Gowers: Giảng dạy và học Toán bậc đại học ở Cambridge

Trong loạt bài này, TS. Gowers, huy chương Fields năm 1998, giới thiệu việc giảng dạy và học Toán cho 3 năm đại học ở ĐH Cambridge. Những lời khuyên của TS. Gowers không những bổ ích cho sinh viên Toán mà cho cả những người làm Toán hay tự nhận đang làm Toán (UVL chưa phải là người làm Toán, chỉ là cánh bướm nhỏ dạo chơi vườn hoa dại và sau đó lại bay đi).

Sinh viên sư phạm Toán sẽ tìm thấy nhiều điều bổ ích cho việc giảng dạy Toán sau này.

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Welcome to the Cambridge Mathematical Tripos

By Gowers

Introduction.

This is the first of what I hope will be a long series of posts aimed at providing back-up to first-year Cambridge mathematicians. This may seem a strange thing to do, since the Cambridge system of supervisions (classes taught on a one-to-two basis, usually discussing questions set by lecturers) already provides an excellent back-up to lectures. Do Cambridge undergraduates, who already have closer attention than in any other university I know about, really need even more help?

Well, perhaps they are lucky enough to need it less than mathematicians anywhere else, but there are several facts that convince me that even more can be done than is done already. Let me list a few of them.

1. Whenever I have lectured a Cambridge course, I have always been aware that I have to go artificially fast in order to squeeze the material into the number of lectures I am given. With more lectures, I could make many more additional helpful remarks about how to understand the material.

2. Whenever I have supervised, I have tried to explain some of the little points that it is difficult to fit into a crowded lecture course. However, I nearly always find that at the end of a supervision I am left with plenty more that I could have said. (Just occasionally, I teach somebody so frighteningly good that I have the reverse problem and don’t know how to fill the time. But that is very much the exception.)

3. Even if lecturers and supervisors do have time for useful additional remarks, they will probably make them once only. So if you happen not to be concentrating at the right moment, or just don’t understand the point first time, that’s your chance gone for ever. Remarks in written form don’t have that drawback.

4. Almost every textbook I know is written in a rather dry and formal style and doesn’t provide the kind of back-up I am talking about.


Can mathematics be taught?
Let me pause right there and try to explain in more detail what I mean. The usual way of presenting pure mathematics (which is all I’ll be talking about in these posts) is this: you have some definitions and some results; you write out the definitions, you state and prove the results, and perhaps you set some exercises that test understanding of the definitions and results. End of story. Well, perhaps it’s not quite the end of the story: if you’re being conscientious then you usually follow each definition by a list of two or three key examples.

OK, what’s missing from that? Well, for a start it is very common for lecturers and authors of textbooks to take for granted that their topic is an interesting and important one. This isn’t completely unreasonable, as usually the topic is interesting and important. But if you’re trying to learn about it, it can be a huge help to have a clear idea why you are making this very significant effort. (“To do well in exams” is not the answer I’m looking for here.)

But perhaps the biggest thing that’s missing, and the thing I most want to get across, is how to go about proving results for yourself. There are plenty of books about how to solve competition-style maths problems, but what about proving the more bread-and-butter-ish results that you are shown in a typical maths course?

Before I discuss that further, let me explain why it matters. You might think it doesn’t, since if a lecturer explains how to prove a result in a course, or an author in a textbook, then you don’t have to work out the proof for yourself. But, and this is a huge but, if you are studying for a maths degree, then

(i) you do have to remember lots of proofs;

(ii) memorizing things requires significant effort;

(iii) if you can easily work out proofs instead, then you place a far smaller burden on your memory.

So it turns out that being able to work out how to prove things (perhaps with the help of one or two small hints) is hugely important, even if those proofs are there in your lecture notes already. Of course, it also goes without saying that being good at proving things will help you solve problems on examples sheets.

Now I think a very common attitude to this is that doing mathematics (that is, thinking of proofs) is something that you can’t really be taught directly: instead, you read your notes and do lots of carefully designed questions and find that proving results is a skill that you develop with practice, especially if you were born with a mysterious quality called mathematical ability. And undoubtedly there is some truth in the previous sentence — the method described is the method by which pretty well all mathematicians working today have learnt how to do maths. But there is a significant downside to this method, which is that there are also many people for whom it does not work. They go to university full of enthusiasm for mathematics and find that the subject at university level is much harder than they expected, and that they don’t know how to go about developing the skills that I’ve just been talking about. Gradually as the course proceeds, they fall further and further behind, while some of their contemporaries seem not to. It can be pretty demoralizing and also, given how hard it is to get into Cambridge, a real waste of talent. (I don’t think it is a total waste, by the way, since many people, myself included, have had the experience of understanding some mathematics much better a year or two later than they did when they were supposed to be learning it. I think that even people who get left behind by the sheer pace of the Cambridge mathematics course leave Cambridge having had their minds altered in a way that is very valuable in their working lives. I’d be very interested to hear from anyone in that position, to see whether I’m right about this.)

