How should mathematics be taught to non-mathematicians?

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. [Update 31/10/12: MEI has just been given government funding to try to develop a qualification of this kind. This is mentioned in a Guardian article today, which is basically accurate apart from the title, which wrongly suggests that the qualification would be an A-level. It wouldn't.] 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.

Fermi estimation.

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 (1/30)/(1/30+1/10)=1/4. 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? (A related question with a more real-life flavour is question 66 below.)

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.

Optimization.

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.)

Algorithms.

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?

Further questions

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 200-gram range. 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?

63. Guardian journalist Zoe Williams recently wrote an article that included the following paragraph: “Less well known is that, mile for mile, it’s more dangerous to be a pedestrian than it is to be a cyclist, and every journey by public transport generates two journeys by foot (most journeys by car will generate at least one journey by foot – it’s rare to be able to drive directly from one door to another). Pedestrians never object en masse; they don’t self-identify as “pedestrians” and they never say how outrageous it is how many of them die. And yet in 2011, in Greater London, 77 pedestrians were killed (to 16 cyclists); 903 were seriously injured (to 555 cyclists); their deaths were up 33% on the year before, the serious injuries up 6%.” Do the statistics she quotes justify the assertion that mile for mile it is more dangerous to be a pedestrian? Is the assertion likely to be true? (The full article can be found here.)

64. As I write this question (on the 12th October 2012), there is a debate about the merits or otherwise of badger culling in order to reduce the spread of bovine tuberculosis. One proposal is to reduce the badger population in certain areas by 70%. But how can one estimate the population in the first place? Pilot studies have been done to try to assess whether culling works. How would you go about designing such a study (assuming you were willing to cull badgers)?

Some basic facts about these issues can be found on the web.

65. If you are a middle-distance or long-distance runner, then you have to worry about two things: speed and endurance. If you run too fast, you won’t be able to keep it up and will be overtaken. If you conserve energy too much, you won’t be able to catch up with the rest of the field. The standard strategy is to run at a roughly constant speed for most of the race and then to speed up at the end in a sprint finish. Is this likely to be a better strategy than aiming for a slightly faster constant speed and no sprint finish? If so, why? And what factors will determine the best strategy? [To keep things simple, it may be better to imagine that you are racing against the clock rather than against other runners.]

66. A Guardian article on 13th January 2013 began as follows.

“Students educated at state schools do better at university than their counterparts from public schools, yet are less likely to translate their degrees into graduate jobs.

“A study by Bristol University found that 88% of its state-school-educated graduates gained an upper second class degree or better, compared with 85% of those from public schools. Among the Russell Group and 1994 Group universities, more than 20% of state school pupils who graduated between 2009 and 2011 achieved first-class degrees, against 18% of those from the independent sector.

“However, superior academic performance is not matched by similar access to the jobs market, with just 58% of state-school-educated graduates finding a professional job, compared with 74% of independently educated graduates in the same period.”

Assuming that the statistics in the second and third paragraphs are correct, do they prove the assertion in the first paragraph?

One thing I would be looking for here is the obvious one that the percentage of students who get a IIi or better is just one of many possible measures one might choose. A subtler point is that even if we accept this measure, the statistics may be misleading for reasons related to the batting-averages paradox. It may be that state-school-educated students are more likely to do subjects where it is easier to get a IIi. (I’m not saying that this is definitely the case: I’m just saying that further information is necessary before one can draw the conclusion that state-school-educated students are better at getting at least a IIi than privately educated students. To obtain this information, one would need to look at the statistics on a course-by-course basis.) Similarly, it may be that state-school-educated students at Russell Group and 1994 Group universities are more concentrated at universities (and in courses) where it is easier to get a first.

Finally, are the percentages who get professional jobs percentages of Bristol graduates, graduates from Russell and 1994 Group universities, or all graduates? It is important to know this to be able to make sense of the statistics.

67. From time to time there is a news item saying that the oldest person in Britain has just died. How often would you expect this to happen?

68. Recently, a report said that the death of a toddler who was starved by his mother could not have been predicted by social workers. How might you interpret this statement more precisely (given that anything can be predicted if you don’t mind your prediction being wrong)? Should we accept it as inevitable that there will be a certain number of deaths of this general kind?


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 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.

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186 Responses to “How should mathematics be taught to non-mathematicians?”

  1. Gil Kalai Says:

    Dear Tim, to some extent I taught a course like that for non mathematicians at HU. (To some extent – because it was also about other things and not all directions from the post were there. Also the questions from the post look quite hard to me – technically hard in some cases and problematic in some others. I would probably recommend easier problems. I hope a post with answers or at least hints will follow.) In my course, for example, one question in the exam was if it makes sense to buy toaster-insurance. The exam was an ordinary exam.

    • gowers Says:

      One thing I perhaps didn’t make clear is that I wouldn’t expect people who took this course to master enough mathematics to be able to give mathematically sophisticated answers to all the questions. I’d be happy if they came out of the course with a good sense of what mathematics can be used for. That way, people who found themselves in later life with complicated policy decisions to make would have a better idea of the traps one can fall into if one relies on instinct only.

      To give an example, somebody might think that to measure the average global temperature it is enough to have a lot of measuring stations and simply take the average measurement. After taking the course, they might not be able to give a good explanation of what to do instead, but at least they would understand why the naive approach is unsatisfactory.

    • gowers Says:

      I should add that I would very much welcome a good supply of easier questions, since that would make it easier to find out what works. I like the toaster-insurance question.

  2. anonymous moose Says:

    Here is a suggestion for an Optimization question.

    Every year, your country gets a certain amount of taxes from its citizens. In the first year the tax revenue is 500 million dollars. In each year, you can partition these taxes among three purposes: you can put any amount of money. You can spend money on education, in chunks of 100 million dollars: for each 100 million dollars you spend, your yearly tax revenue increases by 0.1%. You can also spend chunks of 100 million dollars on waging war, each chunk has probability 10% of increasing yearly revenue by 100 million, and probability 5% of decreasing yearly revenue by 50 million.

    Your goal is to draft a spending plan for the next X years. How, qualitatively speaking, will the best spending plan change as a function of X, when X ranges from 1 year to 1,000 years? There’s no need to find the exact cutoffs, but rather to explain what the plan will look like, and how it will change as X increases.

    • anonymous moose Says:

      Ah, and I have another, a pretty exciting one in fact! It was asked to me by my flatmate, who spends a lot of time doing flight simulations. He says he’s flying the plane, and he knows where the destination airport is, say w.l.o.g. straight up north, 100 miles away. Now, his instruments also tell him that there is wind, at say 30 degrees angle east, at velocity 30 mph. The question is how should he aim the airplane such that it keeps going north. I was surprised that his software doesn’t give him this (it’s obviously a pretty easy calculation for a computer to do, but a bit grueling for a human).

      After thinking of it a bit, I realized that possibly the best way to do it, besides writing software, is having him draw vectors on paper and subtract them using the trapezoid rule. If he draws his intended velocity vector, and draws the wind vector (norms would correspond to velocities, of course), and then draws the negation of the wind vector, and adds this negation to the intended velocity vector, then he’ll get the direction and velocity that he must take. I asked him is he needs an approximation (would probably use the fact that sin(x) is roughly equal to x together with the fact that the wind is typically an order of magnitude less than the airplane’s velocity) but he was happy with the method I suggested. I think developing this method would be a nice question. I tried to do the trigonometry behind this and it was pretty painstaking for a human without a calculator. So, the right question to ask might be to ask the students to suggest a way to calculate the direction and velocity that they should set the instruments, where they’re only allows to use a ruler (with distance lines) and goniometer, but without a calculator, and are allowed small inaccuracies.

  3. esme Says:

    This is a wonderful blog post that I will be bookmarking for further reference.
    Of course, since this is the internet, I will complement my one sentence of praise for the majority of the content with lengthy critique of two aspects that I don’t like:

    1) “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.”

    Choosing this particular topic (at least as a major focus) for math class makes me shudder. It is self-referential and would qualify as a classical example of how to invoke stereotype threat.
    I have spent a lot of time investigating this topic because I have been forced by teachers, other students and colleagues to do so. I know a lot about it and the whole topic makes me sick.
    A contentious topic is fine, but it has to be chosen with care.
    And I have had the experience that my students find the question which door to open in the Monty Hall problem contentious enough (I had the gratifying experience of having a class of 30 students get out of control by yelling at each other “NO, YOU HAVE TO CHANGE THE DOOR!!!”, “NO, YOU ARE WRONG!!!”.)

    2) I think that this list of questions is more adapted to “mathematician visits high school”-programs than to regular high school teaching (and I intend to steal some of them for this purpose), and while I certainly know *some* teachers who would do well with such a course, I don’t see them working well with average teachers.

    I think that the first step is to do problems like this in a Socratic way with students who want to be teachers and, more importantly, with experienced teachers who take some additional training courses.

    • gowers Says:

      You’re probably right that the boys-versus-girls question is overdoing the contentiousness. Maybe one could soften it by simply talking about group A and group B. That way, people might well spot the contentiousness, but it would be more abstract and less personal.

  4. Matt Leifer Says:

    If you are serious about getting this sort of approach adopted at the national level then you need to think seriously about the framing of some of the questions. Questions about house prices in leafy suburbs and air travel are not going to appeal to students who have no experience of such things, and that is before I have even mentioned the word cricket. It would be good if there were some way of sourcing problems that are relevant to the communities that students actually live in.

    • gowers Says:

      I agree with that. Maybe questions that rely on particular cultural knowledge or experience could be given in several essentially equivalent forms.

  5. László Sándor Says:

    How would you relate the reforms or your proposals to the (pocket?) movement of Computer-Based Math?
    http://www.computerbasedmath.org

  6. Tim Says:

    An excellent list of questions!
    Here’s a suggestion for a trigonometry question.

    Tim and Tom go to the 14th floor Sky Bar of their hotel in Helsinki. Tim looks to the sea with his binoculars and says: “Hey, I think I see a building over there! Maybe it’s that 25-storey hotel in Tallinn where I went last week?” Tom thinks a while and says: “No way, Tallinn is 80 km away! It has to be a ship or a mirage!” Who is correct, Tim or Tom?

  7. Terence Tao Says:

    Benford’s law might be a good basis for a lecture or two for a course like this. The fact that about five times as many “real-life” numbers start with 1 as with 9 is always unintuitive when one first sees it, but I like illustrating it by surveying birthdays (how many students in the class are born on a day of the month starting with 1 vs 9) and birth months, both of which exhibit a Benford-like weighting. (Birth years are, of course, considerably more biased in this regard.) Once the students figure out these examples, they would be ready to start discussing why the law is applicable to more general statistics.

    • Terence Tao Says:

      (Minor correction: Benford’s law predicts about six times as many 1s as 9s, not five times as many.)

    • gowers Says:

      I seem to remember reading that it was obvious that the elections that led to protests in Iran were rigged because the results didn’t obey Benford’s law, and then a retraction by the person who originally made that claim. But having some fictitious election results and asking whether they were (probably) rigged could be interesting.

    • Richard Thomas Says:

      A variant on this would be to get the students to turn out their pockets and count pound coins and 50 pence pieces. Why do they have more pound coins, on average ?
      (I don’t see a nice variant of this for the US, unfortunately.)

  8. Spencer Says:

    This is really fantastic and I will certainly spread the message that more intelligent questions are needed and start trying to think of some myself. I agree in particular with your sentiment that such a course would benefit hard scientists and mathematicians as much as anyone: The real-world aspect of these questions aside, solving them is much closer in spirit to how mathematicians solve unfamiliar problems than the questions on the current A-levels!

    Regarding Objection 4, I agree that the prospect of examining such a course is a tricky one. Also, I agree that Objection 5 is the most serious and in fact it relates to some of the issues of Objection 4.

    Firstly, I think you would certainly need teachers with a very different kind of training that that which is deemed enough to qualify you as a teacher nowadays. The teachers wouldn’t necessarily need more or better training as teachers per se, but probably more or better training as thinkers and problem-solvers themselves. For example, as is obviously the case for yourself, *I* can imagine going into a classroom and discussing one of these questions without actually having a concrete “solution” ready to fall back on, but I worry that this is a result of a more serious mathematical training than that wich is generally gotten by those who become school teachers. It seems to fall into the general theme of `working effectively in a state of confusion’, which I see as a skill that comes from solving lots of initially-unfamiliar problems or from doing research.

    The reason why I say that this point relates to Objection 4 is roughly this: That ideal teachers would be competent and confident enough to continually assess their pupils and say, write course material or write and mark an open-ended exam in this style. However, what could happen is that it is in fact impossible to get enough teachers with the right skills and therefore the system falls back on more `traditional’ assessment i.e. a list of questions that more or less have correct answers. This way, a weaker teacher can feel confident because he or she can simply know the answers because he or she has read the special course book provided by the exam board (who just want to make sure the school chooses their board and so have made everything as straightforward as possible within the confines of the syllabus, thus eventually leading to lower standards for everyone – but I guess that’s another debate).

    Having said all that: Generally I think this is really exciting and think something like this could work. From my point of view, it even makes being a school teacher sound a lot more exciting.

  9. Richard Brown Says:

    I just saw a great talk on energy by David MacKay (thanks to Azimuth (https://plus.google.com/104603011082997519952/posts) on Google+), which is filled with Fermi estimation problems and practical numbers about energy use.

    For example, if all the fuel used to power the cars driving down a road was replaced with biofuel, how wide would the plantation beside the road that supplied the vegetative source for the fuel have to be?

    Or, if every household switched off their mobile phone charger would the impending energy crisis go away?

    Talk here: http://www.ted.com/talks/david_mackay_a_reality_check_on_renewables.html
    Free book here: http://www.withouthotair.com/

    Other related ideas: How much do you save by having your hot water cylinder turned on only at night time? How is it possible that a heat pump can be 400% efficient at heating?

