Choosing math courses is one of the most important aspects of being a math major, though these choices are often quite difficult. Upon entering Princeton, many math majors do not know which areas of math to explore. Even for those who have decided this question, it is often not apparent which courses to take and in what order. Moreover, there are always questions of which courses it is feasible to take simultaneously, which courses overlap in material covered, what knowledge does one need before taking a course, and many others.
The goal of this course guide is to provide information to help math majors make these decisions. Before this course guide was compiled, the available sources of information were the registrar's Course Offerings and the math department's undergraduate courses page. These two are important information sources, and every math major should consult them. This course guide supplements these sources by bringing in the student's perspective. All of the information presented here is taken from the experiences math majors have had taking these courses.
Princeton's Math Department is often lauded as one of the very best in the world. Now you can see how they approach the study of the subject there.
(Score: 3, Insightful) by zugedneb on Sunday December 04 2016, @01:13AM
...are the people who had parents and older siblings in higher education.
cant beat that wwith hard work.
then there were those who were part of demo teams for the amiga and early pc.
they were amongs the brightest people i have ever met.
you need a good tutor, or have to find the right context that makes you ask the questions math is the answer to.
then there is hard work for the sake of hard work, but that is the category "people with good grades that one cant talk to".
all else is bullshit.
old saying: "a troll is a window into the soul of humanity" + also: https://en.wikipedia.org/wiki/Operation_Ajax
(Score: 2) by The Mighty Buzzard on Sunday December 04 2016, @02:24AM
Or you can just have that sort of a brain. Like martyb's brain just spits out absurd ways to treat my wonderfully crafted code that no sane person would even think of. It bothers me a little when he has a low success rate. Means he's been reawakening my inner blackhat or the bugs he's looking for would be there.
My rights don't end where your fear begins.
(Score: 0) by Anonymous Coward on Sunday December 04 2016, @01:39AM
Wen devving is applied applied math++ plus u can mak BEEEEEELLLLIONS doing it.
(Score: 2) by Snotnose on Sunday December 04 2016, @01:54AM
Math major here. Graph theory is one of the few things I learned that turned out to be useful. Wasn't even that hard a class.
When the dust settled America realized it was saved by a porn star.
(Score: 2) by fubari on Sunday December 04 2016, @04:30AM
Seeking suggestions for online math classes like described in the fine article: linear algebra, multivariable calc, differential equations. I haven't found anything online yet that looks good; for example, I'm surprised Coursera [coursera.org] is sparse on more serious math.
I'm a long time programmer and am looking to step up my analytics game. Eventually I want to get to work in machine learning or computational linguistics; I know I need to make some math progress before I get there.
I am close to finishing working through all the math on Khan Academy [khanacademy.org] and am astonished at how much I've forgotten. :-) Some comes back quickly, some slower, some I realize I never really learned the first time around at all.
Anyway, getting ready for next my steps and would welcome any suggestions.
(Score: 2) by Snotnose on Sunday December 04 2016, @05:17AM
Linear algebra is dirt simple. Can you add 3 lines of 3 numbers in your head? Can you add 5 lines of 5 numbers in your head? If yes, Linera algebra is a sinch.
Diffy Q's were a bitch, but when I got to the math classes after them they were pretty easy.
I majored in math because by the 3rd semester we were rotating things in the 3rd dimension, and I could visualize it. 3rd semester calculus was easily the easiest math class I ever took.
/ your mileage may vary,
// but really, linear algebra is more how anal you are than how good you are at math
/// wanna add 27 numbers? Linear algebra is for you
//// Graduated in '90, things may have changed since then.
When the dust settled America realized it was saved by a porn star.
(Score: 2) by fubari on Sunday December 04 2016, @06:31AM
thanks :-) I'm at the awkward stage of not knowing what I don't know.
(Score: 1) by Ethanol-fueled on Sunday December 04 2016, @10:32AM
If you are mentioning Khan Academy and Coursera in the same sentence as advanced math, then you are in for a world of hurt. Both of those services are shit.