Another serious drawback with the attitude described above is that it underestimates the extent to which mathematical ability is something you acquire through hard work. It’s true that some people seem to find the subject easier than others. But nearly always you will find that these mysteriously clever people have spent a lot of time thinking about mathematics. In many cases, their ability is no more mysterious than the ability of a very good pianist who has practised for three hours a day for many years.

I am writing the posts in this series because I am absolutely convinced that it is possible to directly teach people how to do mathematics. (I feel this so strongly that I’m prepared to split an infinitive to make my point more forcefully.) Or at least, there are many aspects of doing mathematics that can be discussed explicitly that are normally not talked about and are left to people to pick up on their own. They will be the main theme of this series of posts.

The blog format.

The great thing about blogs is that they allow comments. A few potential advantages of that for this project are the following.

1. If something is unclear or incorrect, you can tell me. The usual maths-blog etiquette is to thank the commenter for pointing out the error and to change it. So I can produce less polished posts (and therefore more of them for the same amount of work) and you, dear reader, can help with the polishing.

2. Sometimes my discussion will get a bit philosophical, and I’ll probably say things that other mathematicians disagree with or fail to say things that they think I should have said. And they’ll probably make comments to that effect, to the benefit of everybody.

3. If there’s something you don’t understand, you can ask about it in the form of a blog comment. Even if I don’t have time to answer your question — at this stage I don’t really know how that will pan out — there are potentially several other people reading the question who are welcome to answer it. (If you are shy about asking a question, then (i) don’t be and (ii) if, despite that order you still are, then ask it anonymously.)

4. Although, as I’ve said, these posts will be mainly aimed at first-year Cambridge mathematicians, and will be focused on the courses you/they are taking, there are mathematicians all round the world taking similar courses, and a blog format allows me to reach them too, if they are interested.

I may well intersperse the posts in this series with other blog posts. If you want to get rid of the other posts and just look at this series, then you can go to the Categories menu on the right-hand side of the page and click on “Cambridge teaching.” There will also be subcategories if you want to focus on posts about particular courses.

If you are a first-year Cambridge mathematician and have just found this blog post, I would be very grateful if you could tell others about it. I am putting up this post over a week before full term starts, to give people a chance to find out about it in good time. I plan to put up the other posts I’ve already written at the rate of about one every day or two.

General study advice.

You’ll be getting plenty of this, so I won’t say too much. But here are a few things that I have often found myself saying to Cambridge undergraduates.

1. Mathematics becomes hard. Every mathematician will be able to tell you rather precisely when it was that they found that mathematics had stopped being an easy subject that they could understand with very little effort and became a difficult subject that they had to struggle with if they wanted to get anywhere. It isn’t necessarily an advantage if this happens to you later rather than sooner. For example, some Cambridge students find the course difficult right from the start, whereas others largely coast through the first year and then find that they can’t coast through the second year. The people who found it hard in the first year may by this time have developed good study habits that the people who found it easy in the first year do not have.

2. When the going gets tough, it is not some failing of yours. It simply means that, just like everybody else, you have to work. Up to now I’ve said this to many undergraduates, but I have come to think that it is a rather unhelpful thing to say if it is not backed up with instructions about how to work. Just how to spend time, once you’ve decided to spend it on mathematics, will be a major theme of these posts. The aim will be to help you to get the most out of the time you spend. You are an adult now, so how much you decide to spend is your decision … but … if you are lazy while you are at Cambridge then you are throwing away an amazing opportunity that won’t come back.

3. Don’t waste supervisions. The supervision system works much better if you prepare for it even a little better. I’d almost go as far as to say that more work is less work. What does that mean? It means that if you do more work before the supervision, you’ll get so much more out of the supervision that it will save you more work in the future than the extra work you’ve just done. In particular …

(i) Don’t leave examples sheets to the day, or even worse, evening, before the supervision. Why not? Because with an unbudgeable deadline you will find that when you can’t do a question immediately, you are so worried that you’ll have nothing to show for yourself at all that you’ll skip it and try to find some easier questions to do. What’s more, you’ll probably skip it without really thinking about it and getting to grips with the real difficulty. If a supervisor tells you how to do a question that you’ve seriously thought about, then you have an AHA! moment and learn something important about how to do mathematics. If a supervisor tells you how to do a question that you have not seriously thought about, then you usually learn almost nothing.