  10. Juhan Says:

    How wonderful!

    It seems that these ideas are widely in the air already, and I can imagine that a polymath-type-of project incorporating all those who are interested, with their different skills and views, could (in theory) be most successful.

    Some examples of similar ideas:
    1) http://blog.mrmeyer.com/?p=13991 and the related website http://www.101qs.com/ – here visitors watch a video and come up themselves with mathematical questions related to real life situations.
    2) http://www.kickstarter.com/projects/mathalicious/math52-a-fresh-way-to-teach-0

    Also, very much in this direction, we were at some point thinking of building a website somewhat similar in style to “mathoverflow” in order to start collecting these types of life-relevant mathematics questions.
    The mathoverflow-voting style could just help to filter out the very best questions/situations, and the comments would help to offer some “model solutions”. I don’t know if this might be a reasonable idea?

    Best wishes,
    Juhan

  11. Tracy Hall Says:

    I second the inclusion of the Monte Hall problem on the list. I’d like to see a whole category of problems involving dimensional analysis, which seems eminently practical but largely neglected; this might overlap largely with Fermi-type problems. The question of why ants can lift several times their weight belongs in this category, with generalizations such as “If a 3 mm ant can lift 50 times its weight, could a 30 cm ant even walk?” I like the example that the usual units for fuel economy, distance per volume, are equivalent to units of per (cross-sectional) area, and the whimsical observation that this equivalence can be made concrete as the number of passing automobiles it would take to exhaust a petrol supply that took the form of filled drums lying end to end in the shape of a continuous cylinder of cross section one meter squared. Similar (and also whimsical) is this question: suppose a strand of round spaghetti is suspended along a walking trail at mouth height. How thick should it be so that eating it supplies exactly enough energy to keep walking along the trail indefinitely? I also like open-ended questions where the goal is to make sense of (devise a scenario that requires the use of) a given set of (possibly fanciful) units: Meter seconds per capita? Meters to the fourth? Quids squared per fortnight? Potato apples per orange? Kilogram meters squared per second squared?

    • gowers Says:

      I very much agree. In fact, the only reason I didn’t put any dimension-type questions in is that I couldn’t think of any good ones. I’ll try to make something out of the ideas you’ve just given.

    • Michael Carley Says:

      Murray’s Law for division of blood vessels is a nice one to do for dimensional analysis; also, a dimensional analysis of the Jesus Christ lizard followed by the question of how big Usain Bolt’s feet would need to be for him to run on water.

  12. Richard Baron Says:

    There should be scope for something around economic irrationality: tulip bubbles, dot-com bubbles, and the like. I think that few years ago there was an ostrich bubble, with people paying over a thousand pounds for each ostrich. (Was the farm in Belgium?) No-one seems to have thought about how much meat there would be on an ostrich, or the implied price per kilogram. Another example is the game in which people can bid for a 20 dollar bill. The winner and the person who made the penultimate bid before the hammer fell must both pay up, but only the winner gets the bill. Therefore, the price goes way above 20 dollars.

    I suppose the question would be, what sort of lack of information, or lack of power of deduction, makes economic irrationality of these kinds possible? And therefore, what market regulations would you put in place to stop it? I am not sure how far one could go with a properly mathematical analysis, keeping psychology that could not be re-cast in terms of game theory at bay, but there may be something here.

    A good source of scientific and political policy arguments that do not prove what they purport to prove is Ben Goldacre’s blog, http://www.badscience.net/ .

    One type of question would be “How would you estimate the revenue effect of such and such a tax change?” An example of how such estimates can be made, concerning the top rate of income tax in the UK, is:

    http://www.hmrc.gov.uk/budget2012/excheq-income-tax-2042.pdf

    You mention the irrationality of buying insurance for washing machines, because the insurer makes a profit and the consequences of breakdown are not disastrous (or, to the extent that disastrous – massive flooding – probably not covered by the insurance). I feel entirely rational in buying one lottery ticket a week, even though the expected value is negative, because the negative amount is trivial and the consequences of winning a large amount would be rather wonderful. There ought to be scope for a question there about what sorts of attitude to risk and reward you need in order to make it rational to choose certain distributions of positive and negative values, and probabilities of those values.

    Come to think of it, there could also be a question about how to design a lottery’s distribution of prizes so as to maximise the expected surplus for the lottery operator, assuming that people need a reasonable chance of winning something in order to be persuaded to participate, and also need the small chance of a big prize in order to be lured into participation, and not to see participation merely as an activity with negative expected value. If we assume that the lottery is already up and running, the reasonable chance of winning something could be put in terms of “each person has 20 friends, and will only participate if he, or one of his friends, has won something in the past year”.

    On assessment, I think a big problem would be how to give markers the time to read answers carefully, and to recognise merit in answers, even if they went at the problem in a different way from that used in the marking guide. I have heard rumours that A level marking is now so much of a mass production business, that even in the humanities (where there are lots of different ways of answering questions – to a much greater extent than in mathematics and the natural sciences as studied at that level), candidates who do not say the right things according to the marking guide suffer.

  13. Mark Wainwright Says:

    I once had a short list of questions like this, with a vague intention of writing them up in some form, but instead I have forgotten them all. Well, not quite all: my favourite concerned a time I went to get my haircut. You might remember the old chap on All Saint’s Passage who had been cutting hair there for fifty odd years and would proudly tell you he’d never been further from Cambridge than Ely. “Funny thing”, he said to me once, “when two chaps come in to the shop, it’s almost always the one with shorter hair that wants it cutting.”

    The question is how funny a thing this is, and to answer it you have to make some model of how people keep their hair. The simplest plausible one is that everyone has some ideal length for their hair, and have it cut back to this length when it exceeds it by say 10%. If that’s right then the barber’s observation is quite unsurprising: indeed, the greater the difference in hair length, the more likely it is the friend with shorter hair who wants a haircut. Conversely, what implicit (and wrong) model do we have when we first hear the observation that makes one think at first blush it will more usually be the longer-haired friend who wants a haircut? This is an open-ended but rather interesting question, I think.

    Here are some random other things that come to mind that involve mathematical thinking – perhaps in a quite vague way.

    Your question about compound interest is, as you say, dull, but there is a related interesting question or rather discussion point. How much is it worth to you, personally, now to have £1000 in one year? However one approaches this, the crucial observation in my view is that your life expectancy is relevant. After all, even a billion pounds payable in 100 years is worthless to me (unless of course I can sell it to someone else). Your view of investments will be different if your life expectancy is in the hundreds or thousands of years, as indeed it is if you happen to be Trinity College, say.

    Vaguely reminiscent are questions about how you align the interests of different parties, say a company and its employees. Should a software company pay a bonus depending on, say, the number of lines of code written or bugs fixed? (Both are disastrous, of course.) Suppose a hypothetical company sells formula baby milk and is accused of aggressively marketing it to breast-feeding mothers, when continuing to breast-feed is safer (and cheaper). The company claims its guidelines forbid this, which let us say is true, but it pays its sales reps on commission. What’s the expected outcome? What would a better system be? (A very open-ended question depending whose interests are being considered!)

    Another similar can of worms: at what rate should benefits paid to low earners be withdrawn as their income goes up? How does the answer relate to the purpose of the benefit and to ideas about progressive taxation? Are the two in conflict? (Personally I think the case for a basic income is unanswerable, but sadly it is perhaps an idea, like the Alternative Vote, whose time has not yet come.)

    If a researcher publishes a study showing an effect of drug A on condition B, can we conclude the effect is real? There is sample statistics involved that is too much like a straight maths syllabus, but there is also the vital question of whether the study was more likely to be published than one showing no effect. What can be done to (a) detect, (b) ameliorate this problem? (Ben Goldacre often writes entertainingly about this: e.g.)

  14. Mark Wainwright Says:

    Oops, this link went missing from the end of the comment above: http://www.badscience.net/2008/03/beau-funnel/

  15. Andre Henriques Says:

    Dear Tim: the kind of persons you want to get input from for this project are not professional mathematicians (I would say that, on average, a professional mathematician has no idea of what will and what won’t work in a high school classroom), but rather people who fit the following profile: 1) have a PhD in math 2) are enthusiastic 3) have spent at least a couple years teaching at high school level. Ideally, they should also 4) have some experience in the kind of alternative teaching that you propose… but maybe that’s too much to ask for.

    I know two persons that fit the above description in the Netherlands, and who I would trust to come up with very good suggestions/ideas/comments that go in the direction of what you’re looking for. Presumably, they also exist in the UK.

    PS: I’ve recently spent a year teaching at high school (and thus away from research), and so I have some idea what I’m talking about… but I’d say that you want people with more experience than me.

  16. abrahamdavidsmith Says:

    Exploring questions in this way for young students (say, ages 13-18, what we would call middle school and high school in North America) is one of the goals of many of the local “Math Circles” programs in the USA. E.g., https://www.nymathcircle.org/

  17. PJ Says:

    How should mathematicians teach non-mathematicians? The same way adults teach babies, then children, then adolescents, then other adults. Translate the language to your specific audience.

  18. Musings… › Leaving certificate maths. Says:

    [...] is a debate going on in the UK regarding maths teaching up to the age of 18. Timothy Gowers has posted on this today and there is some interesting stuff there too. It’s also worth noting Christian [...]

  19. Anonymous Says:

    You drive over a hill and see a red traffic light in the distance. Should you slow down, speed up, or neither? You come around a corner in the woods and see a red light 100m ahead; same question.

    This leads into a discussion of what you want out of your encounter with a traffic light, how traffic lights work (in some places, there are sensors that turn the light green *only* if there is a car waiting at the red-light line), and to model the uncertainty in how long the light has already been red.

  20. dmoskovich Says:

    I would make the topics as “uncontentious” as possible in a strong sense- nothing at all about race, gender, religion, or nationality. You can have “teams” with evocative names- “Team Talent gets an average of X% on a public examination, while Team HardWork gets Y%”.

    I wonder whether it would be a good idea to have problems involving shapes. For example:

    Topic: Ruritania has a fingerprint database containing N>>0 fingerprints. These fingerprints are taken from fingers pressed at slightly different angles, different resolutions, etc. An English visitor, Rudolf Rassendyll, who arrives in the country, looks familiar. “Haven’t we seen that face somewhere before?” the guards and the border ask one another. They take his fingerprints. How could a computer compare Rudolf’s fingerprints to others on file? In general, how could one go about comparing fingerprints of differing angles and differing resolutions, maybe some with smudges, etc.?

    This opens the door to discuss pattern recognition- how does one identify a shape, or a pattern? What might be the degree of confidence in one method or another? In a certain village, there was a fire in a house, and the shape of the ashes on the ceiling is said to resemble Elvis Presley- is this pure chance, or is it a miracle? How could such a question be approached?

  21. David Says:

    It would be absolutely wonderful if questions like these made their way into school mathematics, and I agree that they’re too good to restrict to those students who are taking “maths for non-mathematicians” courses. At first-year university level, I’d much rather teach students who could tackle an estimation problem intelligently but don’t have their heads full of intricate procedures for differentiation and integration than vice versa… In fact, one of my perpetual struggles with such students is to wean them away from a blind “what’s the formula?” approach and help them learn to analyse a problem in the way you’re suggesting.

    The dispiriting thing I’ve found is how strongly many students object to such problems, especially when there’s any hint of assessment. I think the trouble is that by the time they reach me at age 17-19, these guys have had years of being taught that maths is entirely procedural and all about getting the “right answer” by some occult means. Thus, any approach that seems more discursive or open-ended strikes many students as unfair — an attempt to catch them out by setting questions that aren’t “proper” maths.

    Given this problem, I’d advise that the kind of question you suggest should be introduced a lot earlier — maybe even at primary level? — so that it naturally becomes part of students’ understanding of what maths is about. (There has been some research done on how students tackle “word problems” which suggests that it’s very easy to establish and reinforce a habit of tackling them in a more-or-less ritual manner, without trying to understand the problem — see e.g. Alan H. Schoenfeld’s article “A Modest Proposal” in the AMS Notices, Feb 2012.) I fear that if any project to introduce meaningful real-world problems to mathematics teaching is left until the post-16 level when a lot of students have made their minds up about what maths is and whether it’s for them, it will be doomed to die under the weight of student apathy.

    Having said all that, good luck — if questions like this can be sensibly introduced into the school curriculum, the vast amounts of ink spilled over school maths education in recent years won’t have been entirely in vain!

  22. Fergal Daly Says:

    This case of John Puckett is a good one. The police matched crime-scene DNA against a database of 300,000 sex offenders. He was matched at odds of 1 in a million. Those odds mean there are about 300 other people in the US who would also be such a match. So without any other evidence, his chances of being guilty are 299:1 . This argument was not allowed in the court and he was convicted on the million to 1.

    https://www.google.com/search?hl=en&safe=off&q=%22John+Puckett%22+dna&oq=%22John+Puckett%22+dna&aq=f&aqi=g-K1&aql=&gs_l=serp.3..0i30.967.1098.0.1187.2.2.0.0.0.0.68.134.2.2.0…0.0.WeSAME71sNE

    • gowers Says:

      Thanks for that — as you say, an excellent example, which I’ll add to the list. Another point worth making is that even if you think that convicted sex offenders were more likely to have committed the crime in question, you can’t get away from the fact that the probability of a match, in the evet that all 300,000 were innocent, was 1-(1-10^{-6})^{300,000}, which is about 1-e^{-3/10}, which is roughly 1 in 4.

    • Fergal Daly Says:

      It’s interesting that the 1 in 4 figure was the one that the judge refused to allowed as “irrelevant” and is also the harder one to explain to the jury. Judges in the US seem not to want to reveal DNS database dredging as the source.

      It’s not clear to me from reading about it whether his defence team asked or would have been allowed to ask the DNA expert witness “what does 1 in a million mean in a country of 300m people?”

  23. ysella Says:

    I really like the idea of such a math class.

