As I said would happen in my post about a possible approach to teaching maths to non-mathematicians aged 16-18, I went last Wednesday to Watford Grammar School for Boys to try the approach out. The headmaster there, Martin Post, was remarkably helpful and assembled a usefully varied group of pupils, some from his school, some from the equivalent school for girls, and some from a nearby mixed comprehensive school (I wasn’t told which one) whose pupils receive some of their teaching in scientific subjects from Watford Grammar School. What’s more, some of the people there were doing maths and further maths, some were doing just maths, and some were not doing either. The one thing that was not representative about the group was that they were much brighter than average: for example, the non-mathematicians there had been chosen by their teachers as clever people who could have done maths but decided that they were more interested in other things. For most of the rest of this post, I’ll say what questions I discussed and how the discussions went. All but two of them were taken from the list in the earlier post.
I began with a warm-up: I asked how many times a ball was kicked during the recent European Football Championship. Of course, I made it clear that I wasn’t after an exact answer. My first surprise of the day was that pretty well the moment I had asked the question the room went from almost silent to very far from silent: after a while I got used to this. The conversation was, it seemed, entirely about the question at hand. One person asked whether I was interested in the number of passes or the number of actual kicks. I said the latter.
After a while, and when I had managed to quieten everyone down, I drew a logarithmic scale on the whiteboard, going up the powers of 10 from 10 to 1,000,000. I then asked people for their answers and marked them roughly on the scale. There was a significant cluster around 100,000 (by which I mean in an interval from about 70,000 to about 120,000). I then asked people how they had arrived at their answers.
Everybody seemed to know that there had been 31 matches. (Proof: there were four groups of four, so 24 matches at the group stage, and then eight teams left in a knockout competition.) So I wrote that down on the whiteboard. From that point there were two approaches. Unfortunately, I can remember only one of them, which is the one I would have used: to estimate the average amount of time between kicks. The person who proposed this approach thought that was 1-2 seconds, so I asked what we should do with that information. That quickly led to the calculation , which is about . I don’t know whether they were used to doing things like saying “90 is basically 100” but they quickly got into the spirit of it.
Next up, I said that sometimes one reads of prophetic dreams, and that in order to get a feel for how seriously to take them I wanted to know how significant it would be if somebody dreamt of the death of someone close to them and that person died the next day. So I asked a rather vague question: what is the probability of that happening? I followed that up by asking what information it would be useful to know.
One suggestion that came in quite quickly was the probability that someone dies on any particular day. That led to a discussion of whether we were talking about expected deaths (e.g. if someone is very ill or very old) or unexpected deaths. There was general agreement that if someone is expected to die soon, then an apparently prophetic dream would not be too surprising, since both the dream and the death are more likely. So we decided to restrict attention to unexpected deaths. In the end we calculated the probability by simply estimating the number of days in 70 years — we went for 25,000 — and taking the reciprocal.
What else did we need to know? At this point an interesting discussion arose. One person suggested that a relevant parameter was the number of people that a typical person knows. That led to a subdiscussion about how close an acquaintance needed to be to be counted. I suggested that pretty well any identifiable acquaintance would be surprising enough to be worth counting. One other clarification asked for was whether we were talking about reports of prophetic death dreams or true reports of prophetic death dreams. I said the latter.
I haven’t yet said what the interesting discussion was. It was that some people thought it was relevant how many people people typically know, while other people thought it was not relevant at all. The arguments were roughly as follows. Those who thought it was relevant said that if somebody died, then you needed to know how many people they knew if you wanted to assess how likely it was that one of those people had dreamt of their death the night before. Those who thought it was not relevant argued that if someone dreams of a death, then all you care about is the chances that the person dreamt about will die the next day.
Did we have a paradox? Somebody gave a reasonable explanation of why we didn’t, which I made a bit more precise by saying that we were conditioning on two different events (the occurrence of a dream and the occurrence of a death), so it wasn’t too surprising that we had to take into account different things.
But the problem still wasn’t very clear, which itself became clear when people wondered whether we needed to know how many people there were in the whole world. In the end I stated the problem more precisely: roughly how many times per year would we expect somebody in the UK to have a death dream that comes true unexpectedly the next day? We settled on the following method: estimate how many times a death dream occurs per person per year, multiply by the population of the UK, and divide that by 25,000. I asked people how often they had death dreams per year and there were very different answers: some people said they thought 10 was about average and seemed surprised when I said that I thought that for me it was more like one every two years. We settled on 3, which I still found a bit high, but not outrageously so.
