**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.

*Terence Tao’s multiple choice quizzes.* If you don’t feel comfortable enough about some topic you’ve just been lectured to attempt the relevant questions on an examples sheet, try warming up with one of these quizzes. They are a great way to see whether you understand the basics. Even if you *do* feel fairly comfortable with the topic, doing some easy questions first should make you even more secure, or help draw attention to any misconceptions that you might have.

*An interesting article about study habits.* I stumbled on this link (via a web page for an introductory university mathematics course given by Mark Meckes), which is a thought-proving article in the New York Times, entitled, “Forget What You Know About Good Study Habits.”

September 23, 2011 at 5:16 pm |

Wow, this is going to be an awesome series! I hope you don’t mind if some of us that are not students at Cambridge participate. I received my BS in Math back in 1983 but I still enjoy going back and getting a refresher.

September 23, 2011 at 5:16 pm |

Interesting! I expect these posts to be useful to a far larger audience than Cambridge undergraduates and am looking forward to their eventual existence.

(Also, I hate to be impertinent, but do you by any chance only check your DPMMS email during term? I sent an email and wasn’t sure how long to wait for a response.)

September 23, 2011 at 6:18 pm |

As a 3rd year maths undergrad I wish this had been around when I was in my first year, all very pertinent advice.

September 23, 2011 at 6:52 pm |

This looks like it will be a really interesting series, and I think I’d have found it very useful in my first year – a lot of this post lines up very well with either how I got through the Tripos or how I /should/ have got through the Tripos. (I’m currently a first year PhD student.) A couple of minor points:

– When I saw an unfamiliar word on an example sheet, my concern was never that it would be hard to find the definition (which would indeed be silly). It was that the question wrongly assumed I knew the thing’s definition, and therefore it might also wrongly assume I knew other things about it as well. So my options came down to: getting the definition off Wikipedia, trying the question, and hoping I didn’t run into a brick wall due to lack of knowledge; spending time doing more detailed research on the thing and possibly trivialising the question; or just skipping it entirely and using the time saved on the rest of the sheet. I generally took the latter option on the grounds that since I probably wouldn’t have time to completely finish the sheet anyway, I should focus on the questions I knew I had the relevant background knowledge for.

– When it comes to reading lecture notes: I agree that in theory the best thing to do is to try and prove the results yourself. In practice though, especially when you’re an inexperienced mathematician and you don’t already know the fundamental methods and tricks of the subject, this is extremely time-consuming. In term time I found it was hard enough to find time to read through my lecture notes at all without falling behind on my example sheets, even without spending about 60-90 minutes per lecture. Out of term time, I found I had just about enough time to fully understand the material, but only with a slightly faster method which doubled as a revision aid.

My strategy was basically to read the whole proof, pick out the key steps, separate the novel ideas from the routine parts, and then try and work out how someone could have come up with the novel ideas – effectively reverse-engineering the proof. Then I’d try to prove the results from scratch given those ideas. The framework for this was making a new set of notes for the course, which let me record what I’d learned about the proof and try to reconstitute it from scratch at the same time. I don’t think this is quite as effective as your method since it doesn’t give experience with solving problems, but I do think it at least gives a full understanding of the material. It also doubles as a very effective revision strategy – it’s much easier to learn two sentences and reconstruct the proof from that than to memorise two pages of lecture notes!

(It’s probably also worth noting that I generally only made full sets of notes for difficult, proof-heavy pure courses like Analysis or Numbers and Sets – for most others I found that was overkill and made crib notes instead.)

September 23, 2011 at 7:12 pm |

This is going to be a brilliant series – I am really looking forward to reading it. I completed my maths degree at Cambridge last summer, and often wished that there had been more advice for undergraduates along these sorts of lines (although luckily for me I had friends/supervisors/etc who gave excellent advice).

My piece of advice for new Cambridge mathematicians would be: make friends with mathmos at other colleges and in other years, and gossip. Someone else’s supervisor may recommend a book that turns out to be invaluable; you may find that there are excellent online notes for course X on so-and-so’s personal webpage, and all sorts of other things that will be useful to you. Plus, there’s the great comfort of knowing that you are not the only person who is finding this hard.