    I think the problem with many math classes is that they teach how to solve particular types of questions — which for many people amounts to memorizing procedures– rather than how to think mathematically.

    However, I do worry that the sort of class you propose departs too much from the mainstream, and I think you might have a hard time convincing the schools to adopt a curriculum full of open-ended questions. Perhaps it might be better to have more of a mix, where the open-ended discussion leads to some mathematical exposition?

    Some thoughts I had:
    -I was thinking about how an English class is taught. It has the nebulous goals of teaching the students how to read and write. But it also has the more concrete structure of reading a certain list of books. Maybe this class, too, can be structured around concrete units even while attempting to teach how to think mathematically.
    -It also seems that a class needs to have a sense of progress, rather than just having a long list of problems.
    -Many of the problems you gave do start from a real-world situation, but still seem more like “puzzles,” and might give the impression that it’s still not very useful. What if the class teaches actual applications of math? Math is all around us, and it might be enlightening for people to understand the mathematical underpinnings of things they take for granted or simply haven’t thought about.

    There’s the math of computers (the basic idea of a Turing machine), the math of cryptography, the math of physics or economics (which you gave some examples of). When it comes to algorithms, the problems you gave are fun, but you can also teach about computer algorithms, which makes the computers so fast and good at what they do. I think your stress on probability is important, too, since thinking probabilistically is an important skill that often goes against our common sense. I like the idea of providing real data and analyzing whether it is significant.

    Though some of these topics are less open-ended, the discussion can still be Socratic. Perhaps there can be a mix of “puzzle” problems, and applications– puzzles to get the students thinking, and applications to show them how similar problems arise in major applications.

    You might say that this completely departs from the idea of teaching students how to think mathematically, but I think that just as, by reading good books, one can learn how to write, one can learn how to think mathematically by learning about others’ mathematical solutions to big problems, and that if you combine open-ended discussion with exposition of the math that’s all around us, students will gain an ability to think mathematically as well as a better understanding of how math is used in the world.

  24. Jack V Says:

    Unsurprisingly, I tend to like your views on maths, and be very suspicious of Michael Gove’s. That said, when he says that the vast majority of people should continue maths to 18, I think he’s looking at the problem of a vast majority of people who don’t really know anything beyond basic arithmetic because they learned maths to GCSE, it seemed completely irrelevant, and they forgot it all as a useless academic exercise that was never going to matter.

    I don’t think an extra two years of maths is going to help that problem — what most people need could easily be taught at primary school, it just doesn’t stick, so an extra two years of being forced to parrot results you don’t understand probably won’t make people know the basics they never understood better.

    So we seem to have two different problems. One is, “for people who are reasonably academic, but can’t cope with the abstract symbol manipulation, is there another genuine A-level equivalent course that might suit them better?” Another is, “can we do anything so people actually understand simple real-life questions like:

    * If you have any choice in the matter, when you buy or borrow something, look at the total expenditure, not just what you have to pay weekly to start with
    * If something is 15% off, what would you expect to pay?
    * If a news article describes an experiment, can you tell whether it actually supports the claim in the title or not?”

    The last is not exactly A-level material, but it seems like what people need to be taught at some point in their lives, that they’re unfortunately not getting.

  25. Weekly links for June 10 « God plays dice Says:

    [...] Gowers asks How should mathematics be taught to non-mathematicians? The post is motivated by certain proposed changes to secondary education in the UK, to introduce [...]

  26. ogerard Says:

    Dear Mr Gowers, among possible resources, you might be interested by “Street-fighting mathematics” by Sanjoy Majahan.

    It is available here: http://mitpress.mit.edu/books/full_pdfs/Street-Fighting_Mathematics.pdf

  27. Alan Crowe Says:

    Comparative advantage is simple enough to be taught at 16 and goes a long way towards explaining the adult world to any children who are curious about the pickle they will soon be dumped in.

    The template goes like this: Mr A is trying to make x X’s and y Y’s. His productivity is p of X per hour and q of Y per hour. How long will this take?

    Mr B is trying to make x’ X’s and y’ Y’s. His productivity is p’ of X per hour and q’ of Y per hour. How long will this take Mr B?

    Now for question three, the one that is interesting. Suggest a way of sharing the work that reduces work times for both Mr A and Mr B.

    In the case that p>p' and q<q' there is an obvious swap. This is called absolute advantage. Provided that p/q is different from p'/q' there is comparative advantage and it remains possible for Mr A and Mr B to both reduce their work hours a bit by swapping
    some of the work, even in the case that p >p' and q >q'.

    This is part of the reason why there are jobs. Working through numerical examples one soon notices that the swap always involves being more specialised. Children are heading towards an adult world where they are expected to take a job: engineer, teacher, lawyer, builder, plumber, etc. This arithmetic goes some way towards telling them why (But why are they gains from trade so large that this is compelling? There is an extra story to be told about process improvement.)

    In the case that p > p’ and q > q’, it is often worthwhile for Mr C to employ Mr D even if C is better at the job than D. This is another part of the story of why there are jobs. And it is often worthwhile to buy something rather than make it yourself, even if you could make it more quickly than the person you buy it from.

    This exercise can be opened out in all sorts of contentious directions. What if there are three participants? If Mr A improves his skills, will Mr B change his trading, to Mr C’s detriment?

    More contentious still: can one push things further, reducing the total hours worked, but sacrificing mutual benefit? Perhaps Mr A loses and Mr B wins big, for a net gain. Yes! But trading was self enforcing, both won. Nobody had to force Mr A to do extra for the common good. That is the start of politics. Perhaps the children will find this boring, but at least the lesson is trying to educate them about what the adults are quarreling over.

    It strikes me that teaching this is a high stakes gamble. Gains from trade are a kind of something-from-nothing magic. If the children take into adult life the idea that they should have faith in something-from-nothing magic that will be bad. If the children take into adult life the idea that they can (and are supposed to) check the arithmetic of any proposed something-from-nothing magic themselves, that will be good.

  28. Stijn Says:

    I am worried these questions are actually very hard for people that find mathematics difficult. If I understand correctly, the questions should be approached in a modeling, eye-balling, physics type approach. It requires mapping back and forth from a somewhat open problem description to a mathematical description, hopefully isolating the essence. Especially for the probabilistic/statistic type (but others as well, e.g. optimisation) problems it could be very daunting – so many ways to get off the right track or never even find it (allowing that the right track can be broad and allow different approaches). It requires understanding of mathematics, the world, *and* the map between them. I’ve always felt physics is much harder than mathematics :-) This is not to dismiss this approach, and I am curious what the response is to these sentiments.

    • gowers Says:

      To some extent I’m not bothered if the questions are hard. I think one of the most important benefits that a course of this type could provide is making people aware of what kinds of questions mathematics can and can’t answer. In the positive direction, there is far more to mathematics than arithmetic. In the negative direction, there are situations that are difficult to model convincingly, or difficult to measure well enough to apply a model even if it is convincing. An appreciation of the difficulties, as well as of the fact that with hard work the difficulties can sometimes be overcome, would help people make better decisions in later life. Sometimes those better decisions would take the form, “Ah, this looks like one of those tricky situations where the statistics can be misleading: I’d better get expert advice.”

  29. Emilio 'Mnemonic' Pierro Says:

    Mathematics education is something I’ve spent a lot of time thinking about, partly from having an interest in, and partly because my tutor at University was Tony Gardiner. I certainly agree that mathematics education should be extended to age 18, and (apologies if I’m treading over old ground here) it should serve at least two functions:

    1) For those not continuing with mathematics afterward it should give them important life skills to take away, things like logical reasoning, argument structuring, etc.

    2) For those intending or unsure whether to continue, it should give them a taste of what University mathematics is like.

    One thing to remember which I’m sure is obvious, but worth stating, is that you could have the greatest course of Mathematics for 16-18 year olds in the history of everything ever, but if their preceding 10 or more year of education have left them with a resentment of mathematics, then you will get nowhere. Trying to tackle the relationship between British students and mathematics has to start at day 1 of their education, as well as the perception of mathematics in our culture in general.

    Has there been any talk of adding Logic to the pre-University Mathematics Curriculum? I think a study of some of the more “basic” concepts of Logic would be so beneficial to anyone leaving school since they will then have a much greater understanding of, for example, the difference between two things being consistent with one another, and one thing implying another. Although that’s perhaps motivated by a desire for people to be able to decipher political spin…

    I think an emphasis (for people looking to continue with mathematics) should also be placed on the realisation that mathematics is not just about plugging in numbers to equations, or manipulation, but that it is about rigorous proof. Nothing too hardcore, obviously, but perhaps some gentle introduction to it. Starting with Euclid?

    I was shocked at University when I heard from some Finnish friends of mine (who were not studying Mathematics) that at college (I think they were doing the International Baccalaureate) they had learned things like Set Theory. Maybe that could also be added?

  30. Mark Bennet Says:

    Some questions that come to mind:

    You are building a house. How many bricks will you need to order? How strong will you need to make the foundations? [having lived in two where the calculation has not been done correctly]

    How much will the top of a tower block move in a strong wind?

    How much fuel would you need to send a human being to the moon (and back)?

    How many wind turbines would it take to replace one conventional power station?

    How much material would you need to make a school uniform for every child in the school?

    • Richard Baron Says:

      Mark Bennet’s questions contain interesting subtleties, which I guess is what is wanted.

      Replacing all of your power stations is not a multiple of replacing one of them, because you would then have nothing on which to fall back should the whole country be becalmed. One might also ask students to estimate the probability of that happening, assuming a wide distribution of windmills across the country, in order to decide what other sources of power to retain so as to reduce the risk of a blackout to an acceptable level.

      To get the right answer for school uniforms, we do not want the mean height of the children. But we do not quite want the mean of squares of heights (on the basis that we want the surface area), because we must allow for gangly teenagers who have grown vertically more than horizontally.

    • Mark Bennet Says:

      Very glad, Richard, that you’ve seen some of the complexity. The wind turbines is interesting and topical and also raises questions of how one might store energy and manage peak loads (I remember my dad taking us to a place in Wales where the pumped water uphill to do this). Planning for a base load on the system is clearly important. I remember the three day week and homework by candlelight in the 1970s.

      The uniforms also depend on design and the laying out of patterns on cloth, which brings in some geometrical considerations, and thinking about the most efficient design and pattern.

      The tower block depends on materials, and raises the question of how we know which are the best materials.

      Getting to the moon (or Mars) is interesting because it raises questions about what we know, and how we can extrapolate from what we know to what is unknown. Are we really prepared to trust our mathematical models of the world? It also raises the question of the energy density of fuel and the loss involved in lifting the fuel we need later. It will illuminate why rockets have been made in stages, and why fuel design is a critical element in rocketry.

      The house thing is just a down-to-earth practical thing. You could also ask about the design of bridges (which I mentioned above). How do you test whether a bridge is still strong enough to be safe (after an accident/flood/explosion)?

      The point is to have rich, open-ended questions for reasonably intelligent young people to explore and which motivate them to understand why mathematical models of the real world have been so powerful, and to help them to engage with mathematical concepts which will inform practical applications consonant with their mathematical ability.

      One of my brothers could, I am sure, formulate some excellent questions motivated by the performance of vehicle engines, and also the design of bridges. Another is part of a firm which helps chip manufacturers to minimise energy consumption.

  31. thorehusfeldt Says:

    Computer Science Unplugged (http://csunplugged.org/) has lots of good algorithmic activities, many of which fill the same role as the questions formulated in this blog post: they are engaging and provide platforms for using the conceptual framework of mathematics as a mode of communication and problem solving.

    An example are sorting networks, which make question #34 above both more concrete and more open-ended. Do check out the videos as csunplugged of people running around on sorting network diagrams painted on the floor! It’s hugely entertaining.

    I’ve implemented this myself. Much better than with playing cards, if you have the floor space. After this activity you can engage people of all ages and background in real algorithmic thinking: was this optimal? (What does optimal mean, by the way?) If not, here’s a piece of paper. Draw me an optimal network. Maybe only for n=4. How do you convince yourself (or me) that it actually sorts? Do you need to check all numbers? Is it enough to check only for all permutations of 1,2,…,n? For all bit patterns? Oh, by the way, does the n=4 case help you in solving the n=5 case? n=8? Does it sort backwards? Do all sorting networks sort backwards? I claim the network on the floor is optimal: How did I convince myself of that claim? How would I convince you? How large is the proof? How large sorting networks can we prove optimal in that way before the universe ends?

    • gowers Says:

      Many thanks for that tip. In a few seconds it’s given me an idea for another question, and has the potential to provide several more.

  32. sybillabeckmann Says:

    Nice questions!

    I would like to invite all of you who are interested in math teaching — at any level — to read and contribute postings on math teaching at the Mathematics Teaching Community, https://mathematicsteachingcommunity.math.uga.edu/

    I think that a course whose focus is working on questions like the ones given here could be genuinely interesting and worthwhile for students. But to be successful, the course must be designed so students actually engage in thinking about the questions and so students learn some skills for making progress on such questions. Teachers would need to be quite skilled not only with the math itself, but also with helping students learn productive ways to work on a problem.

    It would be interesting to hear from people who have taught such a course about how they do it.

    In the US, the Common Core State Standards for Mathematics have several standards that can be addressed by working on open ended problems such as the ones given here. There is a standard for mathematical practice on modeling (4 Model with mathematics), which applies to all grade levels. Modeling is one of the conceptual categories in the high school standards; it does not have its own separate set of standards but cuts across the other conceptual categories.

    There is a long history of work in mathematics education on improving students’ problem-solving skills and on centering instruction on problem-solving. One example: the Interactive Mathematics Program is a high school curriculum that introduces mathematical ideas by starting with complex problems.

  33. Charles Crissman Says:

    A few more questions, in order of how much I like them:

    1) You arrive at the Department of Motor Vehicles and there is a long line to see a clerk. You need to be at work in one hour. How can you decide if you should bother staying in line, or if you’d be better off walking across the street for a coffee?

