Ok this is a bit of an unanswerable question, but hopefully someone will answer. As I have been going through college & high school there has been a kind of "path" through which you learn math. For example I went through basic math, prealgebra, algebra, precalc, calculus 1-3. My question is where to go after calculus three?(which is where I am at now btw). It looks to me like math really starts branching after calculus. I worried that I'll end up in classes that I can't fully understand because I didn't have the prerequisite knowledge. Do I go to linear algebra, real analysis, number theory, another branch …?
I'm working towards a degree in mathematics and would like to get the most out of every class, So if anybody would tell me what worked/didn't work for them I would be grateful.
Best Answer
A good History of Math course would probably be enjoyable, and give you a good idea of what things are and where they lie.
From the mathematical point of view, Linear algebra is often a good "first advanced mathematics course", a good jumping-off point: it is still concrete enough that you won't get lost in a sea of abstraction (a possible issue with abstract algebra depending on how it is taught), cover entirely new ideas ("advanced calculus" can feel like you are just re-treading the same ground you already know, and depending on the specific topics analysis might also feel like it), but it should make you work through proofs and concepts in a way with which you probably have not done so far. In addition, linear methods will show up all over the place later on, so it would prove useful.
In that same vein, Number Theory can be a really good "first abstract course in mathematics", while sticking close to things you are very familiar with (the integers and rationals) while also probably delivering some exciting surprises. It often surprises a lot of people just how much of mathematics arises out of number theory (complex analysis and abstract algebra, to name just two).
If you want to stick to the applied side, differential equations is a good place to go as well. Linear algebra would be useful there, though.
So, I would suggest linear algebra or number theory first (if you can also get a good history of math course, do that as well), then decide if you want to go towards abstraction (in which case, head to abstract algebra, mathematical analysis, or whichever of linear algebra or number theory you did not take) or more towards applications (differential equations, a good advanced probability/statistics course, or a discrete mathematics course).
It is indeed the case that mathematics starts branching out, but some of the most interesting things happen where the branches meet; it would be ideal to be able to take a good one semester or one year sequence in the major areas (analysis, algebra, differential equations, topology, logic/set theory, number theory), then go on to more advanced courses in whichever area(s) you find interesting. But the truth is that this is very hard to do: not only would such a wide choice not be available except in the largest universities, but it would also mean a lot of your time. I did my undergraduate in Mexico, where all I did in college was mathematics courses, and it basically took six semesters before that had been covered (in addition to the calculus sequence, a linear algebra sequence, an abstract algebra sequence, a mathematical analysis sequence, differential equations, discrete mathematics, complex analysis, probability and statistics, plus some other stuff to "fill in the corners"; it would be barely possible to do it in two years if you are not taking the calculus sequence, but not if you are also taking other coursework as you would be in most institutions in the United States).