Learning maths as a hobby - tips and techniques not to forget

Question

I learn math in my free time. I do it mainly as a hobby, but it also helps me as a software engineer. My main resources are MathTutor site and Stroud books. The problem I have is I can't work on math everyday, and I always feel I forget what I learnt in the past - from simple things like how "factor by grouping" works or trig identities to a bit more complicated things like what integration technique should be used for a given problem. So, I go over algebra, pre-calc and basic calculus over and over again. I mostly watch videos, read and do the exercises "in my head".

Do you have any tips that could allow me to remember what I learn and continue to more advanced math? For example, is it important to do the exercises on paper or keep you workbooks so you can come back to it later? Do you use concept maps flash cards and things like that?

Any tip would be greatly appreciated,

Thx

Answer 1

Personally, I've never had the patience to watch videos (I'll either fast forward through them or not watch them and opt to read the book instead) and I don't think they are all that effective. Lectures, on the other hand, are in real time and interactive, which is better. So I recommend auditing or enroling in a course at university. Outside of lectures, reading key texts is important. By reading, you've really got to "own the material" (this means definitions, theorems and proofs). Now, when you said you did exercises "in your head", I got a little bit concerned. If you can do the problem "in your head", then the ones you chose were too easy (probably chug and plug) and not worthwhile. To really learn material well, and thus remember it, you have to experience the frustration of not being able to solve a problem, but then still have the tenacity to unravel and ultimately solve it. The key to mastery is doing exercises of increasing difficulty (up to and including harder problems) on a given topic. You should keep record in a notebook. As for "flash cards", no self-respecting student of mathematics uses them; they only work for 'memorizing' but not 'understanding'. I conclude by saying that learning math is a 'marathon' and not a 'sprint'. It is much more about racking your brain about harder problems, then it is about doing the computation for trivial ones as fast as possible.

DISCLAIMER: These views are larged based on personal experience, but there is no one size fits all method for studying. So please hold my opinions at face value.

Answer 2

My approach is typically to find books that are challenging enough, and have lots of exercises. Then I get the hang of the material through two approaches (besides careful reading):

First, do any exercise you can understand. Some category theory books, as an example, often have exercises drawn from areas I'm totally unfamiliar with, so I don't aim to do all of them; but I try to get as close as reasonable.

Second, if you're reading a proof in the text, and you have a hunch that you know how it goes, put it down and work out the proof yourself. I've also done this reading papers aimed at a better educated audience; if the details are sketchy, I try to flesh them out on paper.

I absolutely recommend having a notebook (virtual or otherwise) to work things out on. In one's head, it's easy to skim over details based on what seems sensible. Only when you actually write them out will you discover (to your enrichment) that you don't have quite the grasp you thought you did on some point. Better to get those cleared up sooner than later.

I also recommend keeping around material that's well over your head, but sounds interesting. They can be a great resource for suggesting study programs (by working backwards to figure out what's required to understand it), and as benchmarks to see what you understand now that you didn't before.