I never really learned how to revise for a maths exam at school. I'm sure people tried to teach me, but I generally wound up reading passively through my notes, maybe copying them out, and possibly writing big question marks on them in highlighter.
It worked ok for me, at least until a disastrous second-year exam at uni where I suddenly discovered that I didn't know all the things I'd read. If you really want to get on top of material, you need to engage with it - you learn more from a conversation than from a lecture.
Here are a few ways I recommend:
You know what a textbook is, don't you? It's where teachers find you questions for homework.
Well... partly. But there's more to it than that. Your textbook is full of explanations and worked examples you can follow, study and use to improve your understanding. It's generally a good idea to find a topic you need help with, read through the explanation (looking up anything you don't understand), before following along with the examples - I advocate copying them out - and tackling the questions at the end of the section to see how well you've taken it on board.
Probably 80% of what I do with students in class is based on past papers. If you want to do well in a test, it's just common sense to figure out what the test looks like, what kinds of things you may be asked, how the questions are worded... and, by tackling the questions, see how you're doing and where you need to do some extra work.
When you get something wrong, it's worth trying to figure out why - I know it's frustrating, and mark schemes aren't meant as educational tools, but by trying to see where you went wrong, you can avoid making the same mistake next time.
(Also, you can drop me a message if you're baffled about why a particular past paper question was wrong - I'll get back to you as soon as I can make time.)
One of my favourite explanations of why maths tutoring works so well is that you learn more from a conversation than you do from a lecture. The thing is, though, you can can get almost the same benefit by having a conversation with other students. (Obviously, you don't get quite so many immediate explanations, and the quality of comedy probably suffers, but it's a pretty effective substitute).
The reason it still works is that two (or more) heads are better than one - the chances of everyone in your group having precisely the same strengths is very small indeed, so there will be places they can explain to you, just as there will be places you can explain to them. Frequently, there will be questions where one of you can see how to start and someone else will see how to finish - between you, you figure the whole thing out.
You may not have conveniently located mathematical friends to set up a study group, but you can still get some of the benefits of explaining things. One of the most effective ways to learn a new skill is to write down the steps you have to take - either as a list or as a flowchart.
The key is to make everything as detailed as possible - imagine you're explaining it to an idiot, such as a parent or a younger sibling. You use a different part of your brain when you're explaining things than when you're reading or listening.
Flash cards, for some reason, are much less popular in the UK than in America. It's actually quite hard to get hold of reasonably-priced index cards over here (it's galling to pay £2 for something that costs about 20p in the States), but you can make flash cards out of sheets of paper easily enough. Just fold the paper in half, in half again, and in half again, then cut along the creases - presto, eight cards. You can make more as and when you need them.
The way they work is, you write a question or a prompt on one side (for instance, "What is the derivative of $cos(2x)$?") and the answer on the other ("$-2 sin(2x)$"). You make a big pile of cards with everything you want to learn on it, and leaf through them one at a time. If you get the answer right, put that card to one side; if you get it wrong, put it to the back of the pile.
A student of mine took this one step further and taped flash cards to every door in her house - if she wanted to go through a door, she had to answer the question first. She got two A*s in Further Maths. Just saying.
A cheat sheet is just a big bit of paper with everything you could possibly want to know about written on it. For instance, a GCSE maths cheat sheet might have instructions on how to solve triangles, all of the stupid vocab words you need to remember, a few of the common mistakes that everyone makes...
Make it as fun as possible - drawing pictures or sticking on photos helps you remember things later on ("Oh! quadratic equations! They're next to the picture of Stewart Lee!") - and spend the last few minutes before your exam poring over it, to get as much of it as possible into your short-term memory.
Google. "[My topic] revision games." Any questions?
If you've tried doing questions and just can't see why they're not working out, ask someone - a teacher, a friend, or even me. I'm firstname.lastname@example.org - or you can ask in the comments below.
I'm most likely to reply if you give me details of what you've tried and where you got to (and if you give me permission to write about it on this 'ere blog).
Meanwhile... what are you revising right now? And what are your best tips for someone who wants to revise for a maths exam?