Michael Gove, the UK’s Secretary of State for Education, has expressed a wish to see almost all school pupils studying mathematics in one form or another up to the age of 18. An obvious question follows. At the moment, there are large numbers of people who give up mathematics after GCSE (the exam that is usually taken at the age of 16) with great relief and go through the rest of their lives saying, without any obvious regret, how bad they were at it. What should such people study if mathematics becomes virtually compulsory for two more years?
A couple of years ago there was an attempt to create a new mathematics A-level called Use of Mathematics. I criticized it heavily in a blog post, and stand by those criticisms, though interestingly it isn’t so much the syllabus that bothers me as the awful exam questions. One might think that a course called Use of Mathematics would teach you how to come up with mathematical models for real-life situations, but these questions did the opposite, and still do. They describe a real-life situation, then tell you that it “may be modelled” by some formula, and proceed to ask you questions that are purely mathematical, and extremely easy compared with A-level maths.
One comment on that post particularly interested me, from someone called Joseph Malkevitch, who drew my attention to an article he had written in which he recommended a different kind of question both from the usual sort of symbolic manipulation that most people would think of as mathematics, and from the sterile questions on the Use of Mathematics papers that pretend to show that mathematics is relevant to real life but in fact do nothing of the kind. The main idea I took away from his article was that there is (or could be) a place for questions that start with the real world rather than starting with mathematics. In other words, when coming up with such a question, you would not ask yourself, “I wonder what real world problem I could ask that would require people to use this piece of mathematics,” but rather, “Here’s a situation that cries out to be analysed mathematically — but how?”
Inspired by Malkevitch’s article, I decided to write a second post, in which I was more positive about the idea of teaching people how to use mathematics. I gave an example, and encouraged others to come up with further examples. I had a few very nice ones in the comments on that post.
The difference between then and now is that then there seemed to be a probability of approximately zero that questions of that kind might actually make their way into schools. But it seems that Gove is absolutely serious about getting more people to do mathematics for longer, and that creates an opportunity. Of course, it also creates an opportunity to mess things up badly, and one of my reasons for being interested is that I would like to do what I can to avoid the terrible waste that messing it up would be.
As an indirect result of the earlier blog posts, I have found myself in a position of some influence. I’m not sure how much, but I am in touch with the Advisory Council for Mathematics Education, or ACME, which, as its name suggests, advises the government on matters of mathematical education, and there appears to be some sympathy in ACME for the idea of a qualification of some kind that involves interesting real-world mathematics problems. I have other reasons for optimism that I won’t go into here. But to translate that into results, what I would really like is a longer list of potentially good questions than the one I have so far, which I shall now give, together with a few brief remarks. I’ll try to group them naturally into topics.
This is the process of coming up with estimates of the correct order of magnitude for various real-world quantities. Here are a few examples.
1. How much does a cloud weigh?
2. How many people could fit into the Isle of Wight?
3. How many piano tuners are there in Chicago? [That is a classic example.]
4. If the average temperature of the sea were to rise by a degree, then by how much would thermal expansion cause sea levels to rise?
5. How many molecules from Socrates’s last breath are in the room?
Fermi estimation and probability.
Sometimes estimating a probability boils down to doing a couple of Fermi estimates. The next couple of questions are examples of this.
6. You are about to fly to the US. What are the chances that you will die as a result of your flight crashing? And what are the chances that you will die of natural causes while on the flight?
7. What are the chances that at some point in the last five years somebody in the UK dreamt that a loved one had died, only for that loved one to die unexpectedly the very next day?
Sometimes the probability can be more sophisticated. In particular, conditional probability can come in.
8. In September 2009 the same six numbers were chosen in two consecutive draws of the Bulgarian State Lottery. Was this conclusive evidence that the draws were manipulated?
To answer that last question properly, one needs to estimate how many lottery draws of that kind there have ever been, and how many events one would count as suspicious. (For instance, would it be suspicious if for one draw the numbers 1,2,3,4,5,6 were chosen? It would certainly be very remarkable.) That way, one could obtain an estimate for the probability that a “suspicious” draw would have occurred by now purely by chance. So far that is nothing more than Fermi estimation. But let’s suppose that you come up with an estimate that the probability that there would ever have been an occurrence as suspicious as what happened in Bulgaria in the entire history of lotteries of this kind was fairly small — something like 1 in 30. There is a further step one can take, which is a piece of simple Bayesian analysis. We need to decide also what we think the probability is that somebody would manage, in front of the TV cameras, to get the machine to shake the balls in just the way needed to get the right numbers to come out, and that, given this control, they would go for the same numbers as the previous draw, which was bound to attract attention and in fact meant that many people were winners and the average payout was rather small. It all sounds a bit unlikely. I’m not quite sure how to judge the probability that it would ever have happened, but if we went for, say, 1 in 10, then we’d end up judging that the probability, given that the numbers were the same twice in a row, that that was due to chance rather than manipulation was . And one-in-four probability events do happen.