So that led to the calculation . I think we had already multiplied 3 by 50,000,000 to get the number of death dreams per year, so what I actually wrote down was 150,000,000/25,000. So I naughtily changed that to 250,000,000/25,000 which gave me 1,000 and then compensated for the naughtiness by dividing that by 2 to get 500. That confused one or two people, so I explained it again more slowly.
While I can remember which questions I discussed during the session, I can no longer remember the order in which I covered the next three. However, I think it was either at this point or after the next question I discussed that there was a short break — not that it is too important. Anyhow, at some point round now I described Terence Tao’s airport-inspired puzzle, which is the following question. You want to get from one end of an airport to the other and your shoelace is undone. If you want to get to your destination as quickly as you can, is it better to tie your shoelace when you are on a moving walkway or when you are on stationary ground?
The moment I had finished asking the question, the noise switch was in its “on” position again, which was fine. After I had managed to quieten the room down again (which was hardish but not impossible), I took a vote. The options I gave them were “faster on walkway”, “faster off walkway”, and “it makes no difference”. “Faster on walkway” got a few votes, “faster off walkway” got one vote, and “it makes no difference” came top, though not by miles. There were also a number of abstentions.
The next thing I did was invite people to argue for the answers they had voted for. The one person who had voted for “faster off walkway” did not look at all keen to do this, but after a little while someone else said, “You would need to know the speeds of the walkways, your walking speed, and the time it takes to tie your shoelace,” which prompted me to add a “not enough information” option, which was popular, and to which the “faster off walkway” person transferred her vote.
Here are some arguments I collected. In favour of the “on walkway” answer was that when you tie your shoelace you are moving along rather than wasting time being still. In favour of the “makes no difference” option was that wherever you tie your shoelace it takes the same amount of time. And when I tried to elicit an argument for the “not enough information” option, I failed: I asked under what circumstances it would be better to stop on the walkway and under what circumstances it would be better stop off the walkway, but nobody had much to suggest. I also explained that while knowing all the various speeds and times is sufficient to work out the answer, it isn’t obviously necessary: for example, if you are deciding whether it is better to use the walkway or not use it, you don’t need to know how fast it is.
A notable aspect of the discussion that permeated the whole thing was that people were keen to introduce non-mathematical considerations. For example, they (rightly) pointed out that it would make a difference if the airport was crowded: if the people just ahead of you on the walkway are stationary and you can’t get past, then obviously that’s a good moment to tie your shoelace, and if the people just behind you are in a hurry, then you will annoy them by stopping. So I asked them to assume that the airport was deserted. Other assumptions that had to be spelt out were that your walking speed is unaffected by having a shoelace undone and that there isn’t a robot sensor on the walkway that detects that you are tying your shoelace and stops it. Later on I found out who the mathematicians were and who the non-mathematicians were. It turned out that many of these objections came from the non-mathematicians. They were very helpful to the discussion, since one of the points I wanted to get across was that mathematical modelling involves choosing appropriate simplifications. One of the non-mathematicians begged me to say what the answer was (well before I wanted to do so) so the question had clearly worked its magic.
If I had been discussing this question as part of a genuine course, I would have waited much longer for someone to come up with a good solution. But as I wanted to get through several questions, I hurried things along a bit. I started by offering the advice to consider extreme cases. To illustrate what I meant by an extreme case, I suggested that maybe the walkways could be very very fast. (Actually, I think that would have helped people guess the answer, but it wasn’t the extreme case I had in mind.) Pretty soon after that, someone suggested considering the case where someone starts tying their shoelace just after getting on a walkway. Since that was exactly the case I was interested in, I basically gave away the answer at that point by suggesting also considering the case where someone starts tying their shoelace just before getting on the walkway. Most people could see that that was a proof that doing it on the walkway was better. One or two had a bit of trouble still, so I drew some pictures of two people and what would happen if one stopped just before and one just after getting on.
If I’d had more time I would have tried to get everyone to understand why the “you spend the same amount of time wherever you do it” argument gives the wrong answer.
Another thing I did at around this stage was what I call the dividing-up-the-pot game. I first asked question 41 from my earlier post, which is this. You and a friend are out for a walk, when you are approached by a stranger, who offers the two of you £1000 on one condition: that you agree how to split it between you. After establishing to your satisfaction that you are not about to be kidnapped, you propose a 50-50 split to your friend. To your astonishment, your friend insists on receiving £900 with only £100 going to you, and appears to be prepared to lose all the money rather than accept anything less than this deal. What should you do?
