Get a real pen and paper, get a real physical book, sit and solve problems with pen and paper for hours every day for a few months. Then you will pass the exams.
Exactly. Every time I tutor someone in math, I tell them to use up at least a sheet of paper for every interesting question. When they do, their skills improve quickly. Saving paper is a false economy when it comes to math.
While I agree with you, and love aj7's post, I'm going to push back slightly on the pen and paper.
I used to do all my work (solutions to problems, notes) using pen and (plain! not lined) paper. However I realized a couple of years ago that becoming fluent in LaTeX was a better option for me. The reason is that, with the proof neatly typeset, and the ability to re-work and edit repeatedly without making a mess, I found that I think more precisely and systematically. I still do scratch work on paper, but writing up a clean copy as I go is very beneficial.
In addition to those reasons, the other hugely important one is that my notes are now in git, I can grep them, and they don't add to the pile of objects that must be dealt with when moving to a new home.
For best results you need to make a nice LaTeX set up. I use the Skim PDF reader so that it autorefreshes on file save, and set up a Makefile and make it so the PDF is recompiled on every file save. But whatever works for you, I'm sure there are easier setups.
There's a lot to be said for using computer tools. If you're writing proofs, why not do it formally? [0] If you're working with graphical concepts, why not code them up, or use a drawing program (or hey, a graphing calculator) rather than pulling out a ruler and such (and maybe learning to draw at all if you don't know how)? If you have sloppy handwriting (as I'm sure many of us here do), why not type in something you'll always be able to read later? (Along with whomever you show it to -- I did a lot of college homework using LaTeX. With macros I could do things way more efficiently, with comments I could go back and see what I was thinking at a misstep (if I wrote anything).)
The downside of course is that computers are very capable distraction vehicles, you need a bit of discipline to sit at one and study / do this sort of work at the same time for prolonged periods. Pulling out the ethernet cable can help but may not be sufficient depending on one's level of discipline and access to offline distractions.
A lot of the old methods of learning actually work and so the advice is sound to strictly adhere to them when you're having struggles. Certain modern enhancements are worth a qualified mention though.
> If you're writing proofs, why not do it formally?
Because that requires learning a formal proof-verification language. I'm certainly interested in that, but it is a distraction from learning undergraduate mathematics.
> If you're working with graphical concepts, why not code them up, or use a drawing program (or hey, a graphing calculator) rather than pulling out a ruler and such (and maybe learning to draw at all if you don't know how)?
> If you have sloppy handwriting (as I'm sure many of us here do), why not type in something you'll always be able to read later? (Along with whomever you show it to -- I did a lot of college homework using LaTeX. With macros I could do things way more efficiently, with comments I could go back and see what I was thinking at a misstep (if I wrote anything).)
I'm confused; my post was advocating using software, so I'm unclear why you're suggesting I use software.
> A lot of the old methods of learning actually work and so the advice is sound to strictly adhere to them when you're having struggles.
What is that, a flat contradiction of my post?
Very strange, maybe you meant to reply to a different post?
My post was mainly adding agreement to yours with more specifics, "you" used is the "generic you".
> it is a distraction from learning undergraduate mathematics
Arguably so is LaTeX. But it's desirable that students (or just people learning the same material, later) spend some of their undergraduate time learning new things, right? And not just because it's new, but hopefully because it's better. Learning new/different things is just a small step further beyond learning old things with new/different assistants. And maybe some things will have to be cut out, like 17th century prose-proofs (edit: and even just moving to structured proofs without full formal tools is an improvement...), or square roots by hand (http://www.theodoregray.com/BrainRot/)
One thing that I can add, is that the process of neatly recording something really helps cement the process. My professor for dynamics and mechanics of materials required homework to include diagrams of the problem, neatly drawn, on unlined paper. Often I would find that each problem would take three sheets of paper (I'm a horrible draftsman), but I am horribly glad after the fact that I invested all that time.
It is painful, but I don't think there is any easy way of actually learning without just sitting down and doing problems. Have you considered auditing a course at a community college? Very few people (myself included) are motivated enough to work enough problems without the threat of assigned homework. You need to do enough problems on a topic that you are no longer struggling, then do 4-6 more. Those last problems are, IMO, the most important, they actually cement the concepts in long term memory.
As far as books, I can recommend Schaums Outlines for good examples of worked-through example problems.