This is a (hopefully constructive) protest article about the Mathematical Tripos, or rather the terminus it currently leads to. The reader who finds subjective polemic of rather limited interest is encouraged to stop reading now, with my apologies.
In classic soap-box style, one should begin by quoting a famous but long dead socialist. The title arises from the following passage from Act I of Bernard Shaw’s play ‘Mrs Warren’s Profession’, an Unpleasant play which was immediately censored when it was written in 1894: its message is that women become prostitutes because of hopeless poverty and not, as was conventionally considered, wanton immorality. Mrs Warren’s ‘respectably’ brought up daughter Vivie, the other main character, has just made the acquaintance of Praed, a middle-aged architect:
PRAED: ... It was perfectly splendid, your tieing with the third wrangler. Just the right place, you know. The first wrangler is always a dreamy, morbid fellow, in whom the thing is pushed to the length of a disease.
To Praed’s shock, Vivie then explains that she only did so because her mother had bribed her to do so for £50, and this was certainly too little.
PRAED: But surely it’s practical to consider not only the work these honors cost, but also the culture they bring.
VIVIE: Culture! My dear Mr Praed: do you know what the mathematical tripos means? It means grind, grind, grind for six to eight hours a day at mathematics, and nothing but mathematics. I’m supposed to know something about science; but I know nothing except the mathematics it involves.
People often look back on dreadful old Cambridge customs as quaintly romantic, like the annual reading of the results ceremony in the Senate House: great fun if you can close your ears to the sound of crying. (Although that’s not nearly as bad as the gathering of would-be research students in DPMMS common room immediately after, where ashen-faced people walk one by one into an office and silently emerge, with or without a blue piece of paper.) The Tripos has been reformed a little in the intervening ninety-six years; Cambridge has finally conceded that women may be capable of thought, and nowadays the pretence is maintained that nobody is told exactly where they came in the exams. But, as I hope to demonstrate, the old profession of wrangling for places is very much alive.
At present, after the three-year Tripos is complete, there is a fourth year called Part III, which is exceptionally funded directly by the State so that quite a large number of people are able to read it. It is primarily intended as a preparation for research, although quite a number of people take it in order to spend another comfortable year at Cambridge (despite the miscellaneous selection of diplomas ideal for that purpose). Students register with either the Pure or Applied departments, which have very little to do with each other (so that pure and applied lectures on dynamical systems may clash, for example!). This distance is such that, having been Pure, I have hardly any idea of the situation in the Applied department apart from the general impression that it is more organised.
Teaching is done by offering a choice of a couple of dozen lecture courses on specialist subjects, usually so that members of the department give courses on the subjects they are experts in. Some courses have a few example classes as well, and these can be fairly helpful, but there aren’t any supervisions and in effect lectures are the only source. There is little contact with the departments; Part III students don’t often go to seminars, for example, as would be thought quite normal for first-year graduates at most universities. In effect, for someone going on to a doctorate, this year is the most isolated and cut off of all seven or so; which is unfortunate given how easy it is to be at a loose end during it. Having said this, students are provided with careers advice and are usually made welcome when they go and talk to people in the departments.
At the end of the time, candidates choose six of these courses to take examinations in, although they have the option (which in practice almost everyone exercises) of writing an extended essay in place of one of them. This can be quite rewarding and at least feels like real mathematics.
Unfortunately, many of the courses are very specialized, and the degree to which you have to focus on them is such that you never get time to look at anything else. The effect is like looking through a telescope in five or six different directions and pretending to have seen a landscape. What is perhaps worse is that after this process, you may have the mistaken impression that you are now too specialized to be able to look at any of the things missed out: in Dr Körner’s memorable observation, in a few years you find yourself at the top of an ever narrowing gum tree.
But the really dreadful thing about Part III is the examination itself. The courses are (rightly) about difficult things and most of them are lectured at great speed so as to pack a large amount of material in. The result is that most reasonable questions that could be posed about them might take a research student a week to work out, so that it’s quite hard to set questions which test understanding. In consequence, most of the examinations test rote learning on a grand scale; of the papers I took, only small parts of two questions on one of the papers had answers which did not consist of recital of the appropriate passage from the lecture notes. The hardest part of programming a computer to pass the course would be teaching it to read the rubric and work out what combinations of questions are legal. There are some papers which call for actual thought, but they are a somewhat dangerous proposition since you are bound to get less marks on them than others will get on the memory tests: which matters a great deal if you want to stay in Cambridge, since research students are selected solely on the basis of the mark ordering. (Other universities have been known to treat the results as meaningful, as well.) In general the papers vary greatly in difficulty which must make it difficult for the examiners to normalize.