The main person to make a suggestion suggested that you should call your friend’s bluff, since when the stranger starts to walk away, your friend would be bound to say, “OK OK stop! I’ll settle for £500!” I wasn’t sure what more to say at this point, so I told them that they would now have a chance to see for themselves. We would have five rounds, in each of which people would be randomly paired and would have the chance to share ten points, on condition that they could agree how to do so. At the end of the five rounds we would see who had the most points. To do the random pairings I had brought along (part of) a pack of playing cards, which I shuffled and dealt out once per round, and each person was paired with the person who had a card of the same colour and same denomination.
It took a while to get through the rounds, but once it was finished, I asked whether anyone had 30 or more points. Several did. 35 or more? Nobody. 34? Yes, one person.
I then asked people what their strategies had been. To my surprise, several people had used randomized strategies, the main one being to decide by the toss of a coin (or in one case stone, scissors and paper) who would get all the points. I asked one person why he had adopted that strategy. His response was that the main aim of the game was to come top, which one couldn’t do by sharing points out equally and couldn’t easily do by insisting on lots of points each round. So he judged that the all-or-nothing strategy gave him his best chance of coming top. He turned out to be one of the people doing maths and further maths. I think his strategy got him 30 points. The person who got 34 points got them by, as he put it, “threatening his opponents.”
I then started a discussion of what possible other strategies there might be. I had planned to think about a population of strategies and how they would perform against each other, making a link with things like the evolution of morality, but after a few seconds I aborted it because I thought it would be too long and complicated. But this was something else that I would definitely have tried to do if I had devoted more time to this one question.
Similarly, I had a brief discussion of what real-life situations might be reasonably well modelled by this game. There were some suggestions that I didn’t like and now can’t remember. (The problem with one of them was that there was no analogue of getting zero if you can’t agree.) There was one about haggling, which seemed fine. I gave the example of the Conservatives and Liberal Democrats drawing up their coalition agreement two years ago, and said that any negotiation where there is a range of outcomes that would benefit both parties could be thought of as an instance of this game. If I had had more time I would have held back for much longer before giving these examples.
Next (or possibly previously), I asked question 37(i), which is the following. In several parts of the UK the police gathered statistics on where road accidents took place, identified accident blackspots, put speed cameras there, and gathered more statistics. There was a definite tendency for the number of accidents at these blackspots to go down after the speed cameras had been installed. Does this show conclusively that speed cameras improve road safety?
The same person who argued for the randomized strategy in the negotiation game basically knew the answer to this question already. He said no, since if you pick out the extreme cases then you would expect them to be less extreme if you run the experiment again. I decided to move on quickly from this question since there wasn’t a lot more to say. But I told people about a plan I had had, which was to do a bogus telepathy experiment. I would get them to guess the outcomes of 20 coin tosses, which I would attempt to beam to them telepathically. I would then pick the three best performers and the three worst, and would toss the coins again, this time asking the best ones to help me beam the answers to the worst ones. People could see easily that the performances would be expected to improve and that it would have nothing to do with telepathy. One person said, “Oh I want to do that,” of the experiment.
Penultimately, I discussed the following table, slightly modified from an example of Joseph Malkevitch. There are five options, and the orders of preference amongst 55 people are as follows.
14532 — 18 people
25431 — 12 people
32541 — 10 people
43251 — 9 people
52431 — 4 people
53421 — 2 people
Which option should the group go for? (I gave examples of different film genres for the options in a key below, but we didn’t use those.)
I asked for relevant observations. The group seemed pretty clued up on voting methods, so I was soon writing up that option 1 would win under first past the post, and someone calculated that option 3 (if I remember correctly) would win under the alternative vote. I then suggested doing pairwise comparisons, and was very gratified when everyone thought that transitivity was obvious (though one person did say that we should consider not just the preferences but the strengths of the preferences), so I could surprise them by showing that it failed in this case. I then gave the simple 123/231/312 example. Some people were keen on the system where you award 5 points for a first preference, 4 for a second and so on. We could probably have discussed this for quite a bit longer. I briefly mentioned Arrow’s theorem, without stating it precisely, and moved on to my final question.
I told them that for the last couple of years I have been reading Proust. (I asked how many of them had heard of Proust. One had.) More precisely, I have been reading Remembrance of Things Past, Scott Moncrieff’s translation of Proust. The edition I have is in twelve volumes, and my practice has been to read one volume of Proust for every two or three other books I read (my reading taking place before I go to sleep at night). I’ve now read eight of the volumes, and on page 25 of the eighth, I came up against this sentence. (I didn’t tell them that the eighth volume was the second half of “Sodom and Gomorrah”, which Scott Moncrieff coyly translates as “Cities of the Plain”.)