More probability and statistics.
9. In 1999 a solicitor named Sally Clark was convicted for the murder of her two sons, who had died in 1996 and 1998, both at the age of a few weeks. Roy Meadow, a paediatrician, argued for the prosecution as follows. The probability of a cot death is approximately 1 in 8500. So the probability of two cot deaths is roughly the square of this, or 1 in about 73 million. Therefore it was overwhelmingly likely that the deaths were not due to natural causes. Is this argument valid?
Again there is a Bayesian argument to make here. There are a lot of people born each year, so there are a lot of opportunities for the 1 in 73 million chance to occur. And one wants to have some idea of how many mothers we might expect to kill two of their children (taking account of any additional circumstances that might be relevant — such as that Sally Clark wasn’t mentally disturbed etc.). Another major flaw in Roy Meadow’s argument was the assumption that two cot deaths in a single family are independent events. After criticism from the Royal Statistical Society, Sally Clark’s conviction was overturned, but not before she had spent three years in jail. More about this awful story can be read in the Wikipedia article about it.
10. The batting averages paradox: how can it be that A has a better average than B in the first half of the season and a better average in the second half of the season, but B has a better average for the entire season?
11. How much can we trust opinion polls?
Related to that is the following question.
12. How would you go about obtaining a random sample of 2,000 people eligible to vote at the next General Election?
13. How would one go about determining how the average global temperature is changing?
That last question has an additional interest over random sampling, because the places where temperatures are measured are not random. So one would want to weight the measurements according to some kind of density, and also look out for sources of bias, such as placing the measuring devices in towns — though if it is changes in global temperatures that we are trying to measure, then some of those biases matter less.
14. The average global temperature in 1998 was higher than the average global temperature in 2011. Does this demonstrate that man-made global warming is not a serious threat?
15. Obtain figures for how girls and boys do on some public examination, and ask whether the evidence shows that girls are better than boys (if that is what the figures appear to show) at the subject in question.
What would work best here would be if the distributions were interestingly different: one might have a higher average, but the other have a higher variance and more people at the very top. The idea would be to show that averages do not give a complete picture: for that one ideally wants the whole distribution, though a few extra parameters will be helpful if the whole distribution is too much to ask for.
16. “My grandfather was a chain smoker and he died peacefully in his sleep at the age of 95.” In the light of this kind of argument, why are we so confident that smoking causes lung cancer?
17. If you are batting in cricket, you can increase your expected score for any given ball by playing in a riskier way. The trouble is that you also increase your chances of getting out. In a limited-overs game, with a certain number of wickets in hand, how should you decide, when batting, how much risk to take? (Clearly as the end approaches you should play more riskily, but why and by how much?)
Part of the difficulty with 17 is setting up a simplified probabilistic model.
Something similar could be asked about football: if a team plays in a more attacking style, the chances are increased that it will score a goal, but the chances are also increased that it will let in a goal. If a team is playing in a knock-out tournament and is a goal behind, how riskily should it play, and how does the answer change as the end of the match approaches? Again, it is not obvious how to model the situation in the first place.
Questions with a game-theoretic flavour.
18. In a leafy suburban street, it turns out that if you convert your front garden into a car parking space, you increase the value of your house. However, you also decrease the values of everybody else’s houses, partly because you make it more difficult for them to park (since they have to leave access to what used to be your front garden) and partly because the street becomes uglier. If enough people convert their front gardens, then everybody ends up worse off. What should be done in a situation like this?
The above question is just one example of many tragedy-of-commons situations, all of which could be interesting to discuss. I stole it from Tom Körner’s article about mathematics in everyday life in the Princeton Companion to Mathematics.
19. Suppose you play an iterated prisoner’s-dilemma game. What happens when various strategies are pitted against each other? (For simplicity, let’s say that you both get one point if you cooperate, you get two points if you defect and the other person doesn’t, you both lose a point if you both defect, and you lose two points if the other person defects and you don’t.)
Here I imagine that pupils are invited to devise strategies, and then they play against each other several times, promising to stick to the strategies they have devised. They keep score, see which strategies do well, then change their strategies if they want to, and so on. (I’m basing this idea on the memory of a fascinating article in the Scientific American about three decades ago where Martin Gardner described an experiment of exactly this kind.)
20. The retaliation game. The rules for this are very simple. Two players take turns. Whenever somebody has a turn, they have the option of either stopping the game, or giving themselves a point and taking two points away from the other player.