September 23, 2011 at 7:14 pm |

“

Every mathematician will be able to tell you rather precisely (…)”I’d be interested to know when the point was in your own career when you found maths had stopped being easy and was now a hard subject you had to struggle with to make progress.

Thanks for the stackexchange ref.

The unmoderated Usenet group sci.math is also useful. Questions asked by those who’ve worked at getting a handle on what they need to understand usually elicit sensible and helpful responses. (New users should be sure not to get wound up by other responses though!)

September 23, 2011 at 11:45 pm

When I was at school, I had a fit of enthusiasm, obtained a copy of the Cambridge schedules, and tried to read up on several of the first-year courses. I certainly didn’t manage to cover any of those courses in full, but nevertheless the result was that I was pretty well prepared for my first year. And the result of that was that I managed to delay the moment where it all got hard to about the beginning of my second year. I’m answering this question in some detail because if I simply say “The beginning of my second year” and you find things tough earlier than that, you should understand that that isn’t necessarily a problem. It just means you are discovering one year earlier than I did what learning mathematics is really like.

September 23, 2011 at 7:24 pm |

The section on “Can mathematics be taught?” is really nice.

Every mathematics class I teach, I find myself spending more and more time trying to think of ways to express (either in front of the students, or in lecture notes) ideas of how someone might have thought of the statements and proofs of the theorems in the course (even when those are completely standard…)

September 23, 2011 at 8:37 pm |

With regards to using the internet: this can be useful when doing example sheets, although I had to pull myself back (and didn’t sometimes) from googling various combinations of the words in a question to see if it was a standard proof that I could get off ProofWiki or something. On the positive side of this: sometimes questions on example sheets can see odd and contextless, but in looking online a bit sometimes you find exactly what you are proving and how that fits into things wider than exactly what you’ve been lectured on: which is great and often also helps you to understand the problem.

I’ll say one thing though: as a supervisee in my first year, I did say ” I didn’t know how to get started on this”- but I didn’t mean what you say it means. Often what it meant was “I can see how the argument for this goes, I can see ‘intuitively’ why it’s true and if I could just get that first foothold in, I think I could do it; but for some reason I have a mental block which has just become more entrenched, so that now the problem is a Rubik’s Cube which somebody has poured bicycle oil all over.” Which isn’t really the same as “too lazy to really start”, IMO.

September 23, 2011 at 11:49 pm

OK, I’ll weaken my claim to this: “I didn’t know how to get started” often means what I said it means. I know that is true because very often a little questioning reveals that people who say that have made no effort to remind themselves of the concepts or results that are clearly relevant to a question. But if for some people it means, “I sort of see why this is true but I just don’t know how to convert my thoughts into a proper proof,” then I hope those people will get plenty out of these posts, since that’s exactly the kind of thing I want to try to convey.

September 23, 2011 at 10:25 pm |

This looks like it’s going to be very useful! I’m a Part II Mathmo and I’ve been asked to be talk (briefly) at the Part IA induction lecture on 5th October about ‘what I wish I knew when I was in first year’. Would it be okay with you if I mention your blog, and specifically this series, in the talk? (No doubt you have other means of publicising the series, but it seems to match what I’ll be talking about very well.)

September 23, 2011 at 11:22 pm

It would be more than OK — I’d be very grateful.

September 23, 2011 at 10:37 pm |

Not sure if this would aid in what you are doing Timothy but Andrew Wohlgemuth has some great material on teaching how to do routine proofs: http://andrew-wohlgemuth.com/

Also his book “Introduction to Proof in Abstract Mathematics”, by Dover Publications, copyright 2010 is excellent.

September 23, 2011 at 11:41 pm |

Here is a parable:

When I was reading this post I was not paying full attention.

When I got to “The cat sat on the mouse”, my eyes jumped straight to the next sentence after “The cat sat”,

thinking that the details of the quoted sentence were unimportant.

So when I got to “the ‘ouse’ that follows is quite a surprise”, I had to stop and think “Why should that be a surprise? It didn’t surprise me.” Eventually I worked out that it had not surprised me because I hadn’t read it.

One reading of this parable is that it is nice for lecturers to explicitly highlight the surprises, especially if they lie in apparently arbitrary or boring details.

