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[Imperfect]

I've been studying math for about 9 months now, not especially intensively, as I studied other things as well and the school stuff and exam anxiety also unexpectedly took a out a huge chunk. I sucked at it from 3rd grade of elementary all the way to college, but after some thought, I figured out about a year ago that it was the math teachers, classmates, grading, and the school bullshit in general that I hated and wished to avoid, not math itself, so I decided that I'll be good at it in spite of school. I used khanacademy and I rebuilt my math skills from the ground up, meaning I really started with addition, just to dust it off. I got to about 4th high school math(logs, goniometric functions, eqs with this stuff,...), but then anxiety from the final and the graduation exams and the bachelor's thesis kicked in an I was out of order for about 2 months. I never encountered it before, I didn't know what to do with it, and it just blocked all my thinking and learning capacities. I looked at stuff, I read it, but I just wouldn't take anything from it. The only way to somehow ease it was to work on the thesis and study for the exams. As soon as I did something else, pressure went up, stomach tightened, and something was pushing me towards studying the stuff I'll be examined on.

When I got back to math after those abovementioned two months, I found out that I just forgot everything. I yet again have no idea what to do with goniometric function in equations or what those things really are. I somehow remember the difinition, but I don't know how and when to use them. The same goes for logarithms. I know which number does what, but give them to me in a equation, and I'm stuck. Things get even worse when I need to actually apply those concepts, such as for using logaithms to get a variable out of the exponent, or even just noticing that I could add the variable to get it to he other side and then factor, or soemthing along those line. It seems to me that I just memorized a bunch of rules and tricks and automated their use to some extent(and forgot a good half of them during the stress period), but that I never actually understood the concepts behind them, meaning that while I can easily solve the problems which are exactly like the examples I saw, I'm lost the socond I see something that asks me to apply the underlying concept, to really understand it. It's as if I memorized a few sentences in a foreign langage to use in certain exact situations, but didn't really understand what the words mean or where else are they used, meaning of course that when the situation changes slightly, my "foreign language" skills are useless.

TLDR version, or just the last paragraph
So, how can I actually learn what math is, what the equations really tell me, what to imagine behind them, how to know intuitively which steps to use to solve them, etc. How do I learn the actual meaning of the language of mathematics? What's the psychology behind this? And is there a course or a book somewhere that teaches you this skill? Besides khan, I also tried the "Indroduction to Mathematical Thinkig" course on coursera, hoping to accomplish this goal, but I lost the latter half of it to the anxiety.

[Vector]

Use a better textbook and complete all exercises.

I would start with Walter Rudin's Principles of Real Analysis, or Munkres' Topology, or Birkhoff/Mac Lane's Algebra.

[gimlet]

Agree with get good texts, plus get lots of problems with answers, and books or sets of problems with worked out answers - save those for when you're REALLY stuck.  (These helped me a LOT when I got blocked).

Make sure you're studying effectively - read and reread, review old stuff periodically.  It helps for me to re-write key stuff into a notebook.  It also helped to have access to alternative textbooks so I could read a differently phrased explanation of some topic (library is fine, although old versions of textbooks are really cheap too).

Do lots of problems.  DON'T peek at the answers to problems and think "Oh yeah I knew how to solve this" -  work out the problems one by one, writing down all your work, and only when you're DONE check the answers.  For each one that's wrong,  redo it and try to find the mistake AND make a note to do even more problems of this type, obviously re-reading the text as needed.   Do LOTS of problems, and a few times a week at least, go do problems from earlier parts of the text.  Every once in a while,  find or make up a set of problems that includes problems from every chapter you've studied up to the current point - that way you can't get clues on approach from knowing what the current chapter is about.  DO LOTS OF PROBLEMS.

Spend some time on it every day (or ALMOST every day) - don't try to get it all done in a marathon just before some deadline, spread out the work.

It takes quite a few repetitions to get stuff into long term memory.  If you expect to need the techniques for a later course, keep reviewing the material and keep doing problems - don't just throw the books in a box and forget completely about it through summer vacation,  you'll be wasting a lot of the hard effort you put in to try to learn it.  You WILL forget stuff, everybody does  - don't panic, you learned it once, just go back and re-learn it.

To help with visualizing uses, try to find some texts on practical applications of whatever you're studying - usually titled Mathematical Modeling or something similar (these vary from superficial to really in-depth so look over a few).  Realize that you may go a while before really understanding - I definitely had several courses where I had to just memorize the mechanics and apply them by rote for quite some time before something "clicked" and I understood why it was working.

[Vector]

Basically, OP, you seem to be under the popular delusion that all mathematics is computational.  It's not.  What you need is a different textbook, because the usual ones won't teach you any math at all.  They'll just teach you how to be a computer that runs on french fries instead of a GE hookup.

[Cobbler89]

Basically, OP, you seem to be under the popular delusion that all mathematics is computational.  It's not.  What you need is a different textbook, because the usual ones won't teach you any math at all.  They'll just teach you how to be a computer that runs on french fries instead of a GE hookup.

Elaborating on that, while it's important to be able to run actual mathematical operations yourself accurately (and doing lots of problems is crucial to that), math in the broad sense isn't number-crunching (or even vicarious number-crunching through the variables x, y and z) -- it's quantitative logic. What you really want (or should want anyway) is not just some exercises to get good at doing math, but a good explanation of the definitions involved, all tied to each other without having to look all over the place for the other parts, that you can come back to again and again as long as you need to to understand it. (Some simple examples to go with each couldn't hurt either, though once you've crunched enough problems you can think of any of those as an example.) All those fancy terms they throw around seemingly at random in algebra, and again when they start to apply calculus to abstract problems like whether pi is a transcendental pluperfect magnoliatus number (ok, so I realized I couldn't remember all the things pi's supposed to be and started making some up...)? Actually knowing what those mean is knowing mathematics. The rest is being able to correctly use them to make deductions, either in particulars -- solving individual equations, especially ones with actual quantities or concrete applications -- or in the true abstract, proving theorems about those mathematical things themselves. Both are important -- without the abstract understanding you don't know what you're doing solving all those equations, and without the practice solving them you can't reliably apply the abstract understanding either to particular things or to broadening the abstract understanding itself.

Now, if I were to be truly logical in the broad sense here, I'd have either a citation or a proof for all that. 8^) But I don't off the top of my head, so you'll just have to judge whether it's right (or at least seems right) on your own.