although many AP Calculus classes are taught by complete morons, some programs really are great. Mine was actually taught by a Ph.D who was a fantastic teacher and clearly loved the subject

I dunno. I think the AP program is mostly aimed towards natural science and engineering majors, as it does a fine job of computational calculus. It really is up to the universities to require math majors to retake honors sections of calc I and II, if they are offered

 

[mathwonk]

to see my thoughts on ap calc, i recommend reading my detailed comments early in this thread. of course there are many courses that are well taught, and many of the books are excellent, and sadly the ap course in many schools is a better option than the depressing alternatives offered.

the problems are mainly with the test oriented approach, the fact the test itself does not cover theory and proof, the omission of good solid algebra and geometry (with proofs) to make room for calc in high school, and worst of all, the fact that most ap students go into mediocre non honors calc in college instead of honors calc, because that's how they interpret "advanced placement".

i.e. ap is actually functioning to place them DOWN from beginning honors courses to (later) non honors ones. the previous poster is right that universities could remedy the last problem by denying credit for these weak ap courses, but since students do not understand this, we would lose the best students to schools who continue to offer it. at least that's what my department tells me.

so we are undercut in the market in the same way anyone offering a quality product is undercut by those shoddy products made in an inferior way and apparently costing less.

 

[mathwonk]

so do not misunderstand me, it may be for many of you that ap calc is indeed the best choice in high school. but try to learn the reasons for the calculations, in case your class is purely taught to the computational test syllabus. and try to learn some good fundamental plane and solid euclidean geometry, with proofs. and when you get to college, look for the most challenging honors calc course appropriate for your goals and ability. unless you are perfectly happy in a classroom with non honors students covering a non honors curriculum, do not just skip up into regular calc 2 or 3. and if you only have a 4 or 3 on the ap exam, you should probably begin at the beginning.

but all of this also has to be adjusted to the situation you are in at your particular school. there are definitely schools and professors who have just given into the ap mess, and have dumbed down their classes to that level. in that case even an "honors" course may resemble your high school course.

but if you go to one of the schools that still offer a beginning spivak class, like chicago, or uga, then i recommend you take it, especially if you want to be a mathematician, (remember the title of the thread?)

 

[temaire]

I was just wondering, how much does a mathematician make roughly in a year in Canada and the US?

 

[mathwonk]

the university salaries are given on the AMS salary survey. as i recall, the only people who make over 100K in USA are those at top schools like harvard, or elite profs at middle level schools who get big offers to keep them from jumping ship.

http://www.ams.org/employment/2006Survey-FacSal.pdf

it seems the highest salaries are those at the top 15 or so private schools where full profs had median salary over 120K, in 2006.at UGA, a good state school in the south, extremely few full math profs make 100K. UGA is in "group II" where median full prof salary was 92K in 2006.

I believe anyone capable of getting a PhD in math could earn far more in another field besides academic professorship. The guy whose job I got when I was hired, went into industry and returned a few years later making triple my salary. Another young assistant prof who started with me, jumped to industry at a starting salary almost double mine as i recall, or maybe much more.

The attractions of this life are not in the salary. This is only a problem when you try to pay bills, like college tuition, or home mortgages. This is primarily a problem to people at schools in big cities like LA where the salaries barely enable them to live. A few schools like Columbia relieve the tuition pressure by offering free tuition at Columbia and half subsidies at other schools.

At UGA there is to my knowledge no tuition help of any kind, not even for enrollment at UGA. On the other hand the state of GA gives "Hope" scholarships to strong students. They give them even to weak freshman students, if they have a B average in high school, but most lose them after freshman year, since a B at UGA is a lot harder to get than at a GA high school.

 

[morphism]

I was just wondering, how much does a mathematician make roughly in a year in Canada and the US?

At my university in Ontario, a tenured professor makes 100-150 thousand a year.

 

[temaire]

I believe anyone capable of getting a PhD in math could earn far more in another field besides academic professorship. The guy whose job I got when I was hired, went into industry and returned a few years later making triple my salary.

Can you provide some more details on this guy please?

 

[mathwonk]

I went from a meat lugger earning 3.75 an hour in the Boston South end in 1968, or 7,500 a year, to an assistant prof with a PhD making 14,000 in 1977-78. With (real, not govt figures) inflation this was almost the same salary.

Salaries are higher for statistics and computer science than math. Also jobs as a prof at a school like UGA are extremely competitive. We already have hundreds of applications for a couple of openings, that we are beginning to review now.

I don't know the prospects or the attractiveness, but what the USA really needs is a new generation of decent high school math teachers, if anyone is interested and up for it. The high school students we get are really poorly prepared.

The difficulties there are manifold compared to college of course. My son is a Dean at a NY high school, dealing with cases of discipline including students coming to class late because they just got out of prison. Teaching geometry out of Archimedes is not an option at his school.

I have often thought that older college profs who are slowing down in research should be hired as high school math teachers, but maybe an older person just could not deal with the behavior of US high schoolers. Still it might be an option for some 50 year old math profs, who cared to try it.

 

[mathwonk]

i don't have more details on that guy who made triple my salary in a few years, since i will not give his name, but my impression was he was an average applied math guy in industry. Maybe he was unusual.

another ex student of ours, with a PhD in algebraic topology, went to work at an automobile plant, testing the design of new cars using CAD computer programs, but with no experience at all in that area, and apparently does quite well.