    (This is an application of Little’s law.)

    2) 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?

    We’ve used this question in calculus courses at Berkeley and it’s surprising how many people are passionately sure that you should increase by $10.

    3) You have a biased coin whose bias you don’t know, and you want to choose fairly between two outcomes. How can you use the coin to do so?

    4) How would you figure out if a coin was biased?

  34. Gil Kalai Says:

    The post raises many interesting issues. I join Daniel’s suggestion to avoid contentious questions. I also think it is important to have many easy questions and to avoid overlly hard questions. For example, you need to spend much time to explain what expectation (in probability) is and how it is useful before moving to cases where you need more. You may describe a few questions where mathematics does not give a clear answer or which are controversial but, again, most questions should not be of this type.

    Regarding statistics and law. This is an interesting topic and there are a few interesting cases in addition to those mentioned. But these issues are often quite complicated. Perhaps we can start by asking ourselves: Consider the notion of guilty “beyond a reasonable doubt.” Practically speaking, what “reasonable doubt” stands for:1/1000,000? 1/100,000? 1/10,000? 1/100? 1/20? 1/10? 1/5?. It will be interesting to poll it…

    • Richard Baron Says:

      It may or may not be appropriate to translate the phrase “beyond reasonable doubt” into a numerical probability of reaching the correct verdict. But that question of what probability would be good enough was explored by Condorcet in 1785, in his Essai sur l’ application de l’ analyse à la probabilité des décisions: rendues à la pluralité des voix. His conclusions are set out on pages cxii to cxiv. He asked, by how much would your risk of your being killed within a week by disease increase from age 37 to age 47? The point was that this increase was something that people would regard as negligible. He obtained a value of 0.000007, so the required probability was 0.999993. (He used fractions, 1/144,768 etc.)

      He went on to ask what majority one would require from a jury of a given size, with a given probability of each member’s making a mistake. The jury members are assumed to think independently: Condorcet Jury Theorem, and all that. There may be some questions to be made of that: what does a majority opinion show, and under what conditions?

      I have been back to the source, but I first became aware of this text through Lorraine Daston, Condorcet and the Meaning of Enlightenment, a British Academy lecture from 2006, published in Proceedings of the British Academy, Volume 151. As she said, the calculation shows you everything that is both attractive and repellent about the Enlightenment.

    • gowers Says:

      I agree that having several easy questions is a good idea. I also think that having hard questions is a bad idea if you expect to give complete answers. But I also think that there is an interesting class of questions where, for example, setting up a model and analysing it exactly is hard, but obtaining a good qualitative understanding is much less hard (though not trivial). I would like to see questions of that sort included, together with tips about how to teach them.

  35. Américo Tavares Says:

    Just an example. Suppose you have a power plant in which the temperature of the steam entering the turbine is measured by sensors in order to be controlled. You know the MTBF (mean time between failure) figures of different types of sensors. You have to decide to install 1 or 3 sensors. In the latter case a separate device will choose the 2 more similar measurements out of the 3. What would be the expected reliability of each option? Consider that the rest of the plant is responsible for let’s say 98% of the overall reliability. Can the use of 2 sensors have any advantage?

  36. Américo Tavares Says:

    Correction: You have to decide to install 1 or 3 sensors, but in the latter case a separate device will choose the 2 most similar measurements of the 3.

  37. anthonyquas Says:

    (I just posted this as a response to your posting of 2 years ago, but meant to post it here).I think questions about the political decisions of the day are fairly fascinating. Especially when it comes to questions of what’s good for the individual/what’s good for society at large.

    A recent example from the United States: discussion of health care policy. Everybody wants affordable health insurance coverage, but relatively few people have seriously thought about the link between enforcing a large pool of insured people and the feasibility of requiring insurers to cover all applicants. The overwhelming framework for debate is from the viewpoint of the individual.

  38. Gil Kalai Says:

    Regarding the questions of how reliable were the convictions in the murder cases of questions 9 and 46. Tim, what would you regard as the level required for conviction “beyond a reasonable doubt?” What do you regard as the actual required level in reality? (Say in England or the US).

    • gowers Says:

      I don’t immediately know, so I would try to come up with some general principle for tackling the question. Here is the kind of thing I mean.

      1. It’s clear that 100% certainty is unattainable in many cases.

      2. Therefore, we have to strike a balance between convicting innocent people and letting guilty ones go free.

      3. If you let guilty people go free, it can sometimes have terrible consequences for other innocent people.

      4. For reasons that are very hard to justify fully, we are not pure utilitarians. In particular, we tend to prefer policies that lead to actual benefits we can see to policies that lead to greater benefits of a more statistical kind. For example, if a child’s life can be saved with the help of a very expensive drug, we will instinctively prefer spending a lot of money on that drug to spending the money on a campaign to get people to eat more healthily, even if the latter can be shown to save several lives.

      That is not supposed to be a complete list of relevant principles, but once I had collected such a list, I would then estimate how many innocent people would be convicted if I slightly increased the amount of doubt I was prepared to tolerate, and how many more guilty people I would convict as a result, and what the benefits to society were likely to be. How I would cope with 4 is hard to say, but the fact that the increase in wrong convictions would itself be statistical would perhaps make me more utilitarian than if I had just one case to consider. (In the case of John Puckett, I should say, the level of doubt was way too high and the benefits to society of convicting an old man who had not offended for many years were zero. So in that case I am confident that it was wrong to jail him.)

    • Gil Kalai Says:

      Thanks, Tim. That’s interesting! Still in the context of convicting people for murder what is roughly your own threshold for “a reasonable doubt?” and what is roughly your estimate of the threshold in UK or US judicial systems?

      Usually, it is a strict principle in courts that the threshold for convinction does not depend on consideration like the age of the defender and how dangerous he is. Even if you violate this principle it is not clear which way it should go in cases of people who escaped punishment for a long time.

      Another famous case of statistics in court is the case of Lucia de Berk http://www.math.leidenuniv.nl/~gill/hetero_def.pdf (see the homepage of Richard Gill.)

  39. Gowers’ Blog: “How should mathematics be taught to non-mathematicians?” « Augusto Gerolin's Blog Says:

    [...] How should mathematics be taught to non-mathematicians? [...]

  40. What does “beyond a reasonable doubt” practically mean? | Combinatorics and more Says:

    [...] by two questions from Gowers’s How should mathematics be taught to non mathematicians.) Share this:ShareFacebookRedditTwitterLike this:LikeBe the first to like this post. This entry [...]

  41. Pela blogosfera: “How should mathematics be taught to non-mathematicians?” « Problemas Teoremas Says:

    [...] Post de Timothy Gowers em que discute “interesting real-world mathematics problems”: [...]

  42. Pela blogosfera: “How should mathematics be taught to non-mathematicians?” « Problemas Teoremas Says:

    [...] Post de Timothy Gowers em que discute “interesting real-world mathematics problems”: [...]

  43. Anonymous Says:

    This is a very interesting post. However, I suspect that Mr Gowers is being hopelessly over-optimistic about its feasibility. I feel that there would be very few A level standard teachers who would be confident in analysing some of these problems themselves, with questions ranging from notoriously tricky areas like stats and probability, to physics, to logic and and the construction of arguments.

    To my mind, there is a gulf already between the average modern A level question (very constricted in topic, with much hand-holding, and almost no emphasis on anything like formal proof) and what I’ve seen suggested above. And plenty of A level students struggle with what they have to do now. I can’t imagine that too many non-A level students (at whom this is being targetted?) would be able to cope with such a course.

    And how would it be marked and graded, bearing in mind the current “look at the marking scheme” approach for current maths A levels?

    This kind of course may be feasible when given by professional mathematicians or statisticians or physicists at university level, but, as proposed, for the average British 6th former, in the average British school, it’s pie in the sky. (All IMHO, of course)

    • gowers Says:

      To some extent I’ve addressed these points in the post. However, I’d add this.

      1. You may well be right that it is overoptimistic. My main wish is that something like this should be thought about seriously, to the extent of running pilot projects.

      2. The questions above are of variable difficulty: maybe the easier ones would be OK and the harder ones not feasible.

      3. All I really care about is that people would think about these questions, and as a result of classroom discussion come to think about them in a more sophisticated way — even if they fall well short of a full answer.

      4. There are a lot of other things one might do with a 16-year-old with a C at GCSE, but many of them, though they sound more reasonable, are pie in the sky as well.

      5. If I had my way, it wouldn’t be marked at all. It would just be a series of interesting discussions that people would remember in later life. (I can remember such discussions from my own school days.) However, it is unlikely that a course will be taken up if it is optional and carries no credit, so I think the best that can be hoped for is that if you get a bad grade on the course then it will not have a significant negative impact on your life opportunities.

      6. The thing that’s missing from the post is detailed teaching suggestions. I’ll try to find time to provide some, since that will give a better basis for judging the feasibility of the idea.

  44. Anna Says:

    How likely is it that two people in your social circle will plan their birthday parties for the same day as each other?

    This looks like an alternate phrasing of the famous birthday problem (“in a room of x people, what’s the likelihood that 2 share a birthday?”), albeit one phrased to be a little more relevant to the real world (I have no particular reason to care if 2 random people happen to share a birthday, whereas I have every reason to care if my inability to be in 2 places at once forces me to choose between friends). However, there are a few extra things to consider here, such as:

    (1) Some days of the week are more popular than others for parties: for example, if one friend of mine has their birthday on Thursday and another has their birthday on Saturday, there’s a reasonable chance they’ll both try to plan their parties for Saturday.

    (2) Point (1) above doesn’t necessarily always apply: during the school holidays, people are more likely to be free for parties on any day of the week. During term time, it’s probably limited to weekends only. So how does the school calendar affect the chances of party overlap?

    (3) Similarly, people are unlikely to have birthday parties on days that are already special events in their own right – e.g. Christmas, Halloween, New Year’s Eve. Does this have a significant effect on the party distribution; is it something that needs taking into account?

    (4) People in the same friendship group are likely to talk to each other about party plans, and thus avoid schedule clashes. For example, if two friends from school have the same birthday as each other, they’re likely to either have a joint party or organise it so that their parties are on different days. But a friend from school might still have their party on the same day as a family member or friends from outside of school – someone outside their social circle but in yours. So now you have to estimate how many non-overlapping social circles you’re part of, and what the chances are of two people in different circles sharing a party date.

  45. Anna Says:

    Ooh, also, the Monty Hall problem is always good for getting a maths-centric debate going.

  46. girlsangle Says:

    This is a wonderful blog post. I think that the central issue it addresses is motivation, specifically, how does one motivate people to study more mathematics? It has been my experience teaching at the K-12 level that what will excite a student is unpredictable. For example, the interest rate question that some consider “boring” may actually excite others and be their gateway to the world of mathematics.

    I think that building a list of such questions is of value. The point I wish to make is that perhaps rather than focusing primarily on building a list of questions it is a good idea to (also) focus on how to discover what interests a student has (what particularly excites him/her) and how to create such questions (that would belong to your list) so that they pertain to the discovered interests of the student.

    So I would suggest that a key additional criterion for the teacher of such a class should be that the teacher knows how to seize upon the student’s interests and create such questions in addition to knowing how to handle the teaching of them.

  47. Cardster Says:

    I have two themes of comments.

    One is from ANU’s Professor John Hutchinson: “the idea behind… ” whatever you are trying to engange your audience with is ..,

    The second is what does a mathematicla education offers; quite a lot in my view as it encourages you to examine the elements of a problem and to move back and forth from the particular to the general as a way of understanding a problem to be solved.

  48. Anonymous Says:

    [this is a reply to a TG's reply to my post - for some reason I can't reply directly to his (but I could to anyone else's ??) ]

    “To some extent I’ve addressed these points in the post. However, I’d add this.”

    This is true. I’m afraid that I skimmed your original post. Sorry.

    “1. You may well be right that it is overoptimistic. My main wish is that something like this should be thought about seriously, to the extent of running pilot projects.”

    OK. I think that it is a good idea too, but it would have to be replicable in the typical school setting, with the typical teacher, to have any chance of success.

    “2. The questions above are of variable difficulty: maybe the easier ones would be OK and the harder ones not feasible.”

    Part of my scepticism lies not in the difficulty of the questions per se, but in more basic issues: at whom is it targetted? (A level maths students, or the new maths-to-18-for-all students – these will require different offerings, IMHO). What, specifically, would the students learn? (Estimation? Basic use of stats? Math/Physical modelling?). How much real maths would there be? (Related to my first question, of course). If there’s no real maths (e.g. nothing up to A level) why would an A level maths student take it, given that they have other things to study that unis will want them to study?

    “3. All I really care about is that people would think about these questions, and as a result of classroom discussion come to think about them in a more sophisticated way — even if they fall well short of a full answer.”

    I agree that it certainly wouldn’t hurt to have more open ended discussion (with some underlying mathematical rigour) in the school system, but I feel that exam boards, students and teachers would feel that it’s moving in the opposite direction to what’s been happening over the last 20 years. That’s a big tanker to turn around.

    “4. There are a lot of other things one might do with a 16-year-old with a C at GCSE, but many of them, though they sound more reasonable, are pie in the sky as well.”

    I don’t understand what you mean by this.

    “5. If I had my way, it wouldn’t be marked at all. It would just be a series of interesting discussions that people would remember in later life. (I can remember such discussions from my own school days.) However, it is unlikely that a course will be taken up if it is optional and carries no credit, so I think the best that can be hoped for is that if you get a bad grade on the course then it will not have a significant negative impact on your life opportunities.”

    Hmm, this sounds like a good description of A level General Studies (at least as described by my son, who’s just taken it). More seriously, no one will take an A level unless it is required by a university, which in turn requires that it teaches some hard facts or skills; I don’t think that an open-ended discussion style course could catch on, for this reason alone.

    If you wish to improve maths teaching, I feel that there are more specific things to address first: (e.g. few students can structure a logical mathematical argument, know what a proof-by-contradiction is, know what the contrapositive means; most students quail when faced with a STEP style maths question (there’s too little guidance, and too much originality required); many students have insufficient confidence with basic algebraic manipulation; complex numbers have been relegated to Further Maths, few students have seen a old-style geometric proof of any complexity (etc, etc, etc, ..)

    (In fact, you could make a useful list by scanning the STEP reports, and looking at what the examiners complained about.)

    “6. The thing that’s missing from the post is detailed teaching suggestions. I’ll try to find time to provide some, since that will give a better basis for judging the feasibility of the idea.”

    This would be very useful.

    • gowers Says:

      I wasn’t very clear when I wrote 4 above. What I had in mind was that a more obvious proposal, where you try once again to teach the 16-18-year-olds a kind of remedial course where you go over the same sort of material that they’ve already decided they don’t like, and in roughly the same way as before, is very unlikely to be more than marginally beneficial. I’d rather risk pie in the sky than go for a stale pie on the table in front of me.

    • Colin Says:

      If some level of mathematics is indeed made a requirement for all students at A’Levels then I think Gowers’ idea is good. This certainly will side-step what Programs at UK universities prefer students to study (which can be short-sighted at times). The issue of how it is to be assessed can be modeled on General Studies (which interestingly you mentioned) or the O’Level General Paper. I am sure one can come up with many reasons why these two subjects are relevant to future scientists, engineers and medical students. Similarly, the benefits of Gowers’ outline to students who will not major in mathematics, physics, etc, should not be underestimated for all the parallel reasons. I wish the idea the best in becoming a more concrete program. It does appear to be sufficiently engaging and necessary, fitting well into a gap of the current education curriculum, with long-term benefits to both the student and the society at large.

  49. Gil Says:

    Regarding statistical reasoning and law (which is an exciting subject but very problematic for a mathematics course.) There is a very nice example in the paper “False Probabilistic Arguments vs. Faulty Intuition,” by Ariel Rubinstein http://arielrubinstein.tau.ac.il/papers/02.pdf
    This is a very rare case where the verdict contained a probabilistic argument (a mistaken one).

    About the cases mentioned here, the DNA evidence against Puckett is stronger than say a neighbor firmly identifying him among few hundred pictures presented to him in the police station. In both cases this will not be enough by itself for conviction. From the linked article it looks that Puckett, in three other cases, committed very similar assults to the one led to the murder. This looks like a very strong evidence and it seems that the doubt in his case is probably considerably smaller compared to other cases which ended with a conviction. In the Clark case it is not clear that a crime was committed but it is quite clear that the possibility that a crime was committed should have been examined. (In spite of the tragic situation.)

  50. Catleigh Says:

    Yes and finally ACME will come up with a device to blow up Road Runner

  51. Anonymous Says:

    How about “how does Google work?” You’re never going to get the exact algorithm as an answer, but it’s still interesting to figure out how might start to approach the problem of ranking search results.

  52. Richard Baron Says:

    This may be off-topic, but congratulations on your knighthood in the birthday honours list, Sir Timothy!

  53. Richard Gowers Says:

    Yes, quite.

  54. Gil Says:

    wow! Tim, warm congrats!

    Regarding question 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?

    I wonder, if this is the best question to ask, and what can we gain regarding understanding this situation here “if one models the situation mathematically and analyses the model.”

    By a remarkable coincidence there was a very recent post about it in the new blog windows on theory http://windowsontheory.org/2012/05/31/rigged-lottery-bible-codes-and-spinning-globes-what-would-kolmogorov-say/

  55. Neil Calkin Says:

    Congratulations on the knighthood! Very cool to see a mathematician being honoured that way, and well-deserved that you were the one chosen!

    Neil

  56. Knighted for Services to Mathematics | Combinatorics and more Says:

    [...] include also the Princeton companion and the very short introduction, general-public presentations, Gower’s blog and Tim’s [...]

  57. Jamie Dimon JP Morgan testimony and math tests for legislators | New Math Done Right Says:

    [...] Value at risk is full of good examples for the type of course Timothy Gowers discusses for practical math applications. [...]

  58. Joseph Malkevitch Says:

    I have enjoyed the discussions here. While there is both educational interest and mathematical interest to the kinds of problems discussed above, I think it is also very important to remember the effect that the statements of problems of this kind have on the way the public perceives mathematics and those who do mathematics. The kinds of mathematical problems that typically appear on standardized tests, partly because such problems are easily machine graded, give a very distorted view of mathematics. They invariably reinforce the view that mathematics is only a very hierarchically organized collection of techniques: arithmetic of whole numbers, arithmetic of fractions, arithmetic of decimals, solving linear equations, solving quadratic equations, solving higher degree polynomial equations, manipulation of polynomials, Calculus, etc. What these modeling problems show is that there are better ways for people outside of mathematics to think about the many things that mathematicians think about. One such approach is to emphasize themes that structure some of what mathematics addresses as well as mathematical techniques:

    1. Optimization
    2. Growth and Change
    3. Information
    4. Fairness and Equity
    5. Risk
    6. Shape and Space
    7. Pattern and Symmetry
    8. Order and Disorder
    9. Reconstruction (from partial information)
    10. Conflict and Cooperation
    11. Unintuitive behavior

    It is instructive to go through the problems suggested here to see where they fit in to this point of view, and also when planning K-16 courses, including those not only for non-STEM students but also mathematics majors as well, to take into account incorporating a thematic perspective.

  59. max Says:

    Why is the walkway problem difficult? Have two people kneel at the entry of the walkway — one just on it and one just off it. They will both resume walking at the same time, but will be separated. Tying while on is better.

  60. Latest News Headlines - Nick Clegg vows to block Michael Gove’s plan to scrap GCSEs Says:

    [...] sources said the Lib Dems should read a blog by the Fields-medal-winning mathematician Tim Gowers for a “flavour of what future might [...]

  61. danc Says:

    I’ve greatly enjoyed both this post and the comments. I personally really enjoy the sorts of questions posed above and find them genuinely interesting. I think it’s a good idea to have people in general more interested in/aware of the complexities and non-intuitiveness of much of mathematics, particularly statistics.

    However… does that require two extra years teaching? Most of the suggested questions above are interesting and pertain to ‘real-world’ situations but I really doubt that the overwhelming majority of the population have any interest in the answers to any of them beyond a kind of “oh really? that’s interesting” response. As to how many of them would actually sit down and work out the answers for themselves… I ask myself honestly how often I would have had to answer any of these questions in my job or daily life and the answer is very, very seldom indeed.

    I hope this doesn’t come across as negative – I think teaching maths this way is absolutely the right thing to do (and incidentally I think much more Maths teaching these days is like this) but I just do not think that the vast majority of the population would gain any more skills that are genuinely useful to them in every day life through two extra years of maths teaching. So there we are. Still, I hardly ever comment on blogs and this obviously got me interested enough!

  62. Paola Says:

    I haven’t read it all… but its a great idea. I did maths to quite an advanced level, but one experiment I always remember from school is our maths teacher bringing in a tube of smarties for all of us, and we worked out if the colours in each tube were randomly distributed. Maybe not deeply significant in the grand scheme of things, but hugely memorable and based on real life. Personally I do think that maths could be more accessible to more people, and that by making it real, it will come to life.

  63. Richard Thomas Says:

    What a brilliant resource this is! Giving talks to 6th formers will be much easier from now on. I had been using Q9, Q46 and a provocative variant on Q10 http://en.wikipedia.org/wiki/Simpson%27s_paradox#Berkeley_gender_bias_case, but now have much more material.

    Some more I use (sorry if they’re homotopic to ones you already have; I’ve not read every single question carefully).

    1. Men report having twice as many sexual partners as women. Is this possible ?

    (I think most people’s intuition is yes; there are even evolutionary reasons why men should *want* to have more sexual partners than women. To make it a sensible question, assume that the population is evenly split between men and women, all are heterosexual, and – crucially – we all agree on what constitutes a sexual partner.)

    The best answer illustrates the mathematical technique of looking at it from a different angle: via the number of couplings (rather than partners). Each coupling produces one male sexual partner and one female sexual partner, so the total number of sexual partners of each sex is the same (as pointed out by David Gale:
    http://www.nytimes.com/2007/08/12/weekinreview/12kolata.html)

    2. When Bear Stearns collapsed and the stockmarket followed, David Viniar, Goldman Sachs’ CFO, famously said “We were seeing things that were 25-standard deviation moves, several days in a row.” Discuss.

    This a good one for discussing the differences between mathematical models and mathematical fact.

    3. Natural Selection is the subject of constant challenge and controversy, while Intelligent Design does not contradict a single observation. So surely Intelligent Design is more likely to be correct.

    This touches on the scientific method and what science is. Hopefully it would make scientists keener on such an exam, and get them more on our side.

    4. My car is running out of petrol: should I speed up to get to the petrol station quicker (before I run out) or slow down ?

    5. It is raining, I have no umbrella, and I have to get from A to B as dryly as possible. If I run, I run into more rain per unit time, but if I walk I spend longer in the rain. Should I run or walk ? (Yet another question to illustrate the technique of passing to extremes to deduce the answer.)

    6. Has noone yet suggested a good question based on the fact that if prices rise by 50% then fall by 50%, they have overall fallen by 25% ?

    • Richard Thomas Says:

      I remembered some more.

      7. What’s the smallest strictly positive real number ?
      Then, after at least 10 minutes of discussion: Can you show convincingly there is no such number ?

      8. Why is the area of a triangle “half base times height” — then insist students draw lots of pictures, do lots of examples (right angled, isosceles, scalene), etc.

      9. What’s 1+2+3+…+100 ? (Hopefully none of the class knows the trick…) Approximately ? After some discussion: what’s the *average* of those numbers ? Now what do you think the sum is ?

    • Terence Tao Says:

      I’m fond of the friendship paradox as a variant of 1.: your friends tend to have more friends than you do yourself. Nowadays this phenomenon can even be verified empirically using a social network, such as Facebook. At first glance this seems to violate the symmetric nature of friendship, but it’s an instructive exercise to realise why the paradox happens (in fact, it is almost impossible for it not to happen!).

    • william e emba Says:

      Your #1 should be used as an example of “what mathematicians should be taught about non-mathematics”. Gale used an incomplete model of what the researchers measured, but interpreted his model as the last word. His criticisms do not follow from the theorem whatsoever.

      Yes, all men and all women have near parity in average number of partners, but the men and women were only sampled. If the two groups have distinctly different distributions, then random sampling will bias the reported number away from parity.

      Other biases come from the fact that the graph in question is dynamic, not static. Dead people were not sampled, yet continue to contribute to the averages of their former partners. Gender-based bias in age of first sexual experience leads to other biases in sampling the graph, and so on.

    • Richard Thomas Says:

      william e emba: iI think these are rather sophisticated subtleties that would have a small effect; nothing like the disparity that is observed (i.e. the number of partners differing by a factor of 2).
      Anyway the argument Gale gives is simple, illuminating, educational, and proves a result that is independent of how the sampling is done.

    • william e emba Says:

      If half of the men’s partners are prostitutes, and if the prostitutes are few enough that random sampling overlooks them, then there will be a strong mismatch between the full graph and the sampled graph. 2:1 seems entirely reasonable to me. And even if sampling is large enough to include the prostitutes by population numbers, they will still be underrepresented if on average they die much younger than the rest of the population.

      Yes, Gale’s argument is simple. But the theorem is relevant only if the sampling is large enough that distributional spikes are properly accounted for. Definitely non-trivial, and accurate evaluations depend on fieldwork, not armchair speculation.

      I speak as someone who has professionally had to assess samples for these kinds of biases. You could even say the trillions of dollars that modern finance trades is based on ignoring Gale-style theorem-quoting.

    • Colin Says:

      Seems like question 1 (and similar ones) will make for an excellent discussion (not sure the wording of 1 will pass the censors). Both points are well … correct, and very good for discussion of some of the more subtle and important points of the field.

      Continuing the discussion, how does one cater for lies in such data? I don’t thing many people are “wired” to answer honestly such questions even under anonymity and / or with the option to even not answer (particularly among certain age groups of males and females and made more complicated by many other factors such as beliefs etc). It is not like salaries where one can usually find a paper trail.

    • Richard Thomas Says:

      Colin: I would guess that “lying” is not the main problem. It is probably just differing incentives: on the whole men want bigger numbers and women smaller, so there’s an incentive to classify borderline events differently. So we perhaps learn that 50% of a person’s sexual partners came from a fumble whose status is debatable. (Perhaps this is not such a good question for the classroom then, though it would clearly inspire a huge reaction.)

      william e emba: I understand what you say about sampling, but Gale’s argument does say something useful and interesting, independent of sampling. If you take a billion boys and a billion girls and put them in a room together, the following morning the average number of sexual partners boys acquire is the same as the girls. (I.e., we don’t have to ask them about it! While the question was provoked by people having asked them about it, the question itself is just whether it is possible for the two averages to be different.)

      I see the Economist just carried a short story about Question 6.

  64. Fitzmichael Says:

    I (70) was one of those who “wasn’t good at Maths” or, more exactly, at the upper end of O level Algebra and particularly at finding/envisaging (but not not understanding) solutions to Geometry problems, despite finding the subject, then and now, very interesting but, re the Bulgarian lottery and assuming it is like ours, I thought I had understood that every combination, inc sequences like 1/2/3/4/5/6, is equally likely and therefore unremarkable, no?

  65. A fan of *MORE* discourse Says:

    This is a truly wonderful essay; one which I have taken the liberty of recommending it on Dr. Judith Curry’s well-respect weblog Climate Etc..

    A followup essay upon the theme “How should the mathematics of climate change be taught to non-mathematicians?” would be a significant contribution to a public debate that is crucially important to our planet’s future; a debate that (at present) is regrettably polarized and non-mathematical.

    Many thanks for a wonderful essay!

  66. Dan Says:

    Hi Tim,
    Did you ever read “Freakonomics” by Levitt & Dubner? I seem to recall this book causing a bit of a stir a couple years back. Simple stats techniques applied to controversial areas of sociology, sometimes well, sometimes less well. Might inspire some questions.

  67. Marlowe, PI Says:

    I haven’t read all the answers and I’m pretty sure something like this has already come up, but here it goes: the latest safety measures for cars insure that they are very hard to steal while parked. Does this increase the risk of a car been stole while driven?

  68. Ionica Smeets (@ionicasmeets) Says:

    It would be so great if problems like these became part of the curriculum! I use quite a lot of the examples from your list in popular talks and they always seem to resonate very well with non-mathematicians. It is great that you advocate these ideas to the ACME. I am very curious how they will react to this.

    Some other suggestions for problems:

    1. Statistics show that married men have a higher life expectancy than single men. A newspaper concludes from this that marriage is healthy for men. Is that a reasonable conclusion or could there be another reason that married men live longer?

    2. You want to buy a house and start going to viewings. How many houses should you see to have enough information about the market to know which house is best for you?

    3. It would nice to add something about chaos (since it is misunderstood by so many people), but I have a hard time formulating a problem. Maybe something about weather predictions?

    • gowers Says:

      I particularly like questions of the type of your question 1. Another example is the statistic that children of divorced parents are more likely to get divorced themselves. From that fact alone it does not follow that if you get divorced then you are making it more likely that your children will. That conclusion may or may not be true, but deciding which is hard. For example, there may be character traits that make you more likely to get divorced, and these character traits may be ones that people pass on to their children (either through their genes or through the way they bring them up).

      I skimmed quite a long paper on this question recently, and I think in the end it concluded, after taking account of many confounding factors, that there was a link.

    • Ionica Smeets (@ionicasmeets) Says:

      Do you have a link to that paper? I would also like to see that.

      My favorite example of this type is probably the study that showed that toddlers who sleep with the light on have a much higher probability of becoming shortsighted. Doctors concluded from this that sleeping with the light on is bad for the eyes, but it seems much more likely that shortsightedness is inheritable and that parents who are shortsighted leave on the light in the bedroom of their child.

  69. Claire Postlethwaite Says:

    One question idea:
    How much do you actually save on a “Buy one get one half price” offer? What if you don’t really need the second item? Or if there is some probability that you might use it?

  70. Julian Gilbey Says:

    Really interesting thoughts. I discussed a couple of the problems with two of our top set Year 11 students this week, with interesting results. In response to the archetypal Fermi problem (piano tuners), one of them said “I’m really really bad at estimating”, but was willing to go through the process – they are still much more comfortable with “routine” problems than with unfamiliar problems which don’t have an exact or “correct” answer.

    Problem 14, on the other hand, elicited very brief responses, and needed gentle effort to tease out more than a one-line answer.

    Other types of real-life problems which do not seem to have been mentioned are the ability to read and interpret graphs and tables of numbers. I was once being shown information about two financial products, and needed my wits about me to notice that the scales on two comparative graphs were different. And how many graphs are we faced with in the media? An ability to look at them and to extract significant information is essential in the modern world. A friend just reported to me that some lecturers examining Masters level dissertations in education were unable to understand the graphs and tables presented therein, as they did not have the confidence to deal with them.

    Just another idea…

  71. Felix Says:

    Very interesting thoughts an questions. One further easy, but probably not that interesting question:

    How accurate must a London-based muslim choose his direction so that he really can pray towards the Kaaba in Mecca?

    • gowers Says:

      That’s given me another idea. What does it mean to travel “in a straight line” on the Earth’s surface? The usual answer is that it means to travel in a great circle. But the Earth isn’t exactly spherical. The obvious answer in response to that is to take a geodesic. That raises a new question: if you travel in a geodesic, will you go once round the Earth and come back to your starting point?

      That new question can be answered quite nicely by considering an extreme case where the Earth consists of two parallel discs very close to each other and a smooth rim that takes you round from the edge of one to the edge of the other. If you think for a bit, it’s clear that there are plenty of geodesics on this shape that do not come back to their starting point after one trip (or indeed ever in many cases). A rather deep question, way beyond school level, is why we are so confident that if it fails in this extreme example then it must fail in a less extreme one. (One argument I can think of is that some dependence will be analytic and therefore can’t be constant on an interval. I don’t know whether a more simple-minded approach can work.)

      Assuming the flattened Earth I described above is symmetric about the plane that lies half way between the two discs, one can show that the angle of incidence as you approach the rim is equal to the angle of incidence at which you leave it again on the other side. So it’s like bouncing around inside a disc. Understanding that is another quite nice example of how the passing-to-extremes trick allows you to avoid calculations.

  72. Joseph Templeton Says:

    I have heard that:

    It is better for girls to go to girls’ schools, and better for boys to go to mixed schools.

    Surrey has the highest conviction rate of all counties in England; Reigate has the highest conviction rate of all Surrey courts: hence Reigate has the highest conviction rate of all English courts.

    It is better that your petrol tank be 70% full than that it be 40% full (per the Cabinet Office in 2012).

    Congratulations on the honour.

  73. Tom Heinzl Says:

    There is a recent book collecting questions (and answers) of this type: L. Weinstein and J.A. Adam, “guesstimation”, Princeton University Press 2008. Weinstein is a physicist and Adam a mathematician. One of my favourites (tested in class) is “How many people in the world are picking their nose right now?”.

    Another nice one suggested by a colleague is: “Estimate the speed at which your finger nails are growing (in m/s or mph or…).”

    I read at least one news article stating that Fermi type questions are now being asked during job interviews (as “brain teasers”). A quick Google search produced a link to an article by Anderson and Sherman, Journal of Applied Business and Economics 10(5), 33. The abstract says “that corporations now employ these questions in the job interview process, as a means of gauging applicants’ analytical skills”.

  74. Wiskundemeisjes » Blog Archive » De juiste vragen Says:

    [...] Gowers, een vooraanstaande Britse wiskundige, vraagt zich op zijn weblog af wat het ideale wiskunde-programma is voor de meerderheid van de leerlingen die later níets met [...]

  75. Peter Shepherd Says:

    Here’s another estimation you can probably get a pretty close answer for: “How many people in the entire world lost their virginity in the last 24 hours?”

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  77. Ben Blum-Smith Says:

    Dear Mr. Gowers: I am delighted to find a mathematician of your stature talking about the idea of basing math instruction on socratic investigation of real-world questions that cry out for mathematical analysis, rather than on force-grafting the mathematics onto the problems. As a former classroom teacher, currently working on a maths PhD, who has been following this blog for some time, I have often appreciated the pedagogical thoughtfulness with which you write about math, so it didn’t surprise me that you are attracted to this idea of teaching.

    Since you seem to be approaching the idea somewhat from scratch, I would like you to know that in the mathematics education world this is a well-developed (and actively developing!) conversation. At the level of the university, researchers and teacher-trainers have been actively experimenting with and studying just the idea of instruction you’re describing for some time. Some well-known examples are Catherine Twomey-Fosnot, at City College of New York, and Jo Boaler at Stanford.

    More recently, a very robust on-line conversation has developed among classroom teachers who write blogs. The most prominent exponent is Dan Meyer (http://blog.mrmeyer.com/) who was a classroom teacher and is now a doctoral student in education at Stanford. Dan has pioneered the use of digital media in posing the questions. Here is a video that’s a general introduction to his approach, and here is his most recent project, a website where you can submit a visual image or video intended to inspire a mathematical inquiry, and get feedback on its effectiveness.

  78. Mark Wainwright Says:

    I remembered another one, not that my last suggestions seem to have been of much interest, but anyway: why do tube trains come at uneven intervals? Can you (come up with a plausible model of passenger arrival and loading times and definition for overall optimal journey times and) speed up people’s journeys by forcing trains to wait unnecessarily under some circumstances?

    The first question at least is quite simple though I hadn’t thought about it till someone pointed it out recently: if a train is slightly behind schedule, and assuming people arrive at a uniform rate at each station, it will have an above-average number of passengers to load each time, thus being slowed down still further, while the train behind will have a below-average number, so will gain more and more on the first train. This continues until the second train is backed up right behind the first, and a third train can (if not itself delayed) start catching up. A useful corollary is that if a train is very full, there is very often an emptier one just behind it that is worth waiting for. You can usually spot this bunching from the timings on the platform indicators.

  79. Joseph Malkevitch Says:

    The book: WHY DO BUSES COME IN THREES?
    The Hidden Mathematics of Everyday Life
    By Rob Eastaway and Jeremy Wyndham
    176 pp. • 1999 •, addresses this issue and related problems in the spirit of what is being discussed here.

  80. Andrew Mathas Says:

    I didn’t see my favourite questions of this sort in your list. Namely, questions about encryption and how information is transferred securely over the web. Similarly, there are nice questions that you can discuss with school students about about how error correcting codes and how information is stored on CDs etc. Of course, the full details are qiye complicated but there are some nice baby examples that you can introduce this way.

  81. Diana Says:

    You might be interested in Phillips Exeter Academy’s problem-based math curriculum (in Exeter, NH, USA). About 20 years ago, the teachers there decided to do away with the textbook (which was a well-respected textbook by one of their own faculty, still in use in many schools) and write their own curriculum.

    It is almost entirely word-problem based, and the classes are taught much the way you say would be ideal. They don’t use the word “socratic,” because in the socratic method the teacher leads the student in a pre-determined direction. The goal is for the students to ask questions and figure out the solution as a class. By and large, it works very well. This curriculum and method is used for all levels of high school math, from algebra through multivariable calculus.

    Here is the philosophy and how it is put into practice: http://www.exeter.edu/academics/72_6532.aspx
    and here is the entire curriculum, free for download:
    http://www.exeter.edu/academics/72_6539.aspx

    Certainly, not every problem in the curriculum is as engaging as is your goal, but many are. The fact that the school threw away their curriculum and started fresh with a new one, which they wrote and continually update, is something that might interest you.

  82. Maths A-level could include more ‘real world’ problems to attract pupils | Education News Says:

    [...] blog now lists dozens of suggested questions covering areas such as estimation and probability – for example calculating how many molecules [...]

  83. Maths A-level could include more ‘real world’ problems to attract pupils | My Blog Says:

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  84. LED Lighting News » Blog Archive » Maths A-level could include more ‘real world’ problems to attract pupils Says:

    [...] blog now lists dozens of suggested questions covering areas such as estimation and probability – for example calculating how many molecules [...]

  85. David Harris Says:

    I am really enjoying reading this blog, and the article from today’s guardian. Kindred spirits! I began a project in 1999 – planetqhe.com towards the goals in this blog.

    Planetqhe.com is the latest instalment of a personal journey, a response to these critical incidents from my teaching:

    Critical Incident #1 My Calculus student Hany described my ‘Find the Volume of an American football’ activity during his valedictorian speech at the American International School in Egypt graduation ceremony, 1993. Sitting in the audience, I remembered the buzz of focused creativity this activity sparked in my classroom, and thought “How did this come about and how can I do this more often?”

    Critical Incident #2 In Argentina, my class was completing exercises on probability such as “You choose the name of a girl from a bag containing 10 girls names and 5 boys names written on pieces of paper. You throw away the paper and choose another from the bag. What is the probability that it is the name of a girl?” The students were able to answer these questions by forming simple fractions and did so with little interest. I thought “Who ever draws names from a bag? Can’t probability be made more interesting?”

    It wasn’t until I arrived in Bristol in 1999 to study for an M Ed that I managed to find the time to try to do something about these thoughts. I formed the research question “How can I embed Probability in a more meaningful, relevant and interesting context?”

    read more here;

    http://www.planetqhe.com/beta/information/about%20this%20site.htm

  86. Gove backs development of new maths qualification | RSSeNews Says:

    [...] were among scores of suggested questions in a much-cited blogpost by Timothy Gowers, a professor at the Department of Pure Mathematics and Mathematical Statistics at [...]

  87. Anonymous Says:

    The maths curriculum was developed after the industrial revolution for the skills required at that time. Obviously the needs of society and the economy have changed since then. There will always be those few who love the subject and want to take it further but I am concerned with equality of opportunity in state schools. There are not enough teachers with a passion for maths and a degree in the subject (or a closley related one). Many Post-16 students have probably never been taught by a teacher with a maths degree (or a closely related one). Could this also be a factor in why students hate the subject? The problem solving approach is great (I wish I had you as my teacher), and far more relevant but my point is that more should be done to attract maths teachers in the first place. We are limiting our society and future economy by relying too heavily on private education for mathmaticians, scientists, politicians, actors, musicians…

    • Anonymous Says:

      I am teaching an a level designed for c grade mathematicians currently. they live real world applications, and can see links between topics. some of the suggestions above however are looking at things that wouldn’t interest the average non mathematician, sadly! anyway, I hope the use of maths course won’t get dismissed out of hand, since for once I have a bunch of bright young non mathematicians learning about the maths they need for the real world and enjoying it: their words, not mine!

    • gowers Says:

      I certainly don’t claim that every single question above would be of sufficient interest to make a good classroom discussion. It will be important to come up with further questions, and also to test questions on actual classes to get some idea of which ones work and which don’t (which may very according to the backgrounds of the people in the classes). Such testing is part of MEI’s plans for developing the course.

  88. Darren Says:

    I teach in a secondary school and agree with many of the posts here that to implement a course such as this post 16 would pose significant challenges. Investment would need to be made in training teachers/college lecturers, questions would need to be posed that were accessible to learners at this level and I agree that finding a robust way to examine a course like this might be better served through some kind of coursework assessed model.

    I also wanted to comment that the kind of teaching discussed here is going on at a secondary school level, albeit sporadically across the country and I was particularly encouraged to read here about the use of logic, proof and modelling which I believe are essential skills for all students to learn and should form a greater part of the secondary school curriculum.