(Also for listeners it is probably useful to try and record when the lecturer points out this sort of thing, even though they may do it as an informal remark not written on the board.)

Another reading of the parable, addressed to the reader of lecture notes, would support your point “Don’t be too passive when reading your lecture notes.”

September 24, 2011 at 12:49 am |

This is all excellent advice. Some other related advice, most of which is probably almost too obvious to give, but would have helped me at the start of my time as an undergraduate:

– supervisions are there for the benefit of the undergraduates. They aren’t some elaborate attempt to humiliate you by demonstrating your ignorance; they’re there to give you the chance to learn what you need/want to learn. In particular, trying to give the impression of deep thought for a couple of minutes when asked a question, when you actually don’t understand it, is a waste of time. Similarly, so is handing in answers in which you try to obfuscate some gap. In fact, the best thing to do is to indicate on your solutions which ones you think might be in any way suspect or lacking, and come to the supervision with a clear idea of what you want to cover.

– on a related note, it’s painfully obvious to a supervisor when you’re trying to give the impression of deep thought but are actually waiting for a respectable time to elapse before admitting that you don’t know the answer (I learned this the first time I gave a supervision, and I cringed for all the times I’d done this myself).

– if you can make yourself do it, spend (some of) the afternoon going through the morning’s lectures, writing the notes out again but ensuring that you’ve understood every line. It’s amazing how much more enjoyable lectures are when you’ve understood the previous one (rather than the more usual routine of having become lost after a few lectures and then going to 6 weeks of lectures that you don’t understand but just blindly copy down). This makes example sheets much easier, too.

September 26, 2011 at 10:18 am

lol Nice point about the faking deep though thing. I too cringed when I was on the other side of it.

September 24, 2011 at 1:18 am |

It might be a good idea to be consistent between “Stackexchange” and “Stackoverflow”. I *think* the former is correct. In your post you use both.

Also, you might view this with amusement: http://www.reddit.com/r/math/comments/kp6ew/welcome_to_the_cambridge_mathematical_tripos/c2m51v7

September 24, 2011 at 3:51 am

Thanks — corrected. (That is, I’ve corrected what it was in my power to correct.)

September 24, 2011 at 5:37 am |

I found your advice forthright and aligned with much of what I’ve read others think about how mathematics is best learned. Thinking about how to learn higher mathematics has been covered many times, with respect to

– learning technique (e.g. Exner’s “An Accompaniment to Higher Mathematics”),

– teaching technique (e.g. Polya’s introduction in “How to prove it”),

– mathematical content (e.g Garrity’s “All the Mathematics You Missed”). I hope to get insights into Cambridge’s Mathematical Tripos from what you publish, Thank you.

September 24, 2011 at 6:54 am |

Great post. My thoughts from my experiences on reading it: a nod towards motivation was very important for me. Many presentations of pure mathematics are so abstract it just seems as unmotivated as a sudoko puzzle. I distinctly remember not really getting the point of group until doing representation theory in Part II.

For me, the point where mathematics got really hard was more about the quantity of work rather than how hard each part got (more a problem of breadth than depth). This happened a bit in Part II and very much so in Part III…

September 24, 2011 at 7:42 am |

[i]hey 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]

[i](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.)[/i]

hi, i’ve commented here before.

the first paragraph sums me up totally. i left a-level in 2004 with a set of excellent module results and two grade As at maths and further maths. i had two STEP grade 1s and was thrilled to learn at university that many of the students also at my college hadn’t actually made their STEP offer and had got in by the skin of their teeth.

i never hit the ground running. i was so used to being excellent at maths i found it very hard to accept that i was struggling entirely. i got a third in my first year and i’m pretty sure i only just managed that because there was an almost a-level style mechanics question on one of the papers and i was lucky enough to have studied mechanics to M4 at a-level.

i started the second year depressed, demoralised, and trying to avoid maths. i got threatened by my college that they were going to kick me out and it was my friends who were the only ones who were looking out for me. eventually it took working 15 hour days for 3 months and a lot of work on catam (which i relatively enjoyed, and didn’t involve memorising proofs) that got me to scrape a 2:2 in my second year.

