Allocate roughly equal time to all the essay

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Allocate roughly equal time to all the essay questions you tackle. If your essay is good it will already have gained most of the marks available and extra work can not gain many marks. If your essay is poor any improvement is likely to gain marks. 3 Planning an essay for supervision Since most of the marks for an essay are for specific points rather than for presentation and since any examination question must be done in a hurry you neither can nor should aim very high in the actual examination. There are, however, various reasons for trying to do substantially better when preparing the same essay for supervision. (1) If a similar essay turns up in the exam you will have a good model to guide you. (2) If your supervision standards are high you can relax them in the exam. If they are low you will find it hard to improve them under the stress of examination conditions. (3) Writing a good essay is a good form of revision. It helps you to see how different parts of the course hang together. (4) Remember, however, that in a few years you may be defending a PhD, proposing unpopular changes to top management, giving a talk to a selection board or even lecturing to your own students. In such circumstances, it matters not merely what you say but how you say it. You will need to marshal your arguments and present them attractively. Treat the essay as training for such circumstances. If an essay forms part of a week’s work for a supervision you should start thinking about it at the beginning of the week, jotting down ideas as they occur to them. It is a good idea to run through the essay in your head several times during the week before putting pen to paper. (Compare my advice for doing the essay in the exam.) Now imagine that you have to give a talk to other mathematicians who know nothing about the subject. Part of your job is to do some hard math- ematics but you must do more than this. You must also try to give some idea of where your topic comes from and where it is going. You also wish to convince your audience that the topic is interesting. 5
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One problem that faces you at once is that there is not time to prove all the results that you wish to discuss. This should not be treated as an invitation to prove nothing. Instead you must be selective. Your imaginary audience wants to see the important things proved properly but is happy to take the less important things on trust. Thus your first business is to choose the central theorem of your essay which will be proved fully. Next you need to consider the lemmas you will need in your proof. Some of these you may decide to prove, for some you may decide to give a sketch proof, others you will decide to merely state. In the same way, your central theorem may have associated corollaries, and counter examples and you will need to decide how much detail to give for each.
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