I heard about this game from a blog post of Gil Kalai. I think it could form the basis for a very interesting classroom discussion.
With all the last three questions, a subsidiary question one could (and absolutely should) ask is, “Can you think of real-life situations that are similar to these simple games?”
21. A widow dies and leaves the contents of her house to be divided up amongst her three children. The children do not care much about the financial value of the possessions, but they care a lot about the sentimental value. To complicate matters, the sentimental values they attach to the various possessions are quite different. What would be a fair way of dividing the possessions?
This is similar to one of Joseph Malkevitch’s questions (which was about a divorcing couple).
22. In a greengrocer with just one till, it often happens that one customer has a big basket with many items that will take a long time to process, while just behind them is someone who wants to buy one small thing and has the exact change ready. Try to devise a system that would allow the occasional queue jump in a situation like this but that wouldn’t have obvious defects (such as a person with a lot of shopping being overtaken by a very large number of people with only a small amount).
23. Five options are put to the vote. Seven people put the options in order of preference. The results are tabulated and shown to the pupils, who are asked which option should be gone for. The table is of course set up so that different voting systems give different results. The discussion can then be generalized almost arbitrarily far.
See Joseph Malkevitch’s question E on page 90 for a question of exactly this type.
Perhaps one could make this question more immediate by giving the class an actual choice. For example, perhaps there could be seven DVDs, one of which will be watched. Everybody in the class could put the DVDs in order of preference in a secret ballot, with no conferring allowed, and only after the results were tabulated would they discuss which one should be watched. To make the discussion livelier, three of the DVDs could be very similar (e.g., three different episodes of the Simpsons).
Questions with a physics flavour.
Some Fermi estimation problems involve physics. Here are a few more.
24. How do speed cameras work? How accurate are they likely to be? (The basic technique I’m talking about is taking two photos in quick succession.)
25. Why does a mouse survive a big fall when a human doesn’t? (There are many questions similar to this, such as why elephants have thick legs, ants can carry several times their body weight, etc.)
26. How does a Mexican wave get started?
27. Somebody pours you a cup of coffee but you aren’t yet in a position to drink it. You take milk, and the milk provided is cold. You want your coffee as warm as possible. When should you put in the milk: now, or just before you drink it, or some time in between?
28. You are walking from one end of an airport terminal to the other. The airport has several moving walkways, and you need to stop to tie your shoelace. Assuming you want to get to the other end as quickly as possible, is it better to tie your shoelace while you are on a moving walkway or while you are between walkways?
This question comes from a blog post of Terence Tao, and the response to it provides us with strong empirical evidence that people find it engaging.
29. You have a collection of suitcases, boxes and bags of various sizes, shapes and degrees of squashiness. You want to pack them all into the boot of a car and it’s not obvious whether you can. What is the best method to use?
This question is of course rather open-ended. The aim would be to elicit principles such as packing big and inflexible things first (and understanding why it is a good idea), overriding that principle if you find that you have an object that fits very snugly into a space (but what exactly does that mean?) and so on. It would also be interesting to model the situation in two dimensions, perhaps having a board with a rectangular hole into which you have to put a whole lot of wooden pieces without overlaps. It would be different from a jigsaw puzzle because the area you had to fill would be greater than the total area of the pieces and the pieces wouldn’t fit neatly together. Or perhaps one could set up a 3D situation where it was just about possible to pack some objects into a box, but only if you were fairly clever about it — again with the total volume of the objects strictly less than the volume of the box.
30. You have probably heard that the distance to the sun is approximately 93 million miles. How on earth can we know something like that?
That question could lead to a more general discussion of the cosmic distance ladder, which has been beautifully explained by Terence Tao.
Some of the questions above already involve optimization. Here are some further optimization questions.
31. You have a collection of tasks to perform, each of which has a certain probability of failure. If you ever fail on one of the tasks, then you have to start again at all the tasks. (An example: you want to make a Youtube video in one take in which you successfully perform five tricks of varying difficulty.) In what order should you do the tasks if you want to minimize the expected time it will take to eventually succeed?
For more on this question, including an entertaining Youtube video, see this comment of Julia Wolf on my second blog post. She got the question from a Google Buzz post of Terence Tao.
32. The British organization NICE (National Institute for Health and Clinical Excellence) has the task of deciding which drugs should be approved for use by the National Health Service and which should be paid for. Given that different drugs cost very different amounts, do very different things, and benefit very different people, how should decisions about how much to spend on various drugs (given a fixed total budget) be made?
Some concepts that I’d like to see arising out of a discussion here are the ideas of marginal utility (not necessarily under that name) and marginal cost — the rough idea being that if you’ve got an extra pound to spend, then you want to get the most benefit out of that pound. (However, there could be interesting situations where a local optimum is not a global optimum — for example if there are economies of scale connected with a certain drug.)