my third year was relatively more successful because i could pick more courses that i enjoyed but i was still vastly out of my depth. i need to look back to get confirmation on this, but i think i scraped a 2:2 with one alpha. again the catam saved me somewhat.

i don’t know whether a cambridge education has altered my life although i do feel more enriched by my experience there. i don’t neccessarily believe the cambridge system does enough to look out and help people in my position and i think often it’s a system that works for the high fliers but there are a large chunk of students who need help in trying to think in the right way and lecturers and supervisors are too busy to assist in this. my college’s solution (and i’d rather not say which college it was) was a constant assertion that anyone who was failing wasn’t working hard enough, but mentally it is incredibly incredibly difficult for a student from a normal school to go from an absolute high flier to bottom of the class and struggling. supervisions are great if they are effective, but the supervisors i had really needed a repeat session with me after the original supervision… and hour wasn’t enough for all the extra thinking i needed!

now i teach maths to key stage 4 and key stage 5 students – the end of gcse and a-levels. i know that my experience of struggling has helped my empathy. i know that what i didn’t like about the cambridge system i try to rectify in my own teaching.

when i left university i hated maths. after 3 months of working answering a telephone i realised i didn’t hate maths after all and became a maths teacher. what that taught me was that if you’re not in a job where you get to think all the time you get very bored.

lastly:

i wish i could have those three years again.

September 24, 2011 at 10:17 am |

As someone who will very shortly be supervising first year mathmos (not for pure though!), I’ll definitely be pointing them in this direction! I *wish* I had been able to see a post like this when I first started. Pure maths at university confused me greatly, and it wasn’t until my third or fourth year, by which time I had completely abandoned dpmms for damtp, that I started to get a feel for what it was that I’d missed understanding in IA and IB. I’m really looking forward to reading the rest of these!

September 24, 2011 at 11:48 am |

As someone just going into Part III, I’d have loved to have read this as a fresher. By the time I took my finals I was studying almost entirely as per this advice, but it took two and a half years to work out that it was by far and away the best way to go.

It’s amazing how hard it is to believe that reading through your notes/making key points sub-notes does actually save time until you’ve been doing it for a year, but it’s a habit well worth getting into from first year!

September 24, 2011 at 10:34 pm |

This post is excellent, mainly because it makes clear that an ability in maths is not just something that a few lucky people have, but that it can be gained by hard work.

My experience is basically this: I went to do Cambridge to do maths having always done well relative to the people in my school (S,S at STEP), and then underacheived and wasn’t able to stay for Part III. Apart from my general laziness, I’ve always put this down to a lack of competition when I was at school. I found maths at school easy, which probably means I wasn’t exposed to hard enough problems. One of the things I’ve found is quite common in people I know have done well at Cambridge Maths is that they’ve all been exposed to some kind of mathematical competition whilst at school(and so they have had to work genuinely hard to do well).

This series will also address a concern that my brother has always brought up about maths when I’ve tried to teach him some bits. He always complains that mathematicians take some pleasure in making their subject opaque. In a way, this is true; maths textbooks are almost always a rite-of-passage one has to drudge through in order to understand something that can sometimes be explained much more intuitively. Some of these books are described as ‘classics’ by established mathematicians.

It’s probably a bit of showmanship to make learning maths appear as though it’s just a product of ‘talent’. Even in the first year at Cambridge I remember it almost being a competition to see who could complete the problem sheet in the fewest hours (when these were the same people who had spent the entire holidays reading the lecture notes). One person I knew used to try to compress his notes for each hour-long lecture in one line.

It’s because Maths is less commercial than other arts that there is this problem. Everyone accepts that musical prodigies practice a lot, but if you’re very good at maths, it’s usually attributed to a mysterious ‘genius’.

September 25, 2011 at 11:10 am |

“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.”

At the University of Manchester, we had supervisions in a group of three students but only in the first year.

September 25, 2011 at 12:46 pm |

I did some analysis and algebra courses after I retired. While I did relatively well and enjoyed a number of the courses, With one notable exception, I was struck by how poor the lectures were; the lectures were in general poorly organised as a whole and the lecturers in a world of their own. The worst course without doubt was Analysis III (honours) where the lecture banged on about Borel sets and Algebras for weeks and, after about 16 lectures, one of the brightest people I have ever met said to me “What’s going on here?”.