    I hope there is some positive outcome from the discussion on this board but wanted to say that if nothing else I will be using some of the mathematical problems discussed on this blog in my own lessons and encouraging my colleagues in our school to do the same. In my view we cannot wait until a student reaches 16 years old to start teaching them in this way, we need to inspire them earlier and if we do this then there is a much greater chance that they will be able to succeed on a course like this when they do reach 16.

  89. Mark Bennet Says:

    There is a very realistic question about how much you would have to save each year to provide yourself with a pension, and also comparing the charges on different financial products between initial charges and annual charges. Plenty to explore around the mathematics of finance.

    As for assessment – there is a case in Maths for a ‘hurdle’ test, rather than one dominated by marks – the hurdle being ‘can you give a good enough answer?’ rather than ‘can you say lots of relevant things which don’t really amount to progress or understanding?’

    The case study for the final exam for the Institute of Chartered Accountants in England and Wales was marked (when I was a marker – don’t know about now) by giving ‘a’, ‘b’ and ‘c’ marks for relevant comments, development of ideas and conclusions showing relevant understanding. It would be hard to mark like that on a mass scale.

    On curriculum, the first stage of the review of the primary maths curriculum raised some issues about mathematical problem solving (and whether the draft text had effectively reduced this to solving “word problems”). If one goal of teaching problem solving is to approach some of the problems you’ve suggested, that should feed down into reconceptualising problem solving in the earlier curriculum. [Which cakes shall we make given the ingredients we have available?]

  90. TheMikeN Says:

    This feels too much like a discussion between people who already agreed with each other and are, at most, fine tuning their ideology during the conversation. I’m probably just another one doing the same because, yes, the style of classroom ‘ideas-tennis’ being imagined sounds fabulous.

    But the majority of the Maths necessary to draw any conclusions includes elements that would be beyond most teachers and nearly all the target students in Mr Gove’s planning. Schools don’t do anything long-term that they cannot be assessed in and that they cannot prove themselves to be good at. What does that assessment look like here? I quite like the idea of writing discursive paragraphs about intriguing problems but there isn’t a convincing mark-scheme possible so no assessment would be remotely objective.

    At ‘A’ level in 1982, we had a General Studies session each week called Thinking Skills. Basically we’re in that territory with these questions. We didn’t have the ‘work’ assessed then and it wouldn’t be now. Since then everything in schols has by necessity become much more qualifications-focused so the sessions would be very unlikely to take place at all unless they were fully funded by additional external grant.

    I love the Maths under discussion but the policy likely to flow from it fills me with dread.

    • TheMikeN Says:

      I’ve just discovered how some of my post here overlaps with Anonymous at numbers 43 and 48. I’m viewing it from a bit more of a schools’ point of view but … he/she and I agree quite a lot, I think.

    • gowers Says:

      I agree that both teaching and assessment are major challenges. Let me give some idea about what MEI’s thinking is. On teaching, the plan is to provide plenty of back-up in the form of written materials, training, online resources, pooling of experience, videos of successful lessons, pilot projects, etc. It may turn out that the challenge is still too great for the course to catch on, but I don’t think that’s obvious in advance.

      I also think that it would be a great shame if the need for “objective” assessment were to trump educational value. In many subjects — history for example — you also don’t have a “convincing mark scheme” but they find ways of managing. Charlie Stripp, the chief executive of MEI, told me about something called “banding”, where you sort scripts into something like good, medium and not so good. I can’t remember the exact details, but the general principle behind it is that for certain things, like essays, it is hard to decide how good they are in absolute terms, but much easier to decide, given two scripts, which is better (or that they are of a very similar standard). So quite a good way of marking is to sort the scripts. How you convert the sorted order into marks/grades can be decided later.

      In an ideal world, it would be just like your General Studies — something that people would do and enjoy, with no assessment. I think the next best thing is to have a course where nothing too much hinges on the assessment (it could be an AS-level, for example, to be taken alongside people’s main subjects, or the qualification could have some other name entirely) but having an assessment provides just enough motivation for schools and their pupils to put in a bit of effort.

  91. H. McCreight Says:

    I’d say there are two levels of maths learners(rather obvious really!)
    a) Those who will just need a good core of practical maths for general life skills (measuring, financial calculations, budgeting)
    b) Those who are going to have special careers eg aircraft engineering,scientists

    The latter are always going to have to answer complicated questions eg on vector analysis so can you really restructure questions to that level?
    You definitely can for the first group.

    • gowers Says:

      I think scientists and engineers should definitely take A-level maths. I don’t think it would be practical to teach them what they need to know using a course of the kind described above, though they might find it interesting as something on the side.

  92. W-K Says:

    This is a very interesting blog, BUT I have to ask is it missing the basic problem. I got the impression that this government decree is to get those who fail, or do badly at maths, at GCSE level up to a school leaving standard by the age of 18.
    From my experience in trade training these GCSE failing students gave up on maths long before the age of 16, and therefore most of them have very little knowledge of the basics.

    So either the diagnosis has to take place at a much youger age, and the suggested problems introduced earlier.
    Or this “extra maths” needs to re-assess the students and teach the missing basics again post-16.

  93. Antonia Rubinstein Says:

    All interesting – but I took a look at your blog – a first time viewer and note that Michael Gove praises it – is his knowledge of maths at your level or is his reply that of one of his policy makers ?

    I agree that not enough discussion and work is being done at a younger age. I have a daughter at an excellent private prep school and they failed to engage her or understand that she hadn’t grasped the very basic concepts – taught too early for her and a good many other children in her year of 60. They are trying to grasp language at the same time as maths concepts and for many children it is information overload. I have noted the complaints by teachers of the poor behaviour in class amongst many of the boys in particular – the subject – Reception to Yr 5 is mostly taught in a desk/class room bound way – a good amount of the maths is not taught in a way that is practical to their own lives. Much of the problem stems from the fact that these children are not involved at home in any of the domestic chores – such as budgeting and handling small amounts of money for household shopping or cooking from scratch meals or estimating how much petrol goes into a car. air pressure on tires etc. Or how many bricks are required to to repair a wall or to be involved in the construction of a wall or the making of a brick. Or how much water is needed to wash clothes and the amount of electricity needed for one 40 degree wash. The childrens early involvement with maths and their experience of the teachers – at primary level is far removed from real life.

    Trying to explain to people who get maths and are good at it – that they aren’t always the best people to work out how to teach it and make it relevant isn’t easy. It might be a good idea if your team spent a term being filmed for a documentary at a variety of primary schools to see how you get on in the class room teaching – to see what your findings are. Most teachers at primary level are women – men will have a very different slant. Also charismatic and creative people are needed to teach the subject – finally maths should be cross referenced to other aspects of the curriculum and not taught solely in isolation at primary level.

  94. apiarylandlord Says:

    I agree with so much that is said here. However, I know from experience that many prospective students have a barrier as soon as they are faced with something that sounds vague and numerical and i think the mere idea of much of this will put up the sort of mental barriers that prevent them even beginning to engage. By all means use some of these but also consider:
    1. Maths myths. A section that deals with basic misconceptions or problems because this causes so many people to lose confidence and assume maths is more like a foreign country with rules of engagement that are incomprehensible. It also is likely to cause problems in the workplace. It has partly been covered in teh comments. Helping them to make sense of these gives them confidence in themseleves which can mean they can make more sense of the more ‘what if situations’ discussed above. For example, to divide fractions you turn upside down and multiply. The fact this seems illogical and is typically a serious trigger point to turning people off maths and that is why it is so important. Basic knowledge (such as when you divide a number by something smaller than 1 the answer will be larger than the original number) is one way to make sense of this but there are others such as equivalent fractions which lead to a direct understanding of why the rule has evolved.
    2. Creating some modules that could be used to show proficiency in basic things. For example, it is shocking how many people are using spreadhseets based on simple maths but designed and implemented informally and full of problems. I know of someone with a PH D who has colleagues of similar status who rely on a spreadsheet that has evolved in the workplace BUT CONTAINS FORMULAE THAT GIVE INCORRECT ANSWERS. The basic sense of number has been covered above and could minimise some of these issues but there is scope for some spreadsheet modules each of which would be valued by employers. it could include practical tasks where learners needed to find errors in spreadsheets, or a bit about the theory of testing with ‘typical, extreme and erroneous’ values as well as practical tasks where they had to create spreadsheets or perhaps identify what an unknown spreadsheet actually does.
    3. getting industry to identify exactly what problems are encountered and creating elements directly to prevent or minimise each problem would be good.
    4. Don’t forget the creative. True mathematicians know of the pattern and elegance in Maths but too often talented creative individuals never get to see this because they are frightened off by the number side of things. Build in some elements that cater for the creative people (eg pantograph to enlarge drawings?, some series theory related to perspective? perhaps a touch of Escher and Lewis Carroll)

  95. H. Mccreight Says:

    Two things:

    a. People in other countries must have considered this problem and perhaps come up with a solution. Is this being investigated ?

    b. In the spirit of the questions you want to create:

    “The government wants to introduce a new maths course. It has allocated £275,000 to research this. How do you think they arrived at that figure and do you think it is adequate ?”

    • gowers Says:

      Question b has a simple reduction to a possibly simpler question, since MEI put in a costed proposal and the government agreed to fund it.

  96. John Teal Says:

    Age and incentives concern me. My ten year old daughter enjoys counting such things as unit price of 2 for 1 offers and % increases since our last shop. Bringing numbers into conversation from pre-school years seems to me to be important. Secondly, we live in a world of ‘marketing’ which is often close to fraud, e.g. most topically PPI sales by banks. The satisfaction of getting a rebate or compensation from people who play with probabilities is most encouraging, and not only for adults. When people see that a suitably worded letter with a basis of simple counting can get them benefits their interest in combining maths and English can grow quite remarkably. There is plenty of scope for exercises like, find out if the cost per kg estimates in your local supermarket are accurate, what is the % increase in price resulting from reducing the pack content from 500 to 450 grams, does an average 5% p.a. cost increase over 5 years, compunded, really justify an increase of 60% in an insurance policy?
    To make this work, I think teachers will have to be allowed more thinking time.

  97. adrian smith Says:

    The direction of travel – real world problems requiring and engendering an enthusiasm for maths – is spot on and I’m looking forward to being involved.

    The two threads of importance have been touched upon but need to be brought together: making the problems real and relevant to the target audience and helping teachers deliver learning around these problems.

    There have been some wonderfully interesting problems listed but I’ll wager that only a very small percent titivate the target audience. Take a look at the poblems that Karim Kai Ani offers on http://www.mathalicious.com/lessons/

    These haven’t quite been derived in the purist way originally suggested (think of a problem independently of some specific maths) but they are spot-on in terms of what’s likely to enthuse the audience. We need much more of these types of problem.

    Perhaps it would be wonderful to go a step further, and to get teachers to a place where with any group of students an interesting problem could be generated by the group and then tackled with appropriate mathematics (there is an element of this in the IB extended essay approach). All sorts of issues arise of course about the extent to which a ‘broad and balanced’ syllabus might be covered.

    The reality is that this genuinely is a step too far and so we need problems extracted from typical audiences not from teachers and the most talented mathematicians. The role of those latter two groups is to help build the resources and professional development programmes that enable the widest possible range of teachers to deliver these life and society changing initiatives.

  98. carlo rinaldi Says:

    I think that this approach is excellent. I intend to through some of these questions at my trainee primary teachers as the questions are what the Curriculum for Excellence in Scotland is trying to do with children’s thinking.

  99. H. McCreight Says:

    Has anybody told Phil Beadle about this blog ? Would be interesting to to hear his opinion.

  100. Toby Says:

    Something I was wondering which could be interesting to think about is this:
    I was in a classroom when I was around 16-17 and, as you do in a classroom that isn’t particularly engaging, I glanced at the clock. As I did this, the clock stopped. Being a mathematician I then wondered what the likelihood of someone in a classroom glancing at the clock and it stopping as they do so, or, to take it even further, the odds of someone ANYWHERE glancing at a clock/watch just as it stops.

    Another thing I’d always thought would be interesting to think about is this:
    A random person gets off a bus at a random stop, what is the probability that they turn left upon getting off the bus?

  101. Ideas de Tim Gowers sobre la educación secundaria « Matemático en el instituto Says:

    [...] esta extensa entrada hay una exposición detallada del tema incluyendo una lista de 64 problemas que se podrían [...]

  102. Laurence Oakes-Ash Says:

    I agree that 1) students should study maths until 18 and also that 2) more real world examples need to be used to make maths, and mathematical thinking relevant.

    However, I would make two further points – 1) Real world questions need to be an integral part of mathematics teaching through all ages and 2) The syllabus can still come first, with the questions defining the learning objectives. An open ended question and personal study, much like the IB syllabus is also a good thing and as the student poses the question herself, the question will be relevant to her.

    The truth is that maths is taught from the Platonic philosophy where mathematical objects “exist” independent of the real world. I think a discussion of that philosophy is outside the scope of this post, but I do believe that it is the fundamental reason many students are alienated by maths. In many ways, by simply adopting a historical or humanistic philosophy in the way we teach maths, we can communicate the abstract ideas more easily.

    I think it is great that Dan Meyer has been mentioned already, but his Ted speech and blog are worth looking at.

    http://www.ted.com/talk/dan_meyer_math_curriculum_makeover.html

    Take this discussion on Online maths: http://blog.mrmeyer.com/?p=15398. Getting student involved in the question should be key to every lesson that is taught. The difficulty is that there is not enough investment going into the design of teaching tools. With the right teaching, you can deliver any syllabus and we should use technology to better standardise and monitor our teaching of maths.

    The other travesty, that makes much of the post-16 education discussion problematic, is the huge gap in achievement in the 11-16 age range, discussed at length in the Vorderman report. Without basic numeracy, achieving rigourous learning outcomes from the types of questions you discuss is near impossible.

