How do you go about learning mathematics?

I really like mathematics, but I am not good at learning it. I find it takes me a long time to absorb new material by reading on my own and I haven't found a formula that works for me. I am hoping a few people out there will tell me how they go about learning math so I can try out their systems.

I need to know basic things.

• Should I use one book at a time or should I be reading many books on the same topic at once?

• Do you stop reading when you hit on a fact that you don't understand or do you keep reading?

• Do you read all in one go or do you do a little bit and for how long (1 hr, 2 hr or more?)

• Do you read all the chapters or do you do all the exercises before moving on from a chapter?

• Do you adjust your technique in Calculus (calculation heavy) vs. Analysis (proof heavy)? If so, how?

• When you make notes, what do you make notes about? Do you make notes while you read or after?

• If you think these decisions all depend, can you say what they depend on?

I am really lost here. I would appreciate any input.

Of course everybody has their own learning style. Here are some general suggestions.

Find a teacher. It is hard to learn mathematics on your own until you have reached a certain level of mathematical sophistication; nobody is there to tell you what is important and what is unimportant. Take courses at a university; as Agusti Roig mentioned, video lectures on MIT's OpenCourseWare are a good cheap alternative.

Read as much mathematics as you possibly can, from as many sources as you possibly can. This is not limited to textbooks but extends to popular math books, blogs, expository papers, MO, math.SE... doing this will get you used to not understanding things, which is important. It will also expose you to many fascinating ideas that will fire up your curiosity enough for you to look at the material more seriously. As Ravi Vakil says:

...mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards".

A specific way in learning backwards is easier than learning forwards is that instead of reading the proof of a theorem in a book, you might hear about a theorem without proof, but remember that someone on a blog said something vague about a crucial step, then gradually learn enough material that suddenly you can work out the proof independently. I have done this a handful of times, and it is quite satisfying. For example, the theorem I proved in this blog post is classical and extremely well-known, but I had never seen a proof of it. I juggled around some ideas for about half a year until I figured out how to prove Lemma 6 (which I saw in a paper somewhere, again without proof), and I wrote down a proof. Later I read a proof in an actual book, and although the second half of the proof was similar, it did not use Lemma 6. I have yet to see a proof of Lemma 6 in print, although I am sure it is also well-known.

This might sound like more work. But guess how well I remember this theorem and its proof now!

Do as much mathematics as you possibly can. This is not limited to textbook exercises but includes competition problems, finding alternate proofs of theorems, working out concrete examples of abstract theorems, etc. I try to do this as much as I can on my blog; it keeps me sharp and is also, at least for me, much more fun than reading a textbook, which I can't do for long periods of time. This is also why I post here so often.

Question everything. There are a few aspects to this. If something is unclear or unmotivated to you, ask yourself exactly where it becomes unclear or unmotivated. Find someone to explain it to you (for example, on math.SE!). Read a blog post about it. Write a blog post about it! Ask yourself how things generalize and how they connect to other things you know. (Again, math.SE is good for this.) The worst thing you can do is to accept what a textbook tells you as the Word of God.

Finally, teach as much mathematics as you possibly can. This is the other purpose of my blog, and is an amazing test of how well you understand something. You would be surprised how much you can learn about something by teaching it.

Von Neumann is credited with saying that you never understand mathematics, but merely get used to it. To an extent, I find this to be true. But what is it that you're getting used to when you learn mathematics?

1. You get used to the sorts of objects and properties-of-objects which are the subject of study. An excellent way to get used to them is to have examples or connections to material that you already know --- regardless of whether such examples or connections are supplied by the book itself.
   
   Therefore, as I read a book, I look for examples of the structures or properties which are the subject of a book. If I must, I will scan ahead in the book to hunt for them, or stop reading entirely to try and conjure one myself (or find one on the internet if I find this difficult). If I come across a theorem, I try to see how it holds for these examples --- and how attempts to construct counterexamples fail.
   
   

2. You get used to certain techniques for solving problems. Getting used to them involves getting used to seeing them being used, getting used to applying them, and getting used to the circumstances in which they may be fruitfully applied.
   
   Therefore, in a book, if any of the material is new to me, I do the exercises. This holds regardless of whether the exercises consist of 'calculations' or 'proofs'.
   
   

3. You get used to the notation. That is, you get used to parsing the notation and translating it mentally into an alternative, often more explicit, description of what the new notation represents. Notation is important to me, in that I find a well-chosen notation an aid to understanding --- and conversely, a poorly-chosen notation a barrier. This is especially the case when the standard notation of one field looks a lot like the standard notation of a significantly different concept that I'm already familiar with.
   
   Therefore, as I go through a book, I carefully decide whether to adopt the notation of the book --- or to invent my own notation for the purposes of learning the concepts, without regard to the standard usage. Forcing a change of notation as you take notes has the side-effect of forcing you to reflect often as you do it, which helps to reinforce the new concepts; consciously choosing to reject the standard notation also helps you to be more conscious of the notation you are rejecting, and therefore paradoxiacally to interpret it properly (as you are continually conscious that you must distinguish it from conventions with which you're already familiar).
   
   

4. You get used to taking other people's ideas and revising them --- testing their boundaries, and perhaps even rediscovering a standard generalization --- to suit your purposes, or to arrive at a more general theory. This, of course, is something which a textbook cannot explicitly teach you. But especially as you see connections in other fields, or as you pick apart proofs in the textbook, you can ask yourself certain questions, such as:

   • Given a definition of a function or concept on the domain (0,1), is there a natural interpretation which may apply for the boundary points of 0 or 1?
   • Given a definition of an isomorphism between two objects, and suspend the requirement of invertibility? What new relationships between objects do I obtain?
   • Given a proof by contradiction of a result, can I obtain a direct proof of the same result --- perhaps even an efficient construction?
     
     

Obviously, a lot of this process is facilitated by taking notes as one learns --- so I take notes, not just of the material in the book, but of the work I do and the reflections I make.

I don't consider myself anywhere near an expert at this (there are a lot of things that I've tried to learn for a while but failed). However, I have spent a while trying to learn mathematics independently, so I will weigh in, but this should be taken with the usual truckload of salt.

1. Find a good book! It's easy to waste time with a poorly written textbook. I think this one is really important. Different people have different ideas about what constitutes a well-written textbook, though, so it may be desirable to look at books in a library and get a sense of them than simply following someone else's recommendation.
2. Talk to people. I didn't realize how little I knew about algebraic geometry until I was having an email chat with someone and confused the universal properties of projective and affine space minus the origin. That sorted out a few things in my mind.
3. Don't worry about skipping over things that are unimportant. But try to ask someone what the important and unimportant things are.
4. This may be just a personal preference, but I like to make copious amounts of notes when I learn something, starting from the beginning. This is one reason to keep a mathematical blog. However, creating large sets of notes is often not entirely suitable for blogging.