Luckily, it sort of gelled by the end. I complained to another lecturer and was allowed to sit in on the same course given by him, the one lecturer I found worthy of his job. Borel sets and Algebras were confined to the last lecture, where they belonged. He said time and time again “the idea behind this is” or “the motivation behind this” is and many, regulalry tried to give a visual idea of what was going on.

Maths books are too often like computer manuals; a series of prompts for those who already know the subject being discussed. In printed lecture notes, it helps to be expansive early in the course. Also, pure mathematicians should be encouraged to give sheets with a worked example or two of how a theorem or two can be applied to some practical example. I found that would have been very benificial in a number of places.

Lecturers need to remember what is blindingly obvious to them may be opaque to many students.

September 25, 2011 at 11:19 pm

Tim Bourke’s comments remind me of a story, the source of which I have unfortunately forgotten. A professor was going through a proof in the middle of a lecture, and said “using the following obvious lemma … “. A student said “Excuse me, I don’t find that obvious”. The professor broke off lecturing, took up his pad of paper and started scribbling furiously. Ten minutes, several screwed-up sheets of paper and a few swear words later, he put down the pad, smiled and said “It’s alright, it is obvious”.

September 25, 2011 at 1:49 pm |

Great post! I am one of those mathmos who almost got left behind by the sheer pace of the Mathematical Tripos: it took only one term for my supervisors to see through my passiveness towards Mathematics and supervisions, and around the same time for me to be counselled by the Senior Tutor in my college. I worked hard during Lent and Easter terms with the aim of changing Tripos but after scraping a First in IA, I decided to stay on in Mathematics (and was privileged to have sat in your Further Analysis lectures in IB!). I graduated with a 2.1 and am now teaching Mathematics at IGCSE and IBDP.

I wouldn’t have chosen to live any other way. I left Cambridge appreciating the need to not just passively ingest information but to actively question why and how this information is presented, and how it can be applied. (Why is this theorem worded or proved this way? What kind of situations can it be applied to?) I’m not sure if I would have appreciated these truths as quickly or as deeply if I had studied Mathematics outside of Cambridge.

This might come as a sweeping statement, but I suspect most of us who weren’t traumatised by the whole Cambridge experience (mathmos or otherwise) entered the work place quietly confident that if we could survive Cambridge, we could survive almost anything thereafter, so long as we bring with us the intellectual discipline and the work ethic that most of us had developed.

I’ll share your website with students who intend to apply to Oxbridge or study Mathematics at university.

October 22, 2011 at 2:07 am

SL- You speak about the Tripos as being traumatising and of intellectual discipline. This is the fear I’ve had about universities like Cambridge. It really is an elitist place isnt it? It only favors those who can produce excellence otherwise you are kicked out no matter how hard you work. I went to a secondary school which emphasised competition and excellence but in the end the pressure made me ill and I dropped out. Even though I believe Gowers idea that hard work is necessary for success, I am less certain about the method used by say the Tripos. That kind of method is not suitable or favourable for all students.

November 16, 2011 at 9:16 pm

I am just starting my 2nd level OU maths modules. A lot of the study techniques don’t apply to the OU system, no lectures, no lecture notes etc. however a lot does. It is particularly interesting and useful to see how hard a lot of good mathematicians find maths.

Excellent blog, very helpful, thanks

September 25, 2011 at 9:24 pm |

I have just obtained a M.A. in logic in Paris (so please excuse my english…) and this post is a real revelation for me: real mathematicians (I’m really more a philosopher) do have the same difficulties that I have to proove theorems… I can’t wait to read new posts!

September 25, 2011 at 10:15 pm |

[…] Gowers just started a new series of blog posts for first-year mathematics students. While the blog posts will be centered around Cambridge’s […]

September 26, 2011 at 10:30 am |

Great idea.

I am going to enthusiastically recommend this series to any first-years I come across in October. And I too am prepared to split an infinitive for the cause.