33. You have a product to sell. How should you price it so as to maximize your profits?
Again this is a calculus question in disguise. If you decrease the price, you will increase sales (usually, though a side discussion of Giffen goods could be fascinating here) but also increase costs and decrease the average revenue from each sale. You want to stop when the extra revenue from reducing the price is exactly balanced by the extra costs. (If the extra revenue is negative, then the price is already too low.)
34. Six cards have different numbers written on them and are then laid face down on a table so that you can’t see what the numbers are. You are allowed to select any two cards and ask which has the bigger number. How many questions of this kind do you need to ask before you can put the cards in order?
Here I would recommend a classroom discussion in which the teacher actually has six cards and invites pupils to ask which pairs they would like compared until they are confident that they know the order. Of course, the discussion can then be generalized considerably.
35. You are in a maze. Devise a method that will guarantee that you eventually find a way out.
There are many further questions here. Suppose the maze is made of hedges, the hedges all look very similar, your memory is very bad, and you can’t keep track of where you have visited by (for example) dropping stones from time to time. How does that affect your method? And what if you no longer want a guarantee that you will escape, but simply a method that on average gets you out fairly fast? Is there a randomized strategy that works quickly on average? (Obviously one would not be looking for a rigorous analysis of such an algorithm, but a heuristic discussion, perhaps with reference to a picture of an actual maze, could still be interesting. For instance, what happens if you simply make random choices whenever you have choices?)
36. You are doing a jigsaw puzzle. The pieces are all very similar, so the only practical way of telling whether two pieces fit together is to try to fit them together. The puzzle is well made so it is always obvious that two pieces don’t fit when they don’t. What is a good technique for minimizing the expected number of attempts you will need to make to fit pieces together? For example, do you just want to build up one component, or is it better to build up a lot of small components and then fit those together to make bigger ones, and so on?
These questions are ones that I’ve thought of since first putting up this post, or otherwise come across, or had suggested in the comments below (possibly with small modifications — but I’ll link to the comments).
37. (i) 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?
(ii) In a certain school, the pupils in year 9 take a maths exam at the end of the year. Those whose scores are in the top half are taught by teacher A in the subsequent year, and those whose scores are in the bottom half are taught by teacher B. At the end of that year they take another exam. If you take the whole year together, then the spread of scores is very similar to the previous year, but the average scores of the pupils in the top stream go down compared with the previous year, while the average scores of the pupils in the bottom stream go up. Does this demonstrate that teacher B is a better teacher than teacher A?
(iii) A scientist decides to test the effect of coca cola on telepathic powers. He tosses a coin 20 times and records the results. He then takes 100 people and asks them to guess what the results were by writing out a sequence such as HTTHTTTHHTHTHTHHHTTH. As an incentive, he promises a prize of £100 to the person whose guess is closest to the actual sequence. He then picks the ten people who have done best, gets them all to drink a can of coca cola, and retests them with another sequence of 20 coin tosses. To his surprise, he finds that they do considerably worse the second time. Does this demonstrate that coca cola inhibits telepathic powers?
By the way, I contemplated making the scientist female, but decided that that would be even more sexist than making him male …
38. Another scientist questions 2000 randomly chosen people about their eating habits and then follows their health over the next ten years. He notices that people who often eat organic food suffer from fewer heart attacks. Does this demonstrate that organic food protects against heart attacks?
39. You are taking the trip of a lifetime: a round-the-world cruise. One of the highlights is arriving at Manhattan on a gloriously clear day. Roughly how near do you have to be before you can see the top of the Empire State Building?
That question is inspired by Tim’s question below, which itself would make a good question.
40. You are in the process of buying a washing machine for £250 at Curry’s (a chain in Britain that sells that kind of thing), and are offered a five-year guarantee for £60. The sales attendant tells you that typical repairs cost at least £100. Should you go for the insurance?
This question is stolen (in slightly modified form) from this comment of Gil Kalai. The policy I adopt towards insurance, ever since I read the advice somewhere, is not to take out any insurance unless the result of not doing so could be disastrous, since my expected gain is negative (or I wouldn’t be offered the insurance). If my washing machine breaks within five years, it’s annoying to have to pay the money, but certainly not disastrous, so I definitely don’t insure against it. If this principle were taught to millions in school, it could make a dent in the profits of certain companies …
41. 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?
