University mathematics is a big step up from A-level and it can take a while to adjust. Here are some tips.
Give yourself plenty of time to let ideas sink in. John von Neumann: "Young man, in mathematics you don't understand things. You just get used to them." One can argue about the first part of this aphorism, but the second makes perfect sense. Time and time again, I have suddenly realised that, without my really noticing, a strange concept has become part of my day-to-day knowledge.
Try to get several points of view on a difficult idea. Look it up in books, or online (but books are usually better).
Worry about maths! If you can't do a problem then don't just give up. You'll never make progress unless you put in the thinking time (all this stuff about mathematics being just a series of brilliant insights is garbage). Look for similar examples elsewhere. A particularly useful strategy is to ask yourself questions: Why does the problem start by asking me to do this? What do I know that may be helpful? Can I think of an analogous problem? How many equations are there and how many unknowns? (This last is surprisingly effective.)
Let your subconscious do the work. How often has the answer to a problem popped into your head the next day? How often has an 'inpossible' problem seemed trivial the next day: how could I have been so slow? Time after time, in my case. So don't do your tutorial work at the last minute. Try to set up an environment (in the shower, walking, running) where you mind can disconnect and run free Follow the Masters:.
"All truly great thoughts are conceived while walking." (Nietsche) "
"Methinks the moment my legs begin to move, my thoughts begin to flow." (Thoreau)
Talk to other people about work issues: two heads are better than one. You'll get far more out of your tutorials and classes if you have discussed difficult areas beforehand. That doesn't mean that you copy another's work, or produce joint answers to problem sheets. After all, you'll be on your own in the exams.
Don't try to learn large quantities of 'bookwork' (eg, proofs of theorems) by heart. Instead, realise that most proofs have just one 'key step'. If you are familiar with the general techniques of the relevant area, you will then be able to fill in the gaps. That will make you much more versatile because you'll be used to constructing the steps of an argument as you go.
Build 'error-correction' into your work. We all make mistakes, all the time; but successful mathematicians are good at catching them before they do too much damage. You should constantly be alert for things that can easily be checked. For example, if you are working out a definite integral of a positive function, the answer must be positive, and if it is not you have made a mistake. (You may still have a mistake if it is positive, but that's another story...) Are there special parameter values that allow a quick check of a formula? Is a formula dimensionally consistent? Does a function have the right behaviour at, say, the origin? Stop occasionally to check things like these and you'll save an awful lot of time.
Sam Howison, June 2010.
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