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.

Answer

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.

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