September 28, 2011 at 12:16 am |

[…] solution and Nuit Blanche examined an extraordinary claim. You might also want to keep an eye on Gowers’s Weblog starting a new series for beginning maths students to overcome the restrictions of traditional […]

September 28, 2011 at 7:33 pm |

[…] part (i) difficult, then you are not applying one of the pieces of general study advice I gave in the first post of this series. Advertisement Eco World Content From Across The Internet. Featured on EcoPressed Who Are […]

September 28, 2011 at 10:50 pm |

This will be extremely helpful to the students. Thank you, Gowers!

September 30, 2011 at 12:18 pm |

[…] https://gowers.wordpress.com/2011/09/23/welcome-to-the-cambridge-mathematical-tripos/#more-3154 […]

September 30, 2011 at 12:24 pm |

[…] Hint: if you find part (i) difficult, then you are not applying one of the pieces of general study advice I gave in the first post of this series. […]

September 30, 2011 at 12:30 pm |

After reading the whole post, I took a deep breath and then thought about the differences and similarities between a Cambridge student and me. Feeling lucky that I have started learning mathematics in the age of internet. Atleast, during this journey, I’ll assume myself to be taught by Prof. Gowers.

September 30, 2011 at 9:43 pm |

[…] By Gowers […]

September 30, 2011 at 10:31 pm |

Interesting facts, good advice. Thanks. Also thank for your patience in writing.

October 2, 2011 at 7:28 pm |

[…] Then you’re in luck! Fellow math blogger Timothy Gowers and professor at Cambridge University has started a series of posts on his blog for first year Cambridge math majors. Here is Dr. Gower’s first post to get you started. […]

October 3, 2011 at 3:57 pm |

[…] By Gowers […]

October 6, 2011 at 7:26 am |

[…] as he has written only a few articles after I joined the web. But as he started his series on ‘CAMBRIDGE TEACHING‘, I have become a regular reader of his weblog. One math student must read his posts on […]

October 10, 2011 at 7:32 am |

[…] (This is a comment about this post by Professor Timothy Gowers, in a series for mathematical undergraduates he has started there.). […]

October 12, 2011 at 11:02 pm |

[…] thread begins with his introduction, and then proceeds with a series of posts on basic logic. His latest post is about injections and […]

October 24, 2011 at 8:20 am |

I’ve just stumbled across these posts and think that they are a great idea. I’ve two comments:

1. Any chance of getting a-hold of the sources? I’d love to take the ideas and remix them in a suitable fashion for my students, or at the very least be able to provide them in a fashion that I can add some commentary (as a simple example, things about supervisions are not relevant for them). I’d give full attribution and make it clear what was modified. A lot of my students are encountering “real” mathematics for the first time and much of what I’ve read so far (I’ve a lot of catching up to do) would be relevant for them.

2. I completely agree with the sentiment that doing mathematics is much more teachable than we think (or than we give the impression of thinking). In an attempt to make it a little clearer to my students, I wrote something on proofs: http://mathsnotes.math.ntnu.no/mathsnotes/show/proof

November 1, 2011 at 7:29 pm |

Sincere Gratitude Sir.

November 2, 2011 at 8:35 pm |

I wish there were more thought-*proving* articles in the news. I guess that means that I am the first who has read until the end

Personally, I would like to know if you have any feedback from older students how they perceived your admonishments. I agree with most of what you say, but I find it difficult to convince students. Maybe in the future, I will tell them how important and famous you are and then cite you.

One little critique: The cat sentence is the absolute opposite of culture blind and its meaning rather mysterious. I know lots of English idioms and memes, but as a non-native speaker, English children’s culture is something that I have no access to and I have never seen this sentence in any form before.

This might be a more appropriate example to catch spies who pretend to be English.

November 2, 2011 at 11:02 pm |

[…] he has been posting on topics that are quite close to the content of our course, starting with his Welcome to the Cambridge Mathematical Tripos. I recommend that you take a look at his recent postings (and their comments), as one of […]

January 8, 2012 at 10:24 pm |

Dear Professor Gowers,

I’m a graduate student at Stanford who did the Cambridge Maths Tripos in 2005-2009. Tomorrow I start teaching an introductory linear algebra and multivariable calculus course; your posts will really help my students learn to write proofs. Can I please reproduce some sections of your posts to distribute to them? I will of course credit you and give them the url for this blog.