(ii) If you are not on top of the relevant section of the course, then don’t rush into the examples sheet. Read and understand your notes first. This will save time in the long run. The rough reason is that you will know the definitions and results that you are supposed to be using. Many questions are quite easy if you know what the definitions are and what results to use, but almost impossible if you don’t. It’s silly to struggle needlessly with such questions.

While I’m saying that, let me introduce a notion of fake difficulty. Every pure maths supervisor at Cambridge has had conversations like this:

Supervisee: I found this question rather difficult.

Supervisor: Well, what were your thoughts?

Supervisee: Erm … I don’t know really, I just looked at the question and didn’t know where to start. [By the way, never say that. Ever.]

Supervisor: OK, well the question asks us to prove that the action of G on X is faithful. So what does it mean for an action to be faithful?

Supervisee: Oh … er … no, I can’t remember. Sorry.

Supervisor: Have faithful actions been defined in lectures?

Supervisee: I’m not sure. Yes, I think so.

Supervisor: But hang on, if you weren’t sure what a faithful action was, did you not think to look up the definition in your notes?

Etc. etc. This is a fake difficulty because it is not a legitimate reason to get stuck on a question. If you don’t know a definition, you can look it up. (If you can’t find it in your notes, then type it into Google and the answer will be there for you in a Wikipedia article.) “I didn’t know where to start” is a well-known euphemism for “I was too lazy even to work out what the question was asking.” If you come to a supervision with fake difficulties, then you will waste time (not just yours, but that of your supervision partner) dealing with problems that do not require external help, and you will not pick up the mathematical tips that come from engaging with real difficulties.

4. Read your notes between lectures. Of course, I don’t just mean read them, but do your very best to digest and understand them. This is another practice that saves more time than it takes. If you understand your notes on the course so far, then you will follow much more in lectures, which will mean it takes less time to digest those lectures than it would have done, less time to do examples sheets, and less time to revise for exams.

5. Don’t be too passive in supervisions. I know of no other university apart from Oxford where you get the chance to be taught in a group of two, often by senior members of the faculty. This is such an unusual opportunity that you should do your very best not to waste it. I have already mentioned making sure you prepare adequately, which is by far the most important single piece of advice I can think of in this direction. But it also makes a big difference if you do not just sit there and let your supervisor do all the talking. For example, suppose you have got stuck on a question at a genuinely hard place in the question, and your supervisor says, “If you prove the following statement first, the rest of the proof is easy.” That’s fine in one sense, as it tells you the answer to the question you couldn’t do. But in another sense it isn’t necessarily fine, because there is no guarantee that you won’t get stuck in exactly the same way if you find yourself in a similar situation in future. What you want to get out of the supervision is some kind of general message of the form, “In this kind of situation, the following method often works.” If the general message is clear to you from a single example then that’s fine. But if you’re left thinking, “How on earth did my supervisor come up with that?” then ask. Probably if you are keen to learn in this way you will ask naturally, so the real suggestion here is not to be satisfied with merely being shown answers: you are trying to learn how to come up with answers for yourself.

6. Don’t be too passive when reading your lecture notes. Very much the same principle applies here. Of course, your lecture notes aren’t going to start talking to you, but you can still have a kind of “conversation” with them, by doing what we do naturally (according to linguists) in ordinary conversation, which is try to guess what is coming next. If you read a sentence like, “The cat sat on the mouse,” you are not taking in information at anything like a constant rate. You didn’t know that the word “The” was coming first, but it isn’t a huge surprise. You didn’t know that “cat” would follow, but you were probably expecting a noun (though an adjective was also a possibility). Similarly, “sat” had a good chance of being a verb, but you might have subconsciously judged “is” to be more likely. The past tense strongly suggests that the sentence is not an observation but more like some kind of reciting of a story, and you’ve probably heard “The cat sat on the mat” enough times to be expecting that. Therefore, the words “on” and “the” carry very little information — they are what you almost knew would be said. As for “mouse”, the M sound appears to confirm your expectations, but the “ouse” that follows is quite a surprise, and therefore carries quite a bit of information. To remember the sentence afterwards, you don’t then memorize “The cat sat on the mouse.” It’s more like “Take that well-known silly sentence and substitute ‘mouse’ for ‘mat’.”

