1. You are in an airport and are walking from the main departure lounge to a rather distant gate. On the way there are several moving walkways. There is a small stone in your shoe, which is annoying enough that you decide that you must remove it. If you want to get to the gate as quickly as possible, and if there is no danger of your annoying other passengers, is it better to remove the stone while on a moving walkway or while on stationary ground, or does it make no difference?
2. You want to give £1,000 to somebody as a 21st birthday present. The person in question is just about to turn 16. A savings scheme offers a guaranteed interest rate of 3 per cent for the next five years, provided you save the same amount at the beginning of each year. What should this amount be so that you end up with £1,000?
And which of those two questions did you find more engaging? If you are like almost everybody, you will already be thinking about the first, but the second will make your heart sink.
Recently, the government has expressed a wish that all schoolchildren should study mathematics up to the age of 18, a view that appears to have cross-party support. As a mathematician, I am a firm believer in the benefits, both direct and indirect, that mