A trip to Watford Grammar School for Boys

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 $30\times 90\times 60/1.5$, which is about $20\times 100\times 50=100,000$. 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 $3\times 50,000,000/25,000$. 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’,
…………….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.

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.

50 Responses to “A trip to Watford Grammar School for Boys”

1. Jonathan Phillips Says:

Take your shoe off and run!

2. Tobias Fritz Says:

It sounds like the approach you tried was based on an open discussion in which you ask people questions and see how they respond. I like that method and frequently use it myself. However, it has certain issues with shy students that do not participate in this due to low self-esteem, uneasiness about just blurting something out, or similar feelings. So, I wonder: how many students in that class did actively participate? And how did you try to encourage the less active ones?

• gowers Says:

Two things to say about that. First, I got answers from a wide range of people (girls, boys, mathematicians and non-mathematicians all participated). That doesn’t mean that there weren’t people who were too shy to answer my questions — there probably were — but at least the discussion wasn’t dominated by a small handful of people.

The one thing I did in order to try to encourage participation was to say that if you are doing mathematical research, then not having stupid ideas is stupid. I drew an analogy with photography: a good photographer takes lots of bad photos and then selects the few that are good; mathematicians do the same. Of course, saying that wasn’t the same as identifying shy people and encouraging them to speak. I hope a more experienced teacher than me would be good at doing exactly that.

• Tobias Fritz Says:

Thanks for the reply! The second point is a very nice analogy to keep in mind for future use.

3. Gil Kalai Says:

Very nice post. I missed 37(i) last time around and it is indeed an interesting question.

“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. ”

It is not clear what “knew the answer” refers to. This question deserves some discussion. Here is a simpler version and and analogous question:

37 (i) simplified version.

“In several parts of the UK the police gathered statistics on where road accidents took place, identified accident blackspots, and put speed cameras there. Can we be sure that this actions will save driver’s life”

This is the same question except that we do not even bother to gather more statistics. And here is a new question:

” After 30% of the students failed the mid-term calculus exam special 2-hours classes for the weak students were added. Can we be sure that these additional classes will raise the level of the weak students?”

What do you think?

• gowers Says:

By “knew the answer” I meant that he made it clear that he was well aware of the phenomenon of regression to the mean.

I like your simplified version of 37(i). It has the potential to lead to the more elaborate version, because people would first concentrate on whether having a speed camera was likely to make people drive more safely, then on how one might test this, and then on whether the results of the test would be straightforward to analyse.

4. Gil Kalai Says:

A few remarks:

1) Everything depends on the fine details. My guess would be that speed cameras which force drivers to drive slowly do save lives, and that giving special classes to weak students will help them. So one “danger” with this question would be that the massage to the students will be that speed cameras are useless.

2) For the original problem: when you say “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?” If the statistical tests were done right, “definite tendency” would already include the regression to the mean phenomenon. So the question is rather unclear.

3) “Showing conclusively” is something we mathematicians worry much more than the rest of the world. It may well be the case, that there are results which look definite, regression to the mean can account for them although it does not look that way, and we are quite happy with the not-conclusive-for-mathematician-but-good enough-for-people-with common-sense outcomes.

4) It is better to make sure that the audience understand what “mean” means, before trying to teach them “regression to the mean”.

• gowers Says:

There’s a distinction between how I explain things in these posts (which are mainly aimed at mathematicians) and what I’d actually say in the classroom. So I wouldn’t use the phrase “regression to the mean”, for example.

The point of the bogus telepathy experiment is to give an extreme example where the entire “improvement” is easily explainable by chance. In a class, I would then probably give a different example where it was clear that chance played no role. I can’t think of a really good example, but a slightly silly one is where you get people to run 100 meters as fast as they can. You then ask them to try again, but the best few have to wear heavy backpacks. I think that would be reasonably convincing, since most people’s 100-meter time varies a lot less from attempt to attempt than the amount by which one would be slowed down by a heavy backpack.

Having established that there are some examples where you can conclude nothing and others where there is clear cause and effect, one could then perhaps look at an intermediate example. For example, one could get twenty people who are quite good at sport to play a round of golf and then give coaching to the ones who do worst.

Finally there would be the difficult question: how do we distinguish between all these cases. In Watford on Friday I didn’t have time to discuss this (and in any case had not worked out in detail how I would go about it). I contented myself with pointing out that standard deviations are important. (For people who don’t know what standard deviation is, one could have a qualitative discussion about data that can be decomposed into an expectation plus some noise, and say that what matters is the amount of noise — which is something that can sometimes be measured.)

5. Guy Thomas Says:

A few variants of the airport travelator-pavement puzzle could be discussed. Mathematically equivalent I think – and with some classes it might help to show at the end of the discussion how this emerges from the algebra – but with varying ease of intuition:

1. The pavement is very slippery, so you walk more slowly there than on the travelator. Does your answer change?
2. You have no problem with your shoelace, but you are now permitted to run on one surface only. Where should you run?
3. Instead of tying a shoelace, you are required to walk (slowly) in the reverse direction for a short time on one of the surfaces. (This perhaps makes the intuition easier.)

A possible economic interpretation: choose or allocate your activities in different periods or conditions according to comparative advantage, not absolute advantage. (This principle has many other applications.)

For example, in my (1), suppose you walk at 2mph on the slippery floor, walk at 4mph on the travelator, and the travelator itself runs at 6mph. The absolute advantage of walking is greater on the travelator, but the comparative advantage of walking is lower on the travelator. So you do your shoelace-tying on the travelator.

6. Richard Baron Says:

I think your re-writing of Proust may introduce an ambiguity. I make no complaint about your re-writing, but it is amusing to notice the dangers that lurk in languages that are not governed by strict and complete sets of rules.

Your re-writing seems to allow one of two different things not to have happened at the Verdurins, but to have happened in the practice: (1) the clean cut; (2) the assumption of a shell. The question would be decided in favour of (2), the correct reading, if there were a rule which ensured that “if not at” should be associated with the alternative that was nearer to it on the page, but I am not sure that there is any such rule. Nor can we rely on semantics to disambiguate. The assumption of a shell grounded the clean cut, but there is no requirement that they should have happened at the same time. The clean cut could have happened later, and with no connection to the practice, after intermediate consequences of the assumption of a shell had come to pass.

• gowers Says:

I was in fact slightly worried about that, but wanted to rejoin as many as possible of the clauses that had been chopped in half. I might have another go at some point, perhaps adding a word or two to deal with that problem.

• gowers Says:

On second thoughts, the course of life making a clean cut between the old Cottard and the new is a sufficiently general event that I don’t think one could talk of it having happened at the Verdurins’: no personality change is as clean cut as that, or at least not if it’s caused by something as general as the course of life.

7. Mike Taylor Says:

Thanks for this post. Not only is it an interesting account of an important experiment, it’s also freed me forever from the nagging feeling that at some point in my life I ought to make an attempt on Proust. That’s just plan bad writing.

On the airport puzzle. I walk at x m/s, and the moving walkway at y m/s, so that my total speed when on the walkway is x+y m/s. It takes my t seconds to stop walking and tie my shoelace. If I do this on the walkway my speed drops from x+y to y, where as if I do it off the walkway it drops from x to 0. In both cases it drops my x m/s, which means that I lose tx m.

So it looks obvious to me that it makes no difference where I stop to tie my shoelace. Why am I wrong?

• kalai@math.huji.ac.il Says:

Dear Mike, suppose somebody adds tx m to your daily trip to work. Would it make a difference if this addition comes to the highay where you drive or to the walkway where you walk?

• Mike Taylor Says:

No. Either way, it would take me an additional tx minutes to get to work.

• gowers Says:

Let’s try an extreme version of the same problem. Suppose you are very tired and need to get to New York fast. Is it better to jump on the first flight and then go to sleep for three hours, or to have a three-hour sleep in the airport and then jump on the first flight (assuming, let’s say, that a flight is leaving almost immediately in both cases). It’s a no-brainer. (If you want some movement in the aeroplane, then imagine you enter it at the back and leave through the front.)

Another way of thinking about the problem is this. Let’s imagine a film of your walk through the airport, and let’s suppose that the tying of the shoelace is edited out and ends up on the cutting-room floor. If you tie it on stationary ground, then you’re in the same place after the edit as you were before. If you do it on the walkway then you move instantaneously forwards by quite a bit. So although it takes the same time either way to tie the shoelace, if you do so on stationary ground you completely waste that time whereas if you do it on the walkway you don’t.

• gowers Says:

To be a bit more concise, if you take 30 seconds to tie a shoelace and you do so on the walkway, it doesn’t add 30 seconds to your total journey time because you’re further along when you’ve finished than you were when you started. If you want to waste 30 seconds on the walkway, you’ll need to hop backwards at walkway speed while tying your shoelace so as to remain stationary relative to the airport rather than relative to the walkway — good luck!!

• Jeff Says:

Mike, i had the same first intuition, but i worked out a specific example which clarifies the clue in the post about imagining one ties his laces right before he gets on and one right after.

Here is how I visualize it.

Imagine two people, Tom and Jerry, walking side by side at the same speed.

They get to the moving walkway entrance.

Tom bends down to tie his shoe while Jerry takes one more step and bends down to tie his shoe.

Lets also assume that the additonal step takes a negligible amount of time so they essentially start tying shoes at the same time.

Now they both finish tying their shoes and get up and start walking.

Tom starts walking from the beginning of the moving walkway while Jerry has moved further ahead while tying his shoe. They will be moving at the same speed so Jerry will arrive first.

• Mike Taylor Says:

Thanks, Jeff — your explanation of why it’s better to tie laces on the walkway is the one I find easiest to think through. (No disrespect to the others.)

The problem is — while I can see that you’re right, I can’t see why I’m wrong, in my first comment above.

This is a fascinating problem: so simple to express, yet so hard to think about.

• gowers Says:

What’s wrong is that it’s simply not true that in all situations if you drop your speed by x for a time t then you lose a distance xt. For example, if you drop your speed from x to zero just before stepping on a walkway that moves at speed y, then you lose a distance (x+y)t.

• gowers Says:

Similarly, your comment about time lost is false. I tried to address it in my previous comment but one. Suppose you are on the walkway and you walk backwards in such a way as to cancel out the speed of the walkway and you do so for 30 seconds. Then what you have done is allowed 30 seconds to pass while you make no progress. So then you really have added 30 seconds to the time it takes. It follows that if instead you stand still (relative to the walkway), then the extra time you will take is less than 30 seconds.

• Jeff Says:

Mike, not sure if below clarifies since it is similar to what Dick posted.

If with no shoe lace tying one can travel the distance in time

t = x / d1 + (x+y)/d2

Then if one ties his lace before the moving walkway and shoe lace time takes t_shoes.

t = t_shoes + x / d1 + (x+y)/d2

Note, this time is added because we are not covering any distance while tying shoes.

and if on the walkway

let d3 = distance traveled on moving walkway while tying the shoe
d3 = y * t_shoes

t = x/d1 + t_shoes + (x+y)/(d2-d3)

So while they both have the same drop in velocity, in case one the person covers 0 distance during the velocity drop and in case 2 they cover y*t_shoes distance, leaving less distance for the rest of the journey over which they move the same speed.

• tom denton Says:

I had a nice jump on the answer to this from my bicycling fanatacism. After tracking riding times over a good-long period of time, it becomes intuitively obvious that it’s much better for one’s average speed to just keep riding (albeit perhaps slightly slower) than to stop for a rest.

• gowers Says:

That gives me an idea for another question. I’ll add it to the list now.

8. Dick Veldkamp Says:

Mike,

Use your notation, and call the distances d1 for walking and d2 for being on the walkway, then the travel time required for the case tying your shoes while not being on the walkway is:

t_total = t + d1/x + d2/(x+y)

If you tie your shoes while on the walkway, your speed drops from x+y to y during that time, and the distance travelled in that time is t*y. Hence we now get:

t_total = d1/x + t*y/y + (d2-y)/(x+y) = t + d1/x + (d2-y)/(x+y)

Which is shorter by y/(x+y).

9. Stones Cry Out - If they keep silent… » Things Heard: e230v1 Says:

[…] Pedagogy. […]

10. Monday Highlights | Pseudo-Polymath Says:

[…] Pedagogy. […]

11. Colin Billett Says:

And after that, there was no more talk of Swan at the Verdurins!

12. Peli Grietzer Says:

I think your reworked Proust loses something fairly concrete: in the original, giving us all the qualifications and subtleties ahead of the predicate assures that when the predicate lands its interpretation will be highly constrained. Even if the sum total of information conveyed stays the same either way (which I doubt, since the extent to which we’re motivated to think deeply about a text can vary with stylistics), the feel of a sharply informative predicate is very different from the feel of a vague predicate followed by tweaks.

• Basil K. Says:

I’ve never read Proust, but I could imagine an artistically legitimate passage, proustian or other, which defies any attempt at correct parsing; which is, to say, syntactically false. To try to parse it as a reader is still a necessary function towards finally understanding the passage. Imagine talking with a friend and failing to parse him at a certain point; having noted this and listening further, you might see that he just jumped to another subject having suddenly remembered it.

My point is: parsing Proust (or Joyce, or Musil, or…) in a class certainly doesn’t aim to reduce Proust’s intended meaning, it still is the unavoidable necessary first step. What Gowers did, of course, had other than literary intentions, namely mathematical, and since I presume this was clear to the students, then reconstructing the parsed text in a more linear fashion is at least a mathematically legitimate little exercise. “At least”, but not quite just that: I know that as a reader of literature I’m urged to do this “exercise” very, very often, and not at all because I’m also a mathematician.

Because, actually, there is this other way to look at it: Proust might want us to spend that time parsing and reconstructing, linearizing and going back again, just to make sure that the information on the predicate that comes later imprints itself as much as necessary… 🙂

• Mike Taylor Says:

Proust’s novel ostensibly tells of the irrevocability of time lost, the forfeiture of innocence through experience, the reinstatement of extra-temporal values of time regained. Ultimately the novel is both optimistic and set within the context of a humane religious experience, re-stating as it does the concept of intemporality. In the first volume, Swann, the family friend visits [BONG!]

• Basil K. Says:

Since it came to that, I wonder what would Proust himself have to say on the time lost on the walkway instead of on stationary ground.

• Peli Grietzer Says:

Basil K. — I of course agree completely. My comment was intended purely as a comment on “I think it loses some of its distinctively Proustian character… but why it should be pleasurable rather than just a nuisance is hard to say.”

• Basil K. Says:

Oops, I see, sorry for that!…

13. David Roberts Says:

Are you going to get feedback on your visit, Tim?

For any Australians reading this there is the Mathematicians in Schools program run by the CSIRO, http://www.mathematiciansinschools.edu.au/, which aims at getting opportunities like this set up.

14. Greg Marks Says:

I daresay Mike Taylor has just proved that the shoelace puzzle is a very nice problem!

Professor Gowers’s account of his discussion with the students of prophetic dreams reminds me of a poignant passage in Richard Feynman’s autobiographical “Surely You’re Joking, Mr. Feynman!”: Adventures of a Curious Character wherein he describes a clock that stopped in the hospital room the minute his wife died and discusses the temptation (of some) to offer a supernatural explanation.

15. Eugene S Says:

Dear Dr. Gowers,

With all due respect to anyone who embarks on such an undertaking as translating Proust’s magnum opus, Moncrieff’s English version is needlessly awkward. For example, why translate “au contraire” as “on the contrary” when we are not seeing opposites? “By contrast” or “on the other hand” would have been much closer to Proust’s meaning. Likewise, Moncrieff preserves the plural for “les mêmes défauts” (the same defects) but not for its antecedents “des dehors de froideur, etc.”, changing them to singular (a shell of coldness, etc.). This makes parsing an already difficult text still harder.

Moncrieff worked in the 1920s and died in 1930 (which is why his translation can be hosted on Project Gutenberg for free). However, subsequent translators have produced far superior work.

Yes, the price of a hardbound copy of a recent, award-winning translation will set you back a good chunk. But if you are planning to make this your bedtime reading for a couple of years, it’s just a few pennies a day… much less than some other vices.

Other than that nitpick, I enjoyed very much reading your account of teaching math to secondary school students and the ensuing discussion.

• Colin Billett Says:

Quite agree. I have full sets of, and read, the latest three most popular versions, and often refer to the Gutenberg original. I’m currently reading the Prendergast edited set, and loving it.

16. Some Miscellaneous Awesomeness « Research in Practice Says:

[…] Terry Tao’s airport puzzle. If you have to get from one end of the airport to the other to catch a plane, but you really need to stop for a minute to tie your shoe, is it best to do it while you’re on the moving walkway or not? (I learned this problem from Tim Gowers’ blog.) […]

17. Björn Says:

I teach maths to 16-18 year-olds in Sweden. I just wanted to let you know that I found this (and the previous) post very inspiring. Thanks a lot!

18. Rob Kent Says:

The Proust passage which you cited above reminds me of a much simpler and starker made up example which I think is discussed in Jean Aitchison’s psycholinguistics primer “The Articulate Mammal”, illustrating the difficulty which the human brain has with nesting of sub-clauses. Whilst one can readily follow, in real time, a sentence with one embedding like “the mouse the cat chased likes cheese”, most of us need to stop and think about a pretty unnatural-sounding double embedding like “the mouse the cat the caretaker kept chased likes cheese”, and I think most people would reject as wholly unnatural and (in real time at least) incomprehensible a triple embedding such as “the mouse the cat the caretaker the school employed kept chased likes cheese”.

That’s not very relevant to maths education, perhaps, but then I’m a bit puzzled as to why you tackled the Proust in that context.

• gowers Says:

When I was an undergraduate I was told a similar example by a philosophy student called Tony Halbert, who used it to demonstrate that you can have sentences with arbitrarily long strings of the same word, without using inverted commas. For example, if you want five words in a row, you can go for “The dog the dog the dog the dog the dog over there bit bit bit bit bit me.” With a bit of effort I can just about understand this sentence up to around three “bit”s. (To do so I have to visualize the dogs.)

I suppose I included the Proust sentence because one could use it as a springboard for a more general discussion of parsing sentences, which though it would normally be labelled as linguistics rather than mathematics is nevertheless a highly mathematical process and moreover one that is important in mathematics. So it could help with the general aim of encouraging people to think mathematically, and also of demonstrating that mathematics is about more than just the topics taught at A level.

• Harry Says:

I knew Tony Halbert. I hear he lives in Wales now. He used to be a guitarist for a band called Sweet Sensation in the 70s. A fine fellow and one I miss.

19. abundantmichael Says:

On the airport shoelace problem I assumed that walking with shoelaces undone was slower than walking with them done up (this seems to be the case for me in real life because I can not walk super fast with laces undone because my shoes fall off).

So now lets take an extreme case that there is 1 mile of regular walking before the moving walkway which is also 1 mile long. Suppose I walk at 2 mph with laces undone and 4 mph when they are done up and that the walkway moves at 4 mph. And lace tying takes 30 seconds.

If I do my laces up at the beginning then my journey takes
30 secs (lace tying) + 15 mins (1 mile at 4mph) + 7.5 mins (1 mile at 8 mph) = 23 mins

If I wait until the walkway to do up my laces then my journey takes
30 mins (1 mile at 2 mph) + 30 secs (lace tying on walkway) + (1 mile – 30 secs at 4mph)/8 mph
= 30 mins + 30 secs + (1 – 1/120 * 4)/8 hours
= 30 mins + 30 secs + 29/30/8*60 mins
= 30 mins + 30 sec + 58/8 mins
= 30 mins + 0.5 min + 7.25 mins
= 37.75 mins

So with this set of assumptions it is faster to tie your laces at the beginning rather than wait until you you get to the walkway.

20. Lucas Braune Says:

I could not see at first the problem with the police boasting about how efficient their camera placement was (question 37(i)). I think what made me have to look up the answer is that I implicitly assumed the data used to identify blackspots was gathered over a long period and was a good approximation of the mean.

21. Timothy Gowers « Edina tuition Says:

[…] Timothy Gowers visits Watford Grammar School. […]

22. Tibor Mach Says:

Professor Gowers, would it be OK if I translated this article and posted it in Czech (with a link to the original and a mention of you as the author of course) on my blog (the link is bellow…but the blog is all in Czech, apart from the name that is)? I’m currently writing a few articles about how I think mathematics ought to be taught at schools (it is inspired by Lockhart’s essay A mathematician’s lament considerably) and your approach strikes me as an almost perfect example (the imperfection is that sadly you cannot expect all teachers to be able to teach this way).

23. Tibor Mach Says:

Not that this is very important, but I noticed one typo during the translation. In the dreams question you divide 250 000 000 by 25 000 at one point and you write that the result is 1000, so there is one 0 missing (then again the same thing with 500 right after that).

24. Timothy Gowers | Edina Tuition Says:

[…] Timothy Gowers visits Watford Grammar School. […]