He said that the experience of getting a math Phd alone, even in a pure field, put him way ahead of his peers in ability to learn new things. Other students I knew in stat and computer science easily obtained academic jobs making as much or more than our experienced pure math profs, a decade or so ago.

Adequacy of salaries is partly in the mind. As a young prof with 2 children, I recall chatting with an older man who was subsidizing the private school tuition for his grandchildren since his son was making "only 70K".

When he said rhetorically "How can they live?", I almost laughed since I was earning 14K at the time.

 

[mathwonk]

becoming a mathematician, means becoming someone who practices mathematics, whether in the applied world, the academic research community, or the teaching community, at college or high school or junior high school, and maybe other areas.

There are hundreds and hundreds of colleges and junior colleges in the US, and more high schools. Even David Hilbert took a high school teaching certificate as an employment hedge.

Becoming a mathematician does not mean just becoming a fields medalist, or even a famous professor. But even those of us at the middle or lower levels of the research ladder occasionally make contributions that support the work of top people.

It is a community of people who think about and value mathematics, not a hierarchy of back stabbing scramblers for a brass ring. We all find our level, a community of peers we can enjoy talking and discussing and working with, while we take instruction from the work of those who see further.

Just because we are mostly not going to be like Terence Tao does not mean we cannot try to follow his advice, and try to imitate his learning and research behavior. When I was young student studying sleight of hand and "magic" my favorite book author wrote "If you can do one trick well, you are a magician". If you understand one theorem or principle well, you are a student of mathematics. If you solve one problem on your own, even with help, or prove one theorem, even if it has been done before, you are a mathematician.

a professional magician or mathematician is just someone who earns a living practicing his subject.

there are people posting here who are amateur mathematicians in the spirit of fermat the jurist, who earn their living otherwise but love mathematics and work at solving problems and learning more. if i stop learning and doing math, then even if i am called a professor, those people are more active as mathematicians than I am.

I am not so great at advice on how to earn a living as a mathematician, as I have always cared primarily about doing math and learning math. Much of my most successful teaching has been for free, as here, or as a volunteer at a local high school junior high or grade school, or as a parent.

Surviving as an academic is mostly about publishing. So if you aspire to be a successful university mathematics professor in that sense, i.e. to survive, get promoted, and earn a decent salary or more, be sure to publish absolutely everything you do, and in a timely fashion. And if you want to make more money, find an area where more money is available, like statistics, computer science, or applied math, including biology, or security of transmissions, where even pure number theorists can occasionally strike it rich.

If you have more of a social workers mentality, go into high school teaching and try to do something about the pitiful state of USA math instruction, but be warned it is a long hard road, with little likely success out there. People like Jaime Escalante, the hero of "Stand and Deliver", are very few and far between, and notice he gave himself a heart attack doing what he believed in.

 

[mathwonk]

Ultimately, age old wisdom suggests one should try to chose his profession for its suitability to his inner self. It is more highly recommended to work at a profession in which one is happy doing the daily work, than thinking about what recognitions are possible.

I think this is what it means in the bible where it says do your work for the glory of God, not for its reward. A reward, and recognition, is something someone else determines, and mostly you have no control over it. The satisfaction of struggle and accomplishment is your own forever.

At the same time one has to live in the world, and pay bills. This mixture of reality and idealism in life is a challenge everyone must learn to balance for himself. In my experience however, if you focus on doing good work, publish it, and are willing to learn how to teach, there will be places that will be glad to have you and pay you at least a livable wage.

 

[temaire]

Surviving as an academic is mostly about publishing. So if you aspire to be a successful university mathematics professor in that sense, i.e. to survive, get promoted, and earn a decent salary or more, be sure to publish absolutely everything you do, and in a timely fashion.

I'am not very familiar with university terminology, but when you say publishing, what exactly are you publishing?

 

[mathwonk]

research articles, i.e. the results of your research, the problems you have solved, the theories you have created to solve new problems, sometimes also the new problems you have found and believe to be important.

I am not a big publisher, but in 30 years i have maybe , not counting expository articles, over 500 pages of published research.

 

[uman]

That's what professors do: They advance the state of knowledge in their field.

A chemistry professor is a chemist. He discovers new facts about chemistry, writes them up, and publishes them in academic journals read by other chemists.

A math professor is a mathematician. He comes up with new math and publishes. Etc.

 

[temaire]

Isn't it very difficult to come up with new math though? I mean mathematics is very advanced these days. (Mathwonk, this is why I get the notion that mathematicians are "special.")

 

[uman]

Well I've never been a mathematician, so I wouldn't know.

But from what I've read on this forum and heard from mathematicians, if you are reasonably intelligent and good at math and you are willing to work very hard, you can become a mathematician.

According to some famous psychologist whose name I don't remember, it takes ten years of very hard work to achieve expertise in any field, including math. Which is probably why you think what mathematicians do is amazing (as do I): you haven't been through those ten years and you don't think like they do.

Also math being more and more advanced just means that mathematicians today are more and more specialized.

 

[temaire]

Do you have any of your work online mathwonk? I would be interested to see what you have published, if you don't mind.