TITLE: How does advancing through the math major work?
QUESTION [20 upvotes]: I am an undergrad math major that just completed Calculus 3 last semester. This semester I signed up for Discrete Mathematics, and will be taking Intro to Advanced/Abstract Math next.
Of course-- I expected the numbers and computation to be larger and much more complex but instead am finding that there is hardly any number-crunching at all. Just a lot of proofs and logic skills. What gives? I thought the more advanced math got, the more complex the number-crunching would get.
Is it always going to be like this from now on? For both pure and applied math majors?
And secondly-- after completing four years of this stuff, how in the world do you guys remember every rote memorization technique taught step-by-step during freshmen year for all of your past courses in trigonometry, college algebra, calculus 1-3 classes? There are usually like 4-5 steps per technique, with 3-4 techniques per section w/ 10 sections per chapter of a book!  
Take for instance my College Algebra book is 500 pages long-- I can't even remember every single section's memorization of how to answer the problem even after a year of doing Calculus, let alone the three years it will take me to graduate as a math major!
Maybe I just don't understand but it seems like all of the earlier useful rote memorization technique is going to get lost. IS this supposed to be the case?

REPLY [2 votes]: I thought the more advanced math got, the more complex the number-crunching would get.

Yes, it's so advanced that you crunch arbitrary numbers¹! (Like $n$ or $x$ or $z$.) Sometimes you also crunch $n$-tuples of arbitrary numbers (called vectors), or more advanced number-like things called cohomology classes, or ideals, or...
¹a/k/a variables
And there are so many number crunching techniques that you really have to have a firm understanding of basic logic. You get a glimpse of the real fun in your final year of your undergraduate, but it really starts during your Master's or PhD and just gets better from there.
Enjoying hands-on computation with real numbers is important for understanding how to crunch arbitrary numbers, so I'd say it's great that you enjoy number crunching.


Maybe I just don't understand but it seems like all of the earlier useful rote memorization technique is going to get lost. IS this supposed to be the case?

Oh, rote memorization is for kindergarten. You need to remember very little, because you'll be able to quickly derive the formulae you need. (To derive the sine/cosine equalities – like double angle formulae, etc. – all you need is Pythagoras' Theorem and the law for exponentials $(a^m)^n = a^{mn}$).