As with the prisoner’s dilemma, this problem could be turned into an interesting and thought-provoking game. You have a series of rounds. In each round you pair up the pupils and offer each pair ten points, provided they can agree how to split them. If they can’t agree, then they get nothing. You offer some kind of incentive — perhaps a small prize — to the person who ends up with the most points after fifteen rounds. One could experiment with small variations: does it affect how people play if all the current scores are public knowledge? What about if you know the entire playing history of your opponent? What happens if instead of changing the pairing every round you have several rounds with the same pairing before changing?
42. Three people need to get back home after a party. It’s a long walk, but somebody else has a space in their car. They would all prefer not to walk. One of them has an unbiased coin. Devise a method for using the coin to make the decision, in such a way that they all have an equal chance of getting the lift. What happens if the coin cannot be tossed more than ten times?
43. A doctor tests a patient for a serious disease that one in ten thousand people have. The test is fairly reliable: if you have the disease, it gives a positive result, whereas if you don’t, then it gives a negative result in 99% of cases. So the only problem with it is that it occasionally gives a false positive. The patient tests positive. How worrying is this?
I initially resisted putting in this question, partly because it is very well known, and partly because it is a bit close to standard A-level fare. However, the principle behind it is important, and can be stated qualitatively: if the disease itself is much more unlikely than the false positive, then you shouldn’t be too worried about testing positive. What’s more, this principle can be got across without doing too much formal mathematics: you can do things like getting people to imagine a town with 100,000 typical people. Of those people, 10 have the disease, and roughly 1,000 will test positive despite not having the disease. Once you think of it like that, it is much more intuitive that testing positive doesn’t mean that you probably have the disease.
44. You drive round a corner and see a red light. You want to get to your destination as fast as possible. What should you do?
This question was suggested (in a more detailed form) by Anonymous in a comment below. Funnily enough, it is already a favourite question of mine, which I mentioned in the comment thread on Terence Tao’s shoelace-at-airport puzzle. I had left it out here because it seems too hard — in particular, I don’t know the solution to any version of the problem — but I’ve changed my mind now. I think it would be good to have questions where you can get somewhere by using a bit of maths (for example looking at specific strategies and specific assumptions about how the traffic lights behave) but can also sense that you would be able to get a whole lot further if you thought harder and/or knew some more maths. One of the main things I would like a course like this to give people is a feel for what mathematics can do. In the end, that is what is likely to be useful to someone who does not want to focus on mathematics. It’s a bit like the internet: to use it effectively, you need a good feel for the kind of thing you can find out with its help and you should know how to use Google. What you don’t need is a good general knowledge, though some general knowledge helps give you a feel for what else might be out there. Similarly, the skill needed to see that a certain situation can be analysed mathematically can, I think, be to a considerable extent decoupled from the skill needed actually to do the analysis. Obviously the more mathematics you know, the easier you will find it to recognise situations where that mathematics is potentially helpful. My claim is that although it is more difficult to recognise characteristically mathematical situations if you aren’t good at maths, it is not impossible.
45. A renowned wizard arrives in your town and makes the following offer. In front of you are two envelopes, labelled A and B. You can either open both envelopes and keep the contents, or you can go for just envelope A. But here’s the catch. The wizard claims to be able to predict what people will do, and has been correct every single time so far. If he predicts that somebody will choose just envelope A, then he puts £1000 in envelope A and £100 in envelope B. But if he predicts that they will choose both envelopes, then he puts nothing in envelope A and £100 in envelope B. What should you do?
This question is the famous Newcomb’s paradox. Although it does not involve much mathematics, it has the wonderful quality of leading to heated arguments and challenging people to think clearly enough to find the hidden assumptions behind those arguments.
46. In 1972 Diana Sylvester was raped and killed in San Francisco. Despite one or two leads, the police failed to solve the case. However, they kept some DNA, and in 2006 they checked it against a DNA database of 300,000 convicted sex offenders. They discovered that it matched the DNA of John Puckett, who had spent a total of 15 years in jail for two rapes. There was no other evidence linking Puckett to the crime, but the probability that a random person’s DNA would match that of the sample was judged to be 1 in 1,000,000. On that basis, he was found guilty and sentenced to life imprisonment. How reliable was the conviction?
Thanks to Fergal Daly for drawing my attention to this case.
47. How should a government determine tax rates if it wants to maximize the amount of tax that it collects? What about if it has other objectives?
This is not an easy question, but one could at least hope to raise various issues, such as the effect of tax rates on the incentive to work, to employ others to work, and to spend money, and also the difficulty of measuring this kind of effect. The question was suggested by Richard Baron.
48. This is not a question, but it has questions associated with it. I’m fairly sure that there exists software that allows you to invest virtual money in various different stocks and shares (and perhaps other financial products) and see how you do with your investment. One could give everybody in the class a virtual £10,000 and have a competition to see who has the most money three months later. Then one could run a second competition of exactly the same type. People could do as much research as they liked on the investments they were making. It would be instructive to see whether there was any correlation between the results of the two competitions. (My only worry about this question is that it might give some people a taste for the kind of risk taking that has got the world into so much trouble recently, but the moral is supposed to be quite the opposite.)
49. In 1985 almost nobody foresaw that a mere four years later a process would start that would result in the collapse of the Soviet Empire. However, Werner Obst, a German economist, analysed various economic trends and predicted that it would happen in around 1990. How impressed should we be by Obst’s insight?
Something to introduce into the discussion is the well-known fraud where you send lots of people lots of differing predictions of future sporting results, and then offer to sell tips to the few who received correct predictions. Another factor is that there was in fact a long history of predictions of the imminent demise of the Soviet Union — enough for an entire Wikipedia article.
50. You are given a fairly large bunch of common words. Devise a method for creating random sentences out of these words. The sentences need not be true, but they should make grammatical sense. (For example, “The political dog embarrassed the car,” would be OK, whereas, “Tomorrow the Friday in future,” would not be OK.)
I got the idea for this from a book by Seymour Papert that I was recently given and am in the middle of. He described its effect on a girl, who before this exercise (which was done with a computer and a specially designed programming language) had found grammar pointless and therefore didn’t know any, but who became absorbed by the challenge and ended up inventing several grammatical categories for herself.
51. In half an hour’s time you will be given the lyrics for a song. Your task will be to send a message to somebody else (with whom you have been paired), which will tell them what those lyrics are. You could of course send a message that contains the lyrics themselves, but the aim here is to use as few keystrokes as you can. You are allowed to confer with the other person before you receive the song lyrics but not afterwards.
The idea here is to exploit redundancies in the English language at many different levels. There are obvious ones sch as t’fct tht u cn mss out mny ltrs n rmain cmprhnsbl and slightly less obvious ones such as using context get away missing out entire words. At a higher level still, song lyrics often contain quite a bit of repetition.
The next four questions were suggested in a comment below by Charles Crissman, to whom many thanks.
52. You have a task you want to do that involves standing in a queue, but you also have a time limit. (Charles Crissman’s example is that you arrive at the Department of Motor Vehicles in the US and need to be back at work by the end of your lunch break. One that has happened to me is arriving at Cambridge Station needing to catch the next train and discovering that the queue for tickets is unexpectedly long. Another is cutting things a bit fine at an airport and finding a long queue at the bag drop.) How should you assess whether to join the end of the queue or whether more drastic action is required?
53. You run a manufacturing business, and the cost of one of your inputs suddenly increases by £10 per output unit. Should you increase your prices by £10? More? Less? Now suppose instead that your costs decrease by £10/unit. Should you keep prices the same? Drop them?
54. You are with a friend one evening and have different ideas about how you would like to spend it. So you decide to toss a coin. However, the coin is slightly bent, and neither of you is confident that the coin is fair. Can you nevertheless use the coin to make a fair decision?
55. How would you go about checking whether the coin was biased?
The situation in 54 gives me another idea for a question.
56. You and a friend have to decide between two possible ways of spending the evening: going to the cinema or staying at home and watching a football match. Your friend would prefer to watch the football but is quite interested in the film. You too are interested in the film but absolutely hate football, so your preference is much stronger than that of your friend. What would be a good way of deciding what to do? What if the situation repeats itself every week?
57. You are offered the following opportunity. You start with £1, but if you like you can give yourself a chance to increase your money by playing the following game. You toss an unbiased coin. If it ever comes up tails, you lose all your money and the game stops and you will never get the chance to play it again. But if it comes up heads, the amount of money you have multiplies by 3 and you can have another go with your increased stake. What should you do?
58. I want to weigh my young daughter on my bathroom scales. The trouble is, she can’t stand still for long enough. So I weigh myself holding her and then myself not holding her and take the difference. However, the scales are accurate to the nearest 100 grams, so all I can tell from this process is that my daughter’s weight lies within some range of width 200-grams. Is there any way of using the scales to get a more accurate measurement?
What I have in mind here is repeating the process before and after supper, with and without shoes, etc. etc., to obtain a collection of measurements of which I can take the average. Obviously, it would be unreasonable to expect a 16-year-old who didn’t like maths to think of that idea, but one could give hints such as, “My wife goes through the same process and gets a different answer. What could we do with those two answers?”
59. How much money is it worth spending to keep track of asteroids in case any of them are on a collision course with the Earth? Suppose scientists declared that there was a 1% chance that a particular very large asteroid would collide with the Earth in five years’ time. How much money would it be worth spending to divert it?