Let me give you a tip - Advanced math is all algebra. Symbolic manipulation, isolating variables. If you are comfortable with intermediate algebra, then you can do advanced math. Don't do what I did and spend your intermediate algebra class pinching your girlfriend's braless nipples and playing Led Zeppelin on your guitar unless you want to do what I did and work your way up to discrete math from high-school level plane geometry in college.
(Score: 2) by fubari on Sunday December 04 2016, @08:35PM
Oh yes, I expect it to hurt so good.
Good to know on algebra, fwiw I do feel on solid ground there.
r.e. advanced: hmmm, maybe for me change my description: s/advanced/next step/ for single-var calculus.
r.e. Khan: I've been pretty impressed w/Khan Academy. They do an impressive job with drills and feedback, for computerized instruction I don't know how they could do better. I often find myself wishing Khan Academy was around when I was first learning about math.
r.e. coursera: I don't have enough data points to say good / bad, just seems kind of sparse, like there isn't much demand for online "math trajectories."
At any rate, I will keep your advice in mind - thank you. :-)
(Score: 2, Informative) by khallow on Sunday December 04 2016, @05:13PM
Linear algebra is dirt simple. Can you add 3 lines of 3 numbers in your head? Can you add 5 lines of 5 numbers in your head? If yes, Linera algebra is a sinch.
Linear algebra goes way beyond that. For example, the Legendre transform [wikipedia.org] which is a rather simple bit of linear algebra applied to real analysis leads to powerful characterizations of integration (such as Holder's inequality [wikipedia.org]), and an explanation of how to transform a classic dynamical system from position, velocity, acceleration, to position, momentum, force (in other words, transforming a Lagrangian system [wikipedia.org] into a Hamiltonian system [wikipedia.org]).
Linear algebra is also instrumental to calculation of various topological invariants (such as a complete categorization of knots)
(Score: 3, Interesting) by Snotnose on Sunday December 04 2016, @05:33PM
Wasn't implying linear algebra was useless. Far from it, it's very useful. My point was that it's pretty simple to learn, usually involving nothing more than adding and/or multiplying numbers. Often lots and lots of numbers, but it's still just addition and multiplication.
When the dust settled America realized it was saved by a porn star.
(Score: 0) by Anonymous Coward on Monday December 05 2016, @12:45AM
Yes, but mental arithmetic has nothing to do with it. Linear algebra is the language of geometry. It is also the study of vector spaces in general (abstract and concrete). In first year linear algebra, learning the mechanics of vector and matrix operations should be incidental to learning the general theory of Euclidean spaces and the maps between them (represented by matrices of reals), and learning how to apply the theory to solve problems, both in mathematics and other fields.
(Score: 1, Informative) by Anonymous Coward on Monday December 05 2016, @04:55AM
I would avoid online courses for math. Google for recommendations from mathies on specific subjects (e.g. linear algebra), then cross-reference the book reviews on amazon, and go back an edition or two to buy used at a good price (usually well under $20). Read reviews carefully for what level of mathematical maturity is expected. As with finding partners for tennis or golf, the idea is finding a match for your level - neither above nor below.
One thing I've learned is that it often takes 2 or 3 editions for a professor to get the textbook right, but after that, they're basically following fads and fashion in the textbook biz. Oh, you want more big color photos to help connect students with exciting work done by professionals? And how about support for Matlab, TI-89, and graphics packages. etc
(Score: 0) by Anonymous Coward on Tuesday December 06 2016, @09:40AM
Don't do it online.
If you understand code, I recommend picking up a copy of an early-undergrad course book on analysis + proof (eg. the one by that name). Literally do it cover to cover, including exercises, especially - if you can bear it - the ones marked challenging. You should expect this to take about one half to one work-month, 80-160 hours; just do evens or odds if there's solutions only for those but don't skip to solutions without spending meaningful time trying first. The book isn't thick and the learning is fast.
This will give you The View . The one where you can mentally 'hold' the pieces of a problem, and see what the missing bits are. The one where you by default consider 0 and infinity, not unlike good coders. Ask your mathematician friends, watch their eyes glaze over as they try to describe it.