But if the course of life, by making Cottard assume, if not at the Verdurins’, where he had, because of the influence that past associations exert over us when we find ourselves in familiar surroundings, remained more or less the same, at least in his practice, in his hospital ward, at the Academy of Medicine, a shell of coldness, disdain, gravity, that became more accentuated while he rewarded his appreciative students with puns, had made a clean cut between the old Cottard and the new, the same defects had on the contrary become exaggerated in Saniette, the more he sought to correct them.
I displayed it for them and asked how we might go about checking that it was syntactically correct, and, even better, actually understanding it. It was projected on to a whiteboard, so I was in a position to annotate it. Fairly quickly the suggestion came in to use brackets. Somebody also said that we should work from the inside outwards. So I asked where people would like the brackets to go. The first suggestion was to have a bracket opening before “because of the influence” and ending after “Academy of Medicine”. That looked right, but turned out to be wrong: the opening bracket needed to be before “if not at the Verdurins'”. However, eventually we sorted it all out, and I got them, with a bit of prodding, to say that the general principle we were using was to put brackets round bits of text if the sentence still makes sense with those bits removed. (It was because it initially looked as though one could jump from “where he had” to “a shell of coldness” that we made the initial mistake.)
If it had not been inconvenient to do so, I would have rewritten the sentence with indentations, as follows.
But if the course of life,
….by making Cottard assume,
……..if not at the Verdurins’,
…………where he had,
…………….because of the influence that past associations exert over
…………….us when we find ourselves in familiar surroundings,
…………remained more or less the same,
……..at least in his practice, in his hospital ward, at the Academy of
….a shell of coldness, disdain, gravity,
……..that became more accentuated while he rewarded his
……..appreciative students with puns,
had made a clean cut between the old Cottard and the new,
the same defects had on the contrary become exaggerated in Saniette,
the more he sought to correct them.
That would have shown more clearly just how many levels of subordinate clauses there were in the sentence. I would also have talked about trees and perhaps broadened the discussion into one about parsing trees for other sentences (which would have required them to decide and try to justify what the natural tree structure of a sentence is).
I didn’t do this at the time, but it is quite interesting to try to rewrite the sentence to make it more comprehensible. Here is what happens if instead of chopping clauses in two, you try to put subordinate clauses at the end. For example, instead of, “He decided, because he was in a good mood, to smile at people he would usually have ignored,” you write, “He decided to smile at people he would usually have ignored, because he was in a good mood.” The sentence above can be reordered as follows.
But if the course of life had made a clean cut between the old Cottard and the new, by making him assume a shell of coldness, disdain, gravity that became more accentuated while he rewarded his appreciative students with puns, if not at the Verdurins’, where he had remained more or less the same because of the influence that past associations exert over us when we find ourselves in familiar surroundings, at least in his practice, in his hospital ward, at the Academy of Medicine, the same defects had on the contrary become exaggerated in Saniette, the more he sought to correct them.
It still isn’t easy, but it’s easier. Does it lose something important? I think it loses some of its distinctively Proustian character — decoding this kind of sentence is part of the peculiar pleasure of reading Proust — but why it should be pleasurable rather than just a nuisance is hard to say.
I might add that the sentence is fairly typical of the whole book, though not many sentences go quite that far. And there are similar phenomena at other scales: paragraphs sometimes go on for several pages, and chapters can go on for hundreds of pages. So not only does one have to read twelve volumes of 350-400 pages each, but the reading is slow going. Out of curiosity, I looked up the sentence in the original French. Interestingly, it uses one pair of brackets, which makes it a lot easier to understand. (What I mean by “interestingly” is that I wonder why Scott Moncrieff decided to do without the brackets.) Here it is. I got it from an online version.
Mais si la vie, en faisant revêtir à Cottard (sinon chez les Verdurin, où il était, par la suggestion que les minutes anciennes exercent sur nous quand nous nous retrouvons dans un milieu accoutumé, resté quelque peu le même, du moins dans sa clientèle, dans son service d’hôpital, à l’Académie de Médecine) des dehors de froideur, de dédain, de gravité qui s’accentuaient pendant qu’il débitait devant ses élèves complaisants ses calembours, avait creusé une véritable coupure entre le Cottard actuel et l’ancien, les mêmes défauts s’étaient au contraire exagérés chez Saniette, au fur et à mesure qu’il cherchait à s’en corriger.