Raj, a qualified teacher and Maths examiner, offers tips on how to master Maths! Apart from the obvious answer (through a lot of hard work) here are some other ideas on how to keep on top of things.
Don’t skip lessons! You may think that you can afford to work on your own, or miss a few lessons, but this is highly unlikely. The more lessons you skip, the harder you will find it to cover lost ground.
Get yourself a notebook or a folder. It should be your constant companion throughout your studies. Notes in this book or folder must be neat and in ink. In addition to this, you should have a small jotter. You should take your jotter to all your lessons. In it, you should jot down all the rough notes you can. In it, you can do all the rough classwork set. But keep it apart from your folder. Your jotter will contain many notes unworthy of your folder – but the more important notes in your jotter should be transferred into the pages of your folder as soon as possible. Your folder is your own vital document. You write it in your own way to suit your own studies; it is a product of your personality. And after you have finished it, you read it through many times until you know it back to front. You must read through your notes again and again and again and again and again and again and again. Then read it through again. Read it until the very last moment before the exam. By then you should be able to visualise every single page.
In class, you don’t always have to take notes – but you do have to take note! Never take notes just for the sake of taking notes. Don’t merely be a living piece of carbon paper! Think about your notes in the way that seems most important to you. If the teacher writes important notes on the board and tells you to copy them down, you should do so. Note-taking is valuable because it is an active way of studying; it helps you to keep your mind on your work and it prevents you from day-dreaming. But don’t copy down every word a teacher utters. Only take notes of items which will probably be of use to you in an exam. The idea of note-taking is to get down important facts – not to carry out a difficult dictation test.
Process (i.e. re-write and re-arrange) your notes, as soon as possible after the lesson, preferably within hours. Merely reading them through is almost useless by comparison. Look at formulae and try to see what the different bits mean in a general way. Try to improve on the notes by writing your own, re-ordering arguments and expressing them in a way that suits you better. If you are not writing, then you are probably not working sensibly or as efficiently and effectively as you could. Eventually you will find that much of the original notes consist of redundant material once you have understood what you are doing, and that it can be thrown away. You will probably find this disintegrating at first, but don’t worry; the subject becomes worse before it becomes better.
Remember that you do not go to class just to get basic facts and figures – you can get these from your textbooks. You go to watch your teacher’s expert mind at work! He or she should transmit their own particular way of looking at things, their own enthusiasm, their own way of tackling problems, their own personality. Try to emulate them – or better!
Don’t merely follow your teacher through the course. If possible, keep ahead of him. Go to the library and get ahead of his lessons; try to do the next exercise in the textbook Be one up on your pals, and never, never one behind. It makes classwork simple, sends your confidence soaring, and saves you months of work. Your bit of extra effort will mean that you take everything, including the exam, in your stride. But once you fall behind your teacher and your classmates – even if it’s only one day behind – you can easily be puzzled in your work, embarrassed in your class, overpowered by last-minute exam cramming and defeated in the exam. So, as early as possible, get into the habit of keeping ahead. It’s certain to bring success.
Commit some proofs to memory. Find out why your memory fails you: have you forgotten what it is you’re trying to prove, or what the key point is that you are trying to reach? Always try to locate the key points in a proof; much of the rest will be found by routine algebra (a vital area in Maths).
Keep revising definitions, formulae, theorems, and the really fundamental results. That is, write them out with the book closed – don’t just look at them. Write them out several times until you really know them.
Do the problems you are set, but don’t use them as a bridge to theory. If you have special difficulty with one, write it out again and again until you really understand it. There must be a reason why you thought it especially hard. Never be afraid to ask your teacher anything you don’t understand. One problem fully understood is worth several half understood.
Every so often try to recreate a large section of material via definitions and theorems and try to see how far you’ve got; sort out what parts are fundamental, which show genuinely new techniques, and which items are merely tricks or short cuts. What kind of problems could you now solve, given enough free time? Imagine you were in the process of inventing the topic yourself or were going to teach it. Would you be satisfied with your work; could you convince your friends; where might it go wrong; where might it go next; do the worked examples you have really illustrate anything?
For revision, you should try to reduce the amount of notes to an absolute minimum. Definitions and theorems should (where possible) be reduced to clear statements and contain enough keys to enable a proper proof to be reconstructed. Make up tables of paradigmatic examples, of the simplest type which illustrate the concepts of the topic. For example, you should have concrete examples of several proofs; polynomials which factorise over reals and those which do not; graphs of simple equations and those of more complex equations; examples of odd and even functions and those which are neither; lists of derivatives and integrals; examples of particles on slopes or moving under gravity, etc. The list is almost endless.
Always study past exam papers. You need especially to get to know the type of questions set, the sort of wording used, how to answer an exam question, and what sort of answers examiners look for. Interpreting a question is part of being able to answer it properly.