In any event, the amount of theory required is such that there is no way to stop and try to think how to derive things, any more than school Latin and a rough knowledge of the plot is enough to be able to translate the Aeneid at writing speed. So you need to memorize about 120 lectures of writing, basically symbol by symbol. It might be thought that this is so large a task that you would be forced to understand it pretty thoroughly. There is some truth in this; then again, you also end up learning exactly where every lemma lies on the page in your notes. A year later, I could still leaf through these pages in the mind’s eye. Also, you understand what is in the notes, but little else; it is mostly theory you have no motivation for. To give an extreme example, immediately before the Commutative Algebra paper, several of us who could all have recited all the main theorems and proofs of dimension theory of rings had an argument about whether there were or were not any rings whose dimension was not 1.
Miss Warren’s estimate of the amount of time this takes is definitely on the conservative side. Her modern counterparts resort to devices such as tieing themselves to their chairs with string, wearing earplugs to prevent them from hearing any conversation they would be tempted to join, keeping appalling graphs of their daily twelve-hour workloads, and so on. (In the end people really do become dreamy and morbid; I can distinctly remember holding a toothbrush on the night before one of the papers and not being entirely sure what to do with it.) When it is over, you tend to recoil so much that it may be months before you are prepared to look a maths book in the contents page again.
The above notwithstanding, Part III is a way of learning some useful maths and effectively getting an extra year’s funding before beginning research, and it has some good features, such as the essay. Also, it must not be thought that the Cambridge departments are indifferent to the lot of their students. Anyone who has ever attended the (variably effectual) meetings of the Faculty Board will realise that many lecturers are genuinely concerned about reform. At present, a large-scale reform of the undergraduate Tripos is under way, which aims to remove some of the arbitrary division of teaching between Pure and Applied mathematics (vector spaces and differentiation are Pure, but distributions and vector fields are Applied, and so forth) and to make it accessible to more A-level students. Much time and effort has gone into this thoroughly worthwhile aim. It was interesting to note that in the early stages, many people were very concerned that at almost any cost, Part III had to be maintained, that being the jewel in the crown of the Cambridge Tripos.
Also, two years ago an attempt was made by the Pure department to reform the problem of overspecialized courses. It was decided that the old system of having many courses which only one or two people would study, would have to go; a broad range of introductory courses would be added. (This did not stop them from quietly allowing a few people to take specially set second papers on combinatorics and category theory, however; although as the existence of these papers was not directly communicated even to those who had attended the first courses, this number might have been artificially few.) This was partially successful, although many of the lecture courses seemed to be still fundamentally about specific theories in great detail without much reference to their relevance to each other. The problem of overspecialization in effect remains, except that now there are fewer courses available, it is no longer always possible for this specialization to be in subjects people intend to work in.
Having said this, complaining to the powers that be about Part III is like the guests at Fawlty Towers complaining about Manuel to Basil Fawlty, who replies “I know, he’s awful, isn’t he?” Owing to the rather odd election rules for student representatives on the Faculty Board, it is essentially impossible for a Part III student to be one (if you are an undergraduate but graduate part-way through the time, you must step down; if you are a Part III student, you have no certainty of still being in Cambridge for any more than one term from when the elections take place); so it is a cause with no obvious spokesman. But at least there will be an excellent opportunity to do something about it when the first intake to the reformed Tripos reach Part III.
As for how to reform it, at this point I slide yet further into subjectivity. In many ways a useful way to spend the year would be to take another selection of Part II courses passed over in the year before (for example, General Relativity and Linear Analysis is not that unlikely a combination of courses, but sheer pressure on timetables usually clashes them), but this is unrealistic. At the other extreme, moving over to a full MPhil course would result in the exceptional funding status of Part III being lost, which would mean that very few people could take it and it would not be feasible to offer such a range of courses. I propose that the balance between course-work (i.e. on dissertations of some kind) and examinations be drastically shifted, possibly so that the examination part could be just on a few of the ‘core’ courses; other lecture courses would be to provide background, understanding of which would be necessary to work on the dissertations part of it, as well as being useful to research students. Ideally, students would work on several different essays, but on different subjects.
Shaw would probably have considered these proposals dismally unradical, but the Faculty Board (or, more importantly, the staff meetings) may well disagree. Most of my argument, which has perhaps itself been pushed to the point of a disease, has been to propose reform just to make the course more humane for students, but the departments have just as much to gain: a more sensible system for selecting research students and (more optimistically) some integration of all these potential future mathematicians into the institutions they nominally belong to.
Reproduced from Eureka 51 pages 36-39.
HTML conversion Copyright © 2004 The Archimedeans.
Return to Eureka 51 home page
Return to Eureka home page
Return to Archimedeans home page