    The other thing mentioned in this discussion is the Computer Based Maths movement. I agree with the objectives of making maths less about computing and more about doing. However, I again think this hinges on the learning up to the age of say 13-14. By this stage, a student 1) needs to understand how computation works, and 2) needs to understand how to algebraically express basic ideas. At this point more computer tools could be introduced alongside algebra to solve real-world problems or answer real world questions.

  103. H. McCreight Says:

    I agree with the the last entry. i will leave this blog for a week and see if anything new has happened on return. Really what is the timeline for MEI to make this exam/syllabus a reality?

  104. H. McCreight Says:

    Any new inputs?

  105. Dan Bryer Says:

    I haven’t read all the comments so forgive me if I’m repeating anything. As a Maths teacher I enjoy teaching a number of parts of the Decision Maths course as there is no complicated arithmetic or algebra involved and the problem solving concepts have real life applications.

    Bin packing or putting objects of differing sizes into boxes wasting as little space as possible is one of these. Also network problems such as the Konigsberg bridges and can you draw a house shape without taking your pen off the paper? Route problems such as a postman having to go down each street at least once and wanting the shortest total distance or even how does a satnav calculate the shortest/best route from A to B?

  106. John Bibby Says:

    Have you seen the Open University course MDST242 “Statistics in Society”? It takes the philosophy “Start where the student is at – Ask an interesting question – Teach a ‘Swiss army knife’ of multipurpose techniques that students can use to a wide range of questions – Don’t forget the ‘meta-questions’, which speak about what the mathematics has left out”

    Is this similar to your philosophy Tim?

  107. Dr Tasneem shah Says:

    having spent 30 yeras as a scientist (modelling and simulation), I entered in to applied maths teaching. I agreed with idea but before delievering the proper prepared lecture one should talk about general applications of the related topic and it has to be categorized for diffirent disciplne (computer science, electrical engg, mechanical engg etc).

  108. Anonymous Says:

    I’m not sure that the audience for these questions has been properly thought through here. Those who are able at maths or need to do maths to do their chosen pathway will take A Level maths. Therefore you will inevitably have a class of pupils who have gained a grade C at maths, probably through a lot of hard work. These pupils will not be able to access the mathematics behind those questions you put in your post and expecting them to be interested in ‘keeping your coffee warm” or the cost of washing machines just shows your lack of experience in actually teaching pupils from all backgrounds.

    • gowers Says:

      A few points here. First, I don’t think you have to be directly involved in something to find it interesting. For example, I find questions about the right tactics to use in certain sports that I don’t play interesting. Secondly, I fully admit that I have almost no experience teaching pupils from all backgrounds. However, the examples above are intended to be illustrations of a general type of question rather than being questions that I am certain would work well. It will be essential before introducing a course of this kind to test the questions on actual classes and weed out the ones that cannot be taught successfully. I’m glad to say that that is what MEI plans to do.

      But I also think that it is not realistic to expect that any question is going to interest everybody, and maybe some people will dislike the entire concept. However, the best can be the enemy of the good here. I think the right criterion is whether a course like this would be better for some people than an attempt to cram into their heads some more conventional mathematics. If, for example, 40% of pupils disliked this course intensely and 70% dislike more conventional mathematics intensely, then this course would be doing well.

  109. Zeeshan Mahmud Says:

    The word “Cricket” raised a flag :)

    I posted a question on MSE that is sleeping for a while on predicting batsman’s runs. Also another problem that I wrestled with is if the Fall of Wickets of all matches approach any graph.

  110. Zeeshan Mahmud Says:

    Professor Gowers, re: Batting average paradox, do you have a concrete example in mind? I was just curious.

  111. H. McCreight Says:

    How come it’s going to take MEI 2-3 years to make this course a reality ?

    • gowers Says:

      Because it would be very silly to rush something like this: at this stage it is not clear which questions will work in the classroom, and a lot of work will be needed to provide the support that teachers will need in order to teach in a very different way from what many of them are used to. In addition, designing good methods of assessment is a major challenge. If anything, 2-3 years is a worryingly short time.

    • H. McCreight Says:

      Does Michael Gove know its going take 2-3 years ?

    • gowers Says:

      Yes.

  112. Maths A-level could include more 'real world' problems to attract pupils - Government Tenders, Government News and Information - Government Online Says:

    [...] blog now lists dozens of suggested questions covering areas such as estimation and probability – for example calculating how many molecules [...]

  113. Peter Balacs Says:

    I know very little mathematics but find this fascinating. However, to whom is Prof Gowers addressing himself. For instance, he uses terms, such as Fermi Estimation, which, not only do I not understand, I have never even heard of. He presents intriguing questions as examples of his approach to teaching mathematics, but then leaves the reader guessing. Would it be too indulgent to provide possible answers. In his article in the Spectator of 3 November 2012, he says that the “best answers” will appear in the next issue’s letters page. But only one answer to a question is given, relating to the sharing of a Mars bar. Yours sincerely, Peter Balacs

  114. Tibor Mach Says:

    A question proposal:

    In the second world war, the British Airforce had heavy airplane losses. They needed a way to better protect the fighter airplanes (by armour plating). They could not cover the wole airplane in armour (as it would be too heavy to maneuver to be used as a fighter plane), but instead the aim was to cover the most critical parts. They collected data from damaged planes that returned and it turned out most hits were on the wings of the airplanes.

    Does that suggest the best way to protect the planes is to put armour plating on the wings?

    It doesn’t, of course, the important part of the data is that which is missing – the planes that did not make it back. And those were the planes hit in the fuel canisters at the middle part of the plane. The army actually went to a statistician and expected him to say the wings need more plating. He gave them the right answer though and it helped considerably.

    Also perhaps a good asumption is that the enemy fighters cannot aim directly on specific parts of the plane (quite realistic in WW2 conditions I think), so they cannot adapt to the fact, that some parts now have more plating.

  115. HMC Says:

    What ??????

    • Tibor Mach Says:

      It is surely not a question that will take a long time to go through , but gives a neat example of how sometimes the data you get can actually suggests something opposite of what it may seem as obvious at a first glance. The fact that most returned planes had damaged wings suggests actually that wings are not that critical – those planes got back. But the planes with a hull damaged mostly did not make it back – the hits were fatal and so no data could be collected there. If you consider only the data you have and not the data you don’t have you make a mistake – you assume, you have the data from all the hits, which you obviously cannot have. And there are many non-war situation when your information is only partial and not in a form of an i.i.d. random sample.

      If the students don’t believe the teacher, he could illustrate it on a made up situation with fatality probabilities of certain hits and made up data from the returned planes that may seem to suggest something different than what they actually do.

    • gowers Says:

      FWIW I’ve always liked this fact and think it would make a great question. Thanks for the suggestion.

  116. HMC Says:

    I am none the wiser. Sounds like Donald Rumsfeilds famous deliberation on “knowns and unknowns” ?

    • Fergal Daly Says:

      Look at it this ways. If you only consider the data from the planes that return, you are in danger of concluding that nothing needs to be improved because all the planes returned!

      Let’s say there are only 2 parts to a plane, the head and the body. All the planes that come back have bullet holes in the body. None of them have bullet holes in the head.

      Does that mean you need to put more armour on the bodies? That was the army’s conclusion. The statistician realised that it actually means that when a plane is shot in the head, it just doesn’t come back. So it’s the head that needs more armour.

  117. Anonymous Says:

    Well it is now June 2nd 2013, and I have parachuted back on to this blog because my email said someone had added a comment. I did a quick scan of the page for latest date just to check that there was nothing greater than February. Looks like there isn’t a lot happening.
    I expect by the time that this proposed new course/syllabus actually materializes, Michael Gove just might have achieved his secret ambition to be prime minister! Somebody should invent a course called “Maths,,,Why Do I Need To Know This?” Extrapolate and interpolate from that…..

  118. Gabe Durrant Says:

    school of Hard sums-Dara Obrien, has some excellent examples of these types of questions

    • Anonymous Says:

      Yes I have watched School of hard Sums with Dara O’Briain and Marcus du Sautoy, but despite having a maths degree myself and a general ongoing interest in how best to present maths ideas to others, I was bored watching it after 10 minutes and suspect many others would be to…..it failed the test of “really why would I need to know this?”

  119. Dhruv Patel Says:

    Oh My God! So many real life questions in maths? I could’t even read all 66 questions.

    BTW, What are the chances of dieing me on flight? It’s zero because I never travel by flight.

  120. Peter Balacs Says:

    I have not seen a response to my email of 30 December last year. Although I’m not a mathematician I’m very interested in mathematics and would love to know more about the approach presented in this blog. But I would also like to see how some of the questions are to be answered. For instance, I would guess that it is better to tie up your shoelace while on the moving walkway than between walkways, because at least you’re still moving towards your destination. But I’m not convinced that I would be right. By the way, I agree partly with Anonymous about the School of Hard Sums. The problem for me is in the presentation. Amidst all the joking and often unclear diction, the significance of the questions and understanding of the answers was too easily lost. Best regards, Peter

  121. mathtuition88 Says:

    Reblogged this on Singapore Maths Tuition.

  122. Joshua Says:

    Two points of criticism:
    (a) many strong math students would benefit from working on these types of problems for a while.
    (b) the philosophy can also be applied to younger students, see http://www.inference.phy.cam.ac.uk/sanjoy/benezet/ for a similar approach with primary school students.

  123. anonymous Says:

    I think there’s a problem with the numbers in question number 43, or rather the discussion. Of the 10,000 inhabitants 9990 do not have the disease. Of those, 1% will get a false positive result, thus roughly 100 people, not 1000!

    • anonymous Says:

      Apparently, I’m just really bad at reading. The town was supposed to have 100,000 inhabitants. The number of people with false positives is thus 999 (and thus basically 1000 as claimed by Mr. Gowers) — sorry for the noise

    • Anonymous Says:

      Whatever…..this entire blog achieves very little……a couple of people ego tripping…..academia moves at a snails pace and they like it like that…….a lot of other countries are way ahead on their maths education programs while England continues faffing around with endless curriculum changes….time we got Euclid…or was it Pythagoras ?…..to knock some heads together

  124. almaionescu Says:

    Reblogged this on Science and Mathematics.

  125. telescoper Says:

    Reblogged this on In the Dark and commented:
    This post from the estimable “Gowers’s Weblog” passed me by when it was posted in 2012, but I saw the link on Twitter and decided to repost it here because it’s still topical

  126. how to math problems - News, Politics and Stuff Says:

    […] How should mathematics be taught to non-mathematicians […]

  127. Sarah Says:

    I’m a normal Maths teacher. I have a first class Maths degree. I have taught Maths to Further Maths Alevel for twenty years.
    I must say I agree with lots of what is said here:
    a) We need to get our kids to THINK not just APPLY.
    b) Why should a course in Maths not be assessed like English assesses its essays, if they can we can….so open ended is possible.
    c) Some of these problems are just great fun and clever and interesting.

    Now for my concerns.
    If I am a well qualified, passionate and pretty successful Maths teacher and I am having to think about these problems…how do I get my C to even A grade GCSE mathematicians to start thinking about these problems.
    You must understand I am not asking how to solve the problems….that’s the fun part for me to research and you’d find most Maths teachers agreeing. I am asking “how do I teach my average ability sixth formers how to apply themselves to these problems”.
    I tell you now…walking in and posing the problem and expecting a worthwhile discussion on 90% of these problems ain’t happening. It will take a huge amount of preparation. In the two years since this blog was posted. What has been done and said about that problem.

    Lets end the eternal ….here’s a fun application…academic discussion on the concept.
    Lets have some real discussion about how passionate capable Maths teachers make this accessible to our kids!

    • gowers Says:

      I’ve been working with MEI to develop materials for a course based on these kinds of ideas, and I can assure you that the question of how discussions about questions like this could be provoked is one of the main things we have been thinking about on the advisory group I belong to. Work has been done trying ideas out in schools, and we also realize that a considerable amount of CPD will be necessary for (at least some) teachers. If the course actually gets off the ground (MEI is not an awarding body, so this is not guaranteed) then I will report on it on this blog.

  128. RobinFD Says:

    Thanks for the blog. I am a big fan of this style of teaching in science as well as maths – and other subjects too when a moment arises.

    I’m working and studying in Switzerland and haven’t seen the English maths text books at all. I thought you might be happy to hear that one of the mandatory texts for maths students in years 7-9 is called mathbu.ch (http://www.mathbu.ch) and in their edition for 7th grade they introduce the Fermi Problems in the sixth chapter (or setting as they like to call them). Then, throughout the rest of the series, Fermi Problems are scattered related to the setting’s content. Definitely worth a look. The texts are of course in German, but I’m sure if you got in touch with their authors they would be able to talk to you about their project.

    For the record they offer a really creative and excellent model for teaching mathematics as a whole in the series. I use their books for my students and it just opens up the ideas behind mathematics so well that our maths lessons are equally full of mathematical discussions as they are well-differentiated maths exercises. Certainly a major improvement on the maths I learnt at school in my view.

    Cheers,
    Robin from near Lucerne ;>

    • johnbibby Says:

      This looks like a fascinating project. I went to the website and it has about 30 downloads or more (some are duplicates) – is there one download that will give me the whole lot? I can struggle through on the German, but stuff in English useful too. (Who are the ideas-people on this? How will it be evaluated? Are there French/Italian/Roamsch versions?)

      JOHN BIBBY (“Maths in a Van” – I am interested in developing “Maths is all around you” ideas for community and adult learning out of school)

  129. RobinFD Says:

    Hi johnbibby,

    you have to buy the actual books. i’m pretty sure there’s no way you can get the books by download. if you contact the authors however;

    Walter Affolter
    w.affolter@bluewin.ch

    and

    Beat Wälti
    http://www.phbern.ch/ueber-die-phbern/institute/iwb/dozierende/beatwaelti0.html

    I’m sure they can help you out.

    Cheers.

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