Amy

January 9, 2012 at 8:33 am

Yes of course, you’re welcome to do that.

January 26, 2012 at 1:10 pm |

I am reading maths as a hobby and my contact with the subject is provided by old textbooks- for example ‘Further Elementary Analysis’ and ‘Further Mathematics’ by R.I. Porter, ‘A Short Table of Integrals’ by B.O. Peirce (sic) etc. In a way, this makes me a kind of ‘Fred Dibnah’ of maths- all steam-powered tinkering (/bodging?). DIY.

I haven’t got a clue what I’m doing- but I can’t half talk- so perhaps that makes me the Carl Pilkington of Mathematics, instead.

This may make me a tad laughable to some, but I think there’s summat in my approach, particularly for those who are about to start on their course and have that ‘don’t know what you don’t know’ thing. Seek out the mouldy old 60’s textbook to ‘S’- Level. Throw that calculator away. Okay, put it in a safe place. All you’ve got is a bit o’ paper, a pen and the sweat o’ yer brow! “Oh, &%**%$!” I hear you say. Try it. It’s scary and it’s good.

As for books to help with writing proofs, perhaps I’m underpitching this but there are nice, sympathetic, books out there on the ‘How to do Proofs’ theme – for example there’s a good introductory text on doing proofs by Daniel Solow. I have reviewed it (all by myself, spelling and evrythink) on my Librarything page, http://www.librarything.com/work/175566/book/64258951

February 3, 2012 at 10:23 pm |

i like that sub-topic “can mathematics be taught?” for the comments under has woke me.

March 22, 2012 at 2:01 pm |

Hello, my name is Raymond.

I read the blog’s message, I think it is great.

I am not from Britain though. I am from Canada.

I took some maths courses in a Canadian university.

Currently, I am prparing for a series of actuary exams.

These exams are not for school, they are for a

professional designation. I wrote an exam yesterday —

Probability — I think I flunked it. But I think I will study harder

and take the exam again.

Anyhow, I would like to leave a log at the blog on a periodic basis. This is just to help me keep track on my study progress.

April 25, 2012 at 4:59 pm |

[…] denote the right half plane {(u,v)∈R^2 ∶u>0} First read the following quote from Tim Gower's blog. Supervisee: I found this question rather difficult. Supervisor: Well, what were your […]

June 26, 2012 at 11:32 am |

Dear Professor Gowers,

Thank you so much for this amazing blog and even more for the Cambridge Mathematical Tripos series which I (and many others) have found immensely helpful.

Can you please make such posts for Part IB courses this year?

June 26, 2012 at 11:58 am

Many thanks for your comment. I’m reluctant to promise anything at this stage, in case I don’t fulfil the promise, but I can at least say that I am seriously considering writing some IB-related posts for next year.

June 4, 2013 at 5:36 am |

[…] 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.) —TIM GOWERS […]

December 8, 2013 at 7:10 am |

[…] as he has written only a few articles after I joined the web. But as he started his series on ‘CAMBRIDGE TEACHING‘, I have become a regular reader of his weblog. One math student must read his posts on […]

July 10, 2014 at 5:48 pm |

The link for the web page of Mark Meckes is broken. Can you fix it?

July 10, 2014 at 9:50 pm

I’ve fixed it. Thanks for pointing that out.

July 20, 2014 at 2:46 pm |

I have landed late on this excellent blog via a more recent posting so I hope that I will not be interrupting the more significant flow from current students. In this spirit you will perhaps allow me to embrace the implied acceptability of a more philosophical appraisal of the study of mathematics and the varied pleasures that it can bring. Inevitably this is based on my own experience over a number of years as a “non-professional” of the multifaceted mistress that Mathematics can be. If you find personal accounts too tediously self-indulgent, then please read no further.

The journey started at Oxford at the end of the 1960’s (after I had switched from Natural Sciences and thus already a mongrel mathematician). My extremely romantic and neo-Platonic approach emphasised the aesthetic over the technical, i.e. concept over the ability to actually do anything. The one exception being Commutative Algebra where the mechanics did seem to possess an inner harmony and simplicity. Remember this was the 60s! Lectures (when one had nothing else to do) were to be experienced rather than followed and the psychological tricks referred to above were much in evidence at tutorials (as supervisions were called then). Nobody cared too much and I somehow managed a poorish 2nd. Although this could hardly be called a period of study, I was somehow nonetheless imbued with a love of the subject.

