How do you teach yourself math?

Two things have really helped me in learning math(s) by myself:

1. Just read the definitions/discussions. Try to deduce the theorems and such by yourself. This helps immensely in understanding and remembering why things are true.

2. Start a Math(s) blog. It's amazing how much more you force yourself to understand something when you have to write for/teach to someone else.

For theorems with a lot of conditions I will often try to apply/prove the theorem with one of them missing and see where the whole thing falls down.

This really helps in understanding the proof better and remembering the conditions of the theorem.

Three words: definitions, definitions, definitions. Read these over and over and over. Once you have those down, start going through theorems and see if you can figure the proofs out yourself. Get stumped.

Then look at the theorem and proof and see how every part of the hypothesis is used in the proof. This will help you see why the theorem is defined as it is, why certain conditions must be met for the theorem to be true, etc.

This is something a lot of people don't get... Reading a book isn't going to do you much good. Do problems, as many as you can get your hands on.

Your approach seems to work well. What you should supplement this with is to get somebody else to check on your solutions, and to have somebody to talk to about the things you are studying - human knowledge aqcuisition is, for the most part, a communal process, not a singular one.

Edit: Typos.

I taught myself Abstract Algebra over this past summer and there are a few things I think helped:

1. I REALLY wanted to learn it. I basically had to learn it for the GRE I'm taking in October. I had lots of motivation, and every day I sat down with a couple of books and read them a little bit at a time.

2. I had a good system of learning. I was studying out of a GRE book initially, mainly because that gave me a good outline of what I needed to know for the test. I started with that book and made flashcards of definitions, because that's a large part of Abstract Algebra... and I would do them until I knew the definitions.

3. I used good textbooks to get good ideas of the proof techniques used in Abstract Algebra, and then I would try to develop proofs myself once I got a good idea of how they went, mainly because I already knew the definitions at that point. I paid attention to the proofs, and did many exercises out of the books.

If you have strong proof ability and understanding of analysis, i would suggest inquiry based learning. I learned real analysis this way and feel i have a really good understanding of the subject. The idea behind it is you get the definitions and are asked to prove the theorems in succession. It may be difficult to jump into the middle of the material (as measure theory is the last chapter), but it is a good way to get a deep understanding in a subject

Before I start reading a textbook I usually skim it, paying attention to definitions and how they're used in theorems, but skipping all and most non-trivial stuff. Sketching stuff out and how they interplay.

I believe one day of skimming before starting to actually read would be enough for a book like Rudin's Real and Complex Analysis. I haven't read Folland but I assume it's of a similar magnitude.

That way, when I actually read the book thoroughly I have a better understanding of the work frame, and I find it easier to commit to the text when I know where it's heading, and what's the significance and role of every entity in discussion.

Slowly

idk, I just go to the library and pick up a book. Reading a tensor analysis book right now. It can be hard, but as long as you look up parts you are confused about and or reread it until you understand it then you shouldn't have issues. Most people get stuck when they dont understand something and they skip it... BAD BAD BAD. You will come across it again and in more detail, so never skip anything you dont understand.

Wow, I'm actually interested in picking up some Measure Theory so I can tackle some graduate level measure theoretic statistics. I've been out of school for a while now, though, so I didn't have a lot of professors to ask for help.

Someone pointed me to this book, which seems to be extremely rigorous, but if you know any other good resources I'd love to hear them?

Yup same here. I recently started teaching myself math (Calc 2) at home and I do have real exams on it, just like at school. I asked my teachers for book recomendations, CD's, PDF's, worksheets, the whole lot. If you don't have access to work sheets, Google them, you will be amazed how many excercises you will find on a topic and putting "excercises" or "questions" after it: logarithm excercises

Take as many notes as you can, and when you do, always take down examples. Something that has also helped me, and that has been mentioned here, is, when taking notes, pretend you're writing it for someone else. Your brain will try and take in more info and you'll learn more... it's crazy :)

Lastly, don't give up. :)

I am a college engineering student, so what I am about to say may not mesh with higher mathematics, but for me this works. I have taken your basic Calculus circuit including I,II,III I have also taken ODE as well as PDE. To be honest I feel as if going to class did not really help me as I sat there and did not pay attention. A t then end of the day I learnt the basic therom and then just drove through problem after problem. Nothing worked better for me then just practicing problems.First few may be difficult in a new chapter but once you can grind through the first say 5 you begin to see what you should be looking to do in problems. Im almost sure this works for all lower level math courses as well. Like i said I have not hit the pure/proof based classes that require abstract thinking to solve the problems( which i would love to take someday). Practice problems is all it comes down to for me.

What I like to do is find some result that sounds interesting and you want to understand, and then work backwards until I understand each part of it. You'll learn a lot in the process.

I also am a major proponent of learning at different levels, meaning I usually try and work on an easy thing and a hard thing at the same time. The hard thing pushes you, and increases your mathematical vocabulary and maturity, and the easy thing keeps you motivated, and reminds you why you like math.

Do all the problems.

khanacademy.org

"Dummy's Guide to Math" or any beginning math book has a great review of things you should have learned while in college. Now that , ahem , Im older and wiser its easier to read it with out any pressure and read it for the sake of understanding it.

This is an expensive suggestion (potentially), but I suggest having multiple texts covering the same material. I've been reading up on real analysis using a pdf version of the Rudin book (baby Rudin) and the text written by my old professor from undergrad. When a Rudin proof is over my head I switch over to my professor's book and see if the same topic is covered. It helps to have more than one perspective on the same ideas, especially in the prose explanations of a definition or proof.

Pdf versions are great if you have a tablet, and free if you know where to look.

I read books for a couple of hours, then I smash my head into the books for a couple of hours.... then I stare at a wall and "invent" my own mathematics... or try to do everything myself.

Just pick up a book. Say Rudin, so many fanboys of that here.

Go line by line and read. Classes are a waste of time because often professors go too quickly and you won't be able to follow unless you're really good and absorbing information. Get many books, too. Each book is written by a credible professor at some university so it's like having a bunch of professors for the same class, and you can find out which book works best for you.

If you don't understand something, reread it. No pressure.

I see a lot questions like this on r/math and I think you guys need to start to learn to do math, not just talk about.

It's really not that hard. Read a section, do some problems, maybe come here to actually start some goddam threads about math problems when you get stuck. Or you can go to www.artofproblemsolving.com and go to the community forums and actually do some mathematics. Cause here again, you guys just talk about it like it's some goddam philosophy (and yes, it can be regarded as that, but I'm not going to go there).