Once you have this, you can choose to investigate branches of theoretical math as dalliances, spending more time in what you like. Discrete stuff and number theory? Real and complex analysis? Topology and graph theory? Etc.? All of it will be legible to you, and you'll be able to read and understand some 50-90% of the material about 2-3x as fast as without the initial setup (of the rest, it'll be partly grokkable as fast without, and maybe partly ungrokkable period).
But the learning curve for math is weird. One really must get the foundation for each topic, and most of the foundation for starter-advanced maths is within analysis+proof.
(Score: 3, Interesting) by bzipitidoo on Sunday December 04 2016, @05:06AM
I got a lot of math in my education. Calculus in high school. But I didn't feel that I really understood calculus and opted to take the college level calculus classes, even though I had earned a B in Calculus I with my AP exam score. I thought they might be better taught at the college level. Nope! Was a big mistake. I ended up with a C in Calculus I thanks to the graduate student they had teaching that class being an inexperienced, nitpicking ass. Lost a whole letter grade on a test for forgetting to put dx/dy on all my correct answers. He just mechanically took 2 points off each time I made that same mistake.
Then the school got me with another hole in the rules. My SAT scores were good enough that I didn't have to take any freshman math before Calculus. But, weirdly the school did NOT give us credit. By the time everyone in my major was done with the required math, we'd all had the same amount of math, but most of my classmates had 15 hours of credit, and I had only 12. An advisor told me to take trig, but I rejected his lame advice, and a good thing too, as I would not have gotten credit had I followed it. I was forced to take a more advanced math class, and picked differential equations.
Anyway, my experience was that math teaching is focused too much on trivia, too much rote learning through drills. Definitely not enough understanding, so that students learn where and how to apply it. For instance, what distinguishes an analytic proof from a geometric one? What's the distinction between an irrational number and a transcendental number, and why do we care? Why is "Squaring the Circle" so hard? Transforms such as Laplace and Fourier are pretty useful and cool. And FFT, don't forget FFT. But math classes tend to dwell on the tedious details of how to do them, rather than when to use them and what to do with them. That's particularly annoying now that we have smartphones that can handle the details.
(Score: 2) by fubari on Sunday December 04 2016, @08:41PM
agree r.e. understanding / depth of concept coverage.
And yeah I've had TA's like you describe.
I'm re-studying all the math I've forgotten. My first time around (pre-internet, yeah I'm that old :-) ) the drill was to hurry to keep up and cram for tests (being un-interesting was just icing on the cake-of-apathy... couldn't compensate for that).
As for depth, I've been pleased working through the stuff on Khan Academy; I find myself wishing it had been around when I was starting out.
If the kids over at Khan Academy keep working I wouldn't be surprised if they keep going deeper and further... but for now they haven't gotten into linear algebra or the same.
(Score: 0) by Anonymous Coward on Sunday December 04 2016, @05:37AM
This guide might have been useful for me, but I did pretty well stumbling around on my own. Almost everything I studied has been useful at one time or another, professionally but also because I like to dabble in a bunch of fields. The top five most useful subjects have been: single/multi variable calculus, linear algebra, probability theory, differential equations, and a toss up between real analysis and abstract algebra. Even group theory turned out to be useful. It actually helped me get hired by Apple (efficient solution to take home problem), and became useful when I studied cryptography later on. If I had to do it all over again, I probably would have taken a more applied approach. I ended up taking a lot of second courses in analysis and algebra that have never come in handy. It would have been nice to have a background in Fourier Analysis and the Calculus of Variations right after I was finished, but I found the gaps were easy to fill in due to my training. I think getting a math degree is a great choice for self-learners; it provides a lifetime of entertainment and opportunity.
(Score: 0) by Anonymous Coward on Sunday December 04 2016, @07:00AM
You're not a normal person; your experience of life is not applicable to most others.
(Score: 4, Insightful) by khallow on Sunday December 04 2016, @07:45AM
(Score: 0) by Anonymous Coward on Sunday December 04 2016, @12:10PM
the best math classes are the ones in walking distance of a good cook and a soft bed ...