60. Devise a strategy for never losing at noughts and crosses (=tic tac toe). Try to make it as economical as possible, while still telling you unambiguously what to do for each move. What more could you ask of a strategy?
The answer to the last part is that you could ask for a strategy that always wins when it is in a winning position. It would be an interesting exercise to try to get people to formulate precisely what a “winning position” is.
61. A political blog gets so many comments that good comments are often drowned out by a sea of stupid ones. To combat this, the owners of the blog decide to introduce a reputation system, so that comments by people who have a good record of being interesting appear at the top of the list. What would be a good way of doing this?
62. In a football league of 20 teams each team is supposed to play each other team twice, so it is supposed to play a total of 38 matches. As a result of a players’ strike, each team in fact plays only 10 matches. At the end of the season, decisions have to be made about promotion and relegation, so the teams have to be put in order. The initial proposal is simply to add up the points that the teams have obtained so far, but some teams complain that this is unfair because they have played against much tougher opposition than some other teams with similar numbers of points. Can you devise a fair system that would take the quality of the opposition into account?
How would questions of this kind form the basis for a mathematics class?
I have very strong views about this. What I emphatically would not like to see is teachers learning “the right answer” and giving a mini-lecture about it to their classes. Instead, the entire discussion should be far more Socratic. The idea is that the teacher would go into a discussion about a question like this with a good grasp of the issues involved, but would begin by simply asking the question. An initial danger is that nobody would have anything to say, but one way of guarding against that is to discuss questions that people are likely to care about. For example, the question above about whether girls are better than boys at a certain subject is far more likely to encourage people to think critically about statistics than a mathematically equivalent question about a less contentious topic. If the discussion stalled, the teacher’s job would be to give it a little nudge in the right direction. For example, in the unlikely event that nobody had anything to say about girls and boys and their exam results, one could ask a question such as, “Does the average grade tell us everything we need to know about how good boys are at this subject?” If that still didn’t elicit a response, one could ask something more specific like, “Girls got a higher average score on this paper. Does that mean that the highest score of all must have been achieved by a girl? Does it mean that most girls scored more highly than most boys?” That would start people thinking. If even that failed, one could show the class a couple of silly distributions. For example, one distribution might have all boys with an identical score, and all but one of the girls with a slightly lower score and one girl with a stratospherically high score that lifts the average above the average for boys. (That doesn’t answer the two questions above, but it indirectly gives people a technique — looking at extreme cases — for answering questions of this general type.)
The main point is one I’ve basically made already: the discussions should start from the real-life problem rather than starting from the mathematics. Pupils should not feel that the question is an excuse to force some mathematics on them: they should be interested in the question and should feel the need for the mathematics, the need arising because one can give much better answers if one models the situation mathematically and analyses the model.
For the remainder of this post, I want to consider a few obvious objections to the idea of a course of this kind.
Objection 1. The last thing we want to do is water down the mathematics curriculum like this.
Response. I am not proposing a watering down of the mathematics curriculum. This idea is not aimed at everybody, but rather at the (pretty large) cohort of pupils who are intelligent and motivated to learn, but who for one reason or another do not get on well with the traditional mathematics curriculum. One thing one could do with such people is make one further attempt to get them to learn how to rearrange equations, solve quadratics, solve simple questions in trigonometry, and so on. But do they really need that? And even if they do, is it likely that they will become more receptive to that kind of mathematics if it is presented again in basically the way that they already know they dislike? Is it not far more likely that a completely different kind of course of the kind I’ve just described would help at least some of them to lose their dislike of mathematics? (Obviously that is an empirical question, and one that should be empirically tested if an idea like this is taken seriously.)
Objection 2. Certain very important mathematical skills, such as solving word problems by forming equations and solving those equations, do not seem to figure in the questions above.
Response. How important a life skill is turning word problems into equations and solving the equations? For some people it is undoubtedly useful a lot of the time, and for many more people it could be useful occasionally. But if you think it should dominate the way mathematics is taught, I recommend the essay What is Mathematics For? by Underwood Dudley. In fact, I recommend it anyway. In it, he rips to shreds the argument that things like algebra and trigonometry are necessary in people’s lives (even their working lives). His conclusion is that mathematics should be defended for its own sake. For example, for most people the great benefit of learning how to solve quadratic equations is not that one day they might actually need to solve a quadratic equation, but rather that it is a wonderful example of a non-obvious idea in mathematics: at first it looks as though you have little choice but to use trial and error, but by the clever trick of completing the square you can solve the problem quickly and systematically.
But another (possibly related) reason I have not focused on real-world problems that require algebra and trigonometry is that I have found it very hard to come up with good examples. I can come up with examples of some kind, but not ones that are interesting and engaging.
