The process of learning mathematics
University modules go much faster than A-level teaching. (They make the sixth form seem like a lazy idyll in lotus-land.) Therefore beginning a Mathematics degree is always a shock. We ease the transition for you by taking the first year gently, but it's still easy to fall at any stage into the trap of doing too little work.
Work at those exercises! It is essential to back up every hour of lecture time with at least one hour of private study on the same topic. (Your social or sporting life is important too - nobody denies that - but if you don't limit those pleasant activities, expect only a third-class degree.) When you get an example sheet, the ideal is to do all the exercises within a week, by your own effort. Only the best students will manage this all the time. Often there are Sections A, B and C on an example sheet. Section A is the most straightforward, Section B is slightly harder, and contains the exercises for handing in, while Section C contains exercises that go a little further afield. Some of these are quite hard, while others touch on topics that stray from the syllabus, but that the lecturer couldn't bear to leave out. But you can tell you're really falling behind if you don't even try to do all the questions in Section B! You should certainly do all the questions which you are asked to hand in. (Marks on written assignments tend to average around 70%: be aware that examination questions may be rather harder.) Discuss with the people in your supervision group, or with friends from lectures, those questions which you (or they) can't do. If you are really stumped by a large proportion of the exercises, talk about it to your tutor, who may well be able to help.
Solving is winning. "Each time you succeed with a problem, you have won a small victory over the mathematics." It boosts you psychologically, and your mind stores the mathematical ideas involved without the pain of rote-learning.
Mathematics is not a spectator sport. It's tempting to ask your supervisor to show you how to work out the problems. But it only does a tenth of the good. "Oh, YES, now I see" - but have you really learned anything? Given the same type of problem two weeks later, you may well have forgotten. Some lecturers prepare solutions for posting on the web. Usually only the solutions to Section B are given. If, before hearing how to do a problem, you had worked on it by yourself or with a friend, you'd be much more likely to take it in. Working on a problem yourself helps to make nets for catching ideas with. If you haven't done the work, the ideas just fly right by you.
"I [don't] understand" can be misleading. How often have you said something like, ``I understood everything she said in that lecture''? Doubtless she's an excellent and popular teacher. But can you do the exercises she set you? If not, sorry, but you don't really yet understand what she said. On the other hand, if you can do the questions for some module, then you are understanding it (and so you needn't worry too much).
Mathematics takes time to absorb. The absorption takes place as you do your written work. So write early and write often.
Files and ring-binders are hopeless at learning mathematics. Don't collect printed notes and store them away in a file. Air them, read them, discuss them with your friends, your supervisor, your tutor. Ask the lecturer questions too: he wants to share his enthusiasm with you. Talking mathematics makes it live. Then read your notes again - and then see how any remaining problems have become more transparent.
Personal organization and work
Your most important resource, which to get a good degree you need to draw heavily on, is your own effort and determination. Try to be reasonably organised and systematic. Try to keep on top of your work. Most of your time is not scheduled by the university, but when exams loom you'll find you wish you had done more work earlier. Many maths students found it possible to revise for A-level modules in the few days before the exam, but this is usually a disastrous strategy for university modules. Modules cannot be learnt in a week. You need time to think about the theory and practice on examples.
If you have problems understanding things, ask people: other students (in your own or higher years), your supervisor, your tutor, the lecturer.
Study Skills. New students (and some experienced ones too!) may need to build up their study skills to get the best out of the effort they put in to their work. The university library keeps books on study skills under LB1049 or LB2395; you are encouraged to spend some time looking at these. We recommend books by W. Cassie, R. Freeman, A. Howe, L. Marshall and A. Northedge, and the pamphlet, D. Burkhardt (Ed.), Study Skills in Mathematics. This last contains some good hints on problem solving, and you will get more from G. Pólya, How to Solve it.
Preparing for Exams. On starting a module, your first target is to absorb the lectured material and the lecturer's problem sheets. Later in the term, and in the run-up to the exam, test yourself out on past exam papers, which give a good indication of the standard expected.
In the third term, many lecturers give a revision lecture on their module, which should help you see its overall structure.
There's no point in trying to guess what will be on the exam paper - it may or may not be related to last year's paper, or to hints you think the lecturer dropped, and it's extremely unlikely to be related to the silly rumours that sometimes develop in the heat of Term 3. Rather than worrying about what will be on the paper, you're better off thinking through the material of the module, and making sure you know what the theory means in practical problems such as those on the example sheets. Even if you don't have much time, there's just no point in trying to memorise your notes; aim to analyse a corner of the theory, and work it all out in a case you can understand.