Learning Mathematics

Teaching at Cambridge is through lectures and supervisions: the lecturers give formal expositions of the portions of mathematics that are in the syllabus, and distribute sheets of Examples which you then seek to solve and discuss with your supervisor.

Gradually, as you progress through the university course, you will become increasingly responsible for your own work; and, if you go on to become a research student, this responsibility will increase yet further.

You have acquired a certain conception of mathematics in school, which may be a help or a hindrance to you in Cambridge. University mathematics comes as a shock to many freshmen, and you may be unnerved by having to change your ideas about mathematics. Comfort yourself with the reflection that however irrelevant your present mathematical knowledge may seem to contents of the lecture courses, the mathematical facility you have acquired will still be useful.

"Learning mathematics is like driving through a fog"

Learning mathematics is like driving through a fog. At any moment some things are perfectly clear, some are unclear but discernible, and some are wholly obscure. As you move along the road, objects become clearer, until when they are very near you, you cannot understand how they were once obscure. It is the same with mathematical ideas: at any stage of your career there are some which are very clear, some you know something about, and some are wholly obscure. What brings objects through the fog into clarity is, in the case of motoring, petrol and in the case of mathematics, work.

As an undergraduate you are not expected to do anything more than accept uncritically the material you are asked to learn. But as you develop and become a research student, you will have to take a dynamic view not only of your relation to mathematics, but also of the state of mathematics itself; so that your language, notions, and aims are changing all the time.

Mathematics is conventionally organised into definitions, theorems and proofs.

You must pay careful attention to definitions. Generally a new definition will suggest a host of trivial questions - possible variants which might or might not be strengthenings, simple instances and so on - and you should check all those, to get the definition clear in your mind. Of course with a definition of something such as a group, which will recur repeatedly, your notion of a group will gradually develop, and will not stop once you have learned the definition.

As for learning the statements of theorems, you will find the hypotheses easy to remember if you recall what part they play in the proof.

So how do you learn proofs? The thing to remember is that most proofs contain very few ideas, and that as your mathematical technique develops, you will be able to supply more and more proofs for yourself without having to read the proof the author has given. Therefore you should try to isolate the key idea of a proof and summarise it to yourself in a sentence; then, when you are asked in the examination to prove such-and-such a theorem, you simply repeat the sentence to yourself and rely on your technique to fill in the details. Up to Part II, most arguments are in any case not more than a page in length, but for Part III, five-page discussions are common, and you will then need to develop the knack of making a précis of a proof.

A good way to discover the ideas in a proof is to try to prove the theorem yourself, either before you read the proof, or at least with the book shut. Two things may happen, one encouraging and the other beneficial: either you may find a satisfactory proof by yourself, which is good for your morale, and may be even better for it if you have found a proof that is different from the text; or you get stuck, and where you get stuck will usually be where an idea is needed, and then when you return to the text you will see the point of the given proof.

If you find a particular proof very hard to grasp, an emergency treatment is simply to write it out several times, first with the book open and then with it shut. It is pure parrotry, but it may be a way of cracking the nut, and at worst it will get you through the examination and you can hope that understanding will follow. If you are finding real difficulty with a particular theorem, talk to your supervisor or one of your fellow mathematicians about it.

You may wonder how much detail you should give in a proof. In the examination these two rules may be helpful: give as much detail as you have time for, and state clearly whatever you do not propose to prove. The examiners will then surmise that your knowledge is thorough. In private you should go into details whenever you are uncertain of them or when you feel that the undetailed proof you would otherwise give is too glib.

In the last resort your mathematical development is up to you. Your supervisors and lecturers may try their hardest to explain things to you; but if you can develop the habit of self-criticism, you will be your own best teacher. You know whether you understand something or not; and if you don't, do not shirk the task of clarifying the matter.

If you want to get the best out of your supervisions, do not be frightened of telling your supervisor that you do not understand something. Let him see your written solutions of problems you have done; first, it will cheer him up to see that you can do things by yourself; second, he may be able to suggest shorter methods; and thirdly, you may have found a method that is new to him, which will interest him. But also be willing to take him work with which you have got badly stuck. He does not mind whether you have or have not got stuck; what he wishes to see is that you want to improve yourself.

Finally, be on good terms with the other undergraduates reading mathematics, and talk to them about problems and points in lectures. Mindless copying from others is of course to be avoided; but it is beneficial after you have thought about something to compare your ideas with those of others.

The above remarks were written by the distinguished mathematical logician, Adrian Mathias, when he taught at Peterhouse.

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