Here is a boring question, just to give an idea of what I don’t like. You know that you will need two thousand pounds in five years’ time, and a bank account offers you a fixed rate of interest of 5% per year over the next five years. How much do you need to invest so that you will have two thousand pounds at the end of the five years?
That problem certainly leads to an equation. But it also makes my heart sink. I couldn’t go into a classroom and enthusiastically start a discussion about it. Nor could I expect people who don’t like maths to have an opinion one way or another about what should be done.
Perhaps part of the problem is that the answer to the question is a number. The answers to the questions in the list above are things like “The best thing to do is X” or “The verdict was wrong because Y”. The Fermi estimation problems do ask for numbers, but the numbers are approximate, you are not asked to solve equations, and the questions are somehow fun.
I am not in principle against questions that naturally lead to algebra, but in practice I find them very hard to come by.
Objection 3. Future biologists and chemists would be hugely helped by being more competent mathematically. A course like this does not help to develop that competence.
Response. I have two complementary responses. One is that future chemists and biologists do not have to take a course like this. In general, people would be encouraged to take A-level, or maybe just AS-level, if they are good enough at mathematics to benefit from it. A course of the kind I am describing would be good for a large number of people, but not necessarily the best course for a large percentage of people.
However, I actually think that future biologists and chemists and even future mathematicians could benefit greatly from a course like this, that encouraged them to think mathematically rather than simply applying some methods they’ve been taught to standard problems.
Objection 4. It’s not enough to think about what to teach: you must also think about how it would be examined.
Response. Unfortunately that’s true. However, one thing we have in our favour here is that results on the course would not have the kinds of important consequences that A-level results have. People taking the course would be doing it as a sideline. I would hope that universities would be more interested in a prospective history student (say) if they had done well on a how-to-think-mathematically course, but that it would be regarded as a fringe benefit rather than one of the main criteria on which the student was judged.
I am told by people I know at ACME that they would be happy to recommend some form of continuous assessment or project work. Normally, I am not keen on that at all, but for a course like this, I think it could work. Suppose, for instance, that a news story breaks that, like the Sally Clark case, has a significant mathematical content. To expect somebody who isn’t all that good at maths to comment intelligently under exam conditions does not seem reasonable. But it seems much more reasonable to give that person a project that is done over several weeks and requires them to look up facts on the internet and draw conclusions from those facts with the help of a bit of mathematical reasoning, perhaps after mathematically similar questions have been discussed in class.
I actually quite like the idea of a very open-ended exam that feels a bit like a general paper: you could pass the exam with almost no mathematics, just by discussing the questions reasonably intelligently, but to do well you would need to come up with mathematical models, Fermi estimates, abstractions and the like.
In general, I think that assessing a course of this type would be challenging, but not impossible. And therefore I would not want the need for assessment to distort what is taught (which was the basic problem with the Use of Mathematics A-level).
Objection 5. You’d never find enough teachers who were capable of teaching a course like this. To do it well, you need to have a very sophisticated understanding of probability, statistics, game theory, physics, multivariable calculus, algorithms, etc.
Response. This is to my mind by far the most serious objection. The best I can offer is ways of mitigating the problem. A few ideas are the following.
1. Produce copious teaching materials — for example, a book with the questions above, and many more, each with a detailed discussion of the mathematical (and other) issues it raises, and detailed suggestions for how to elicit ideas from the pupils rather than simply “telling them the answers”.
2. Identify a few outstanding teachers, video them giving successful classes on these questions, and make the videos available to other teachers.
3. Set up a forum where teachers can exchange ideas and report back on what worked well for them and what worked less well.
4. Thoroughly road test questions before letting them loose on the nation’s schoolchildren. In fact, that applies to the entire course: make sure one has something that definitely can work before encouraging too many schools to teach it.
I myself am planning to do a bit of testing myself. In just under a month’s time I shall be giving a talk at Watford Grammar School for Boys, and I intend to take one or two of the questions from the list above and see whether I can get a good discussion going. I’ll report back on how I get on.
If you agree that something like this could work and want to increase the chances of its becoming a reality, then I would be very grateful for any questions you can think of that would go well with the above ones. I don’t mind if they are mathematically very close to questions already on the list, as long as they don’t look too similar on the surface. An ideal question is one that is based on a recent news story, or on an experience that we all have (or can imagine having) from time to time — such as Terence Tao’s shoelace-in-the-airport question — since then its real-world relevance cannot be questioned. But any question that is interesting, that does not explicitly mention mathematics, and that can be profitably analysed with the help of some mathematics that is not inappropriately difficult (so I’d rule out differential equations, for example), will be gratefully received. I am told that the more questions I can come up with, and the more varied they are, the greater my chance of convincing ACME that they can convince people further up the chain that this could work.