Something similar happens with mathematics. If your notes contain the statement of a lemma, then try to guess what the lemma is going to be good for. And then try to prove it. Yes, I really did say that. Don’t read the proof as it is in your notes, but try to do it yourself. Get past all the fake difficulties until you feel genuinely stuck. At that point you can either decide that you will put in a lot of effort (for a correspondingly large reward) and attempt to find the proof even though there is a genuine difficulty, or you can put in just enough effort to convince yourself that the difficulty is genuine and then peep at your notes for a hint. If you do this, then afterwards all you have to remember is the hint, and not the entire proof, just as with the sentence above all you have to remember is mat–>mouse and not “The cat sat on the mouse.”

In short, be organized.

I do not want to pretend that following the advice above is easy. I do maintain that if you follow it then you will save time rather than spending more time. But there is no getting away from the fact that saving time in this way depends on being well organized. It is psychologically easier for many people — OK, I’d better admit that as an undergraduate I was one of them, so I am urging you to do as I say rather than do as I did — to leave things to the last minute, working only when a looming deadline, such as a supervision or an exam, lifts the level of worry high enough. But if you do this, you will need a lot more time to get to the same standard as you would if you were more organized, or a similar amount of time to get to a much lower standard, or some combination of the two.

I am hoping to explain in much more detail in these posts how to use your time well if you do decide to spend it reading your lecture notes. Once the courses actually start, I will be able to illustrate what I say with examples of actual definitions and results that you have recently come across, which will make the discussion a lot less abstract.

The general plan for this series of posts.

I don’t know how much I will have the time or energy to write. However, if I write as much as I hope to write, then the main focus will be on how to come up with relatively routine proofs. To elaborate a little, there are a lot of arguments in mathematics that experienced mathematicians find very easy to think of, but beginners find much more difficult. What is it about the brains of experts that makes them find it so much easier? How can you convert your brain into that kind of brain? That is what I want to try to explain.

Unfortunately, it is not possible to do a good job of explaining this without first making sure that you have a good grasp of a few basic logical principles. At the time I am writing this paragraph, I have written several posts about basic logic. (I was expecting to write one or two, but they just expanded and split and expanded and split, and I have ended up with far more of these introductory posts than I thought would be necessary.) I hope you will find even the basic-logic posts interesting and helpful, but if you don’t, then bear in mind that they are not what this series is all about. What it is all about is the courses you will be taking this term, the results and definitions in those courses, the proofs that you are expected to understand, how such proofs get discovered, what is interesting about the definitions, and so on.

Online resources.

If you haven’t understood part of a lecture course, you have options that I could only dream of when I was an undergraduate, except that I didn’t, because, to my great shame, I didn’t predict the Internet. (To give an idea of what life was like, if you felt like seeing another undergraduate, the standard method was to walk to their room and knock on their door. If they weren’t in, they would usually have left a pencil and some paper blu-tacked to the door, on which you could write a note. And if you wanted to telephone somebody outside the Cambridge area and didn’t have their number, you could go to the Porters’ Lodge, where there was a huge bank of telephone directories. And so on. You don’t know how lucky you are.) Here are a couple. I may add to the list later.

Wikipedia. There is a lot of very good mathematical content on Wikipedia. It isn’t perfect by any means, but, as I mentioned above, if you don’t know what a word means in a question on an examples sheet, that is not an excuse for not doing the question, even if your lecturer hasn’t yet defined the term in question. Just type the word or phrase into Google and the basic information will be there in a Wikipedia article, which will usually be one of the top two or three entries to come up on Google.

Mathematics Stackexchange. This is a carefully moderated question-and-answer site. If there is something you don’t understand in one of your courses, then see whether you can formulate a precise question that encapsulates what it is that you don’t understand. Often just the effort of doing that will help you to sort out your difficulties, but if it doesn’t, then post the question on Mathematics Stackexchange and the chances are (if you’ve done a good job asking the question) that it will be answered very quickly — possibly within minutes. And if you’re feeling public-spirited, maybe you can answer someone else’s question too. It takes a little while to explain exactly what makes a suitable question, which I won’t try to do here. I recommend that you visit their FAQ page to get an idea of what the site is all about. Of course, questions that are directly related to the content of the posts on this blog are probably better asked here.

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