In subsequent years, perhaps not surprisingly, I came to regret this naively superficial approach (I was occasionally heard to remark that “education was wasted on the young”). So some 20 years later I embarked on an MSc at the OU (whilst working full-time in industry). This took 6 years, one module at a time, each of which was based on a pretty good textbook and OU notes. However it was the TMAs that were a revelation. With study-time embedded in so much elapsed time there was considerable scope for mulling over a problem, whilst doing the washing-up say, until some key aspect was revealed. (More in keeping with Grothendieck’s description of the action of the “Rising Sea” than Poincare stepping onto a bus). The crowning pleasure was then the leisurely crafting of the technical solution to be as clear and elegant as possible. How different from the pressures of the Tripos!

Whilst the OU experience used bits of mathematics as a vehicle for intellectual satisfaction, it did not provide that coherent experience of the modern subject to undo my youthful profligacy. So in 2009 (aged just 59) I took a sabbatical from work and enrolled for Part III of the Tripos (then designated CASM but later commuted to yet another degree of MASt. The fact that my son was at Kings at the time gave an irresistible impetus to this venture). This was then a third type of study experience, focusing almost exclusively on lectures. With memory and speed of thought being laughably poor by now, keeping up was out of the question. The idea was to take comprehensive notes whilst hanging on to a sense of the direction of travel. In addition the wonderful library at the CMS provided for as much background browsing of classical texts as you could wish. It goes without saying that I was in the fantastically privileged position of this year having absolutely no relevance to my career! (I really did feel sorry for some of the young students, several of whom already wore look of years of stress and singularity of focus).

One discipline that I imposed upon myself was that I would try and somehow pass the exam at the end. Once again that frisson of trying to store away retrievably a few bits of technique, underpinned by a modest essay meant that I scraped through, very near the bottom of the list. The ceremony at Senate House with the doffing of caps and fluttering of lists was a definite high point as I waited with almost as much excitement as those hoping for distinction and a road to PhDs.

Needless to say, I found the material in the various (pure) courses that I took impressively difficult and supposed a lot of technical knowledge that I didn’t have. No matter. My lecture notes have fulfilled their purpose and five years into retirement I am still working on them, back-filling where necessary (often via Wikipedia and the produce of a carefully constructed Amazon wish list over the years) and sometimes enjoying an “ah ha!” moment after a few days contemplation.

So I offer this personal endorsement of the joys of mathematical study, certainly in my case for the less able, as a counterpoint to the hothouse of full-on competition. Yer makes yer choice and then takes yer pleasure where you can find it.

March 17, 2015 at 12:04 pm |

Sorry. I don’t find this dump of ideas helpful at all.

If it distracts a student from doing his daily work of attending to the relevant exercises to be done,, it is extremely unhelpful.

Certainly the professor has a lot of good ideas. But he also has a couple of strange ideas which I do not accept..

gowers Says: September 23, 2011 at 11:45 pm

When I was at school, I had a fit of enthusiasm, obtained a copy of the Cambridge schedules, and tried to read up on several of the first-year courses. I certainly didn’t manage to cover any of those courses in full, but nevertheless the result was that I was pretty well prepared for my first year. And the result of that was that I managed to delay the moment where it all got hard to about the beginning of my second year. I’m answering this question in some detail because if I simply say “The beginning of my second year” and you find things tough earlier than that, you should understand that that isn’t necessarily a problem. It just means you are discovering one year earlier than I did what learning mathematics is really like.

It seems clear to me from what gowers says that he found an effective method.of preparing for first year. Why doesn’t he think that this is “what learning mathematics is really like”? Why doesn’t he thin that the same method could be used to prepare for second year and third year?.

Richard Mullins

September 2, 2015 at 6:00 pm |

[…] I could make many more additional helpful remarks about how to understand the material.’ (https://gowers.wordpress.com/2011/09/23/welcome-to-the-cambridge-mathematical-tripos/) [11] Their impact on awareness, however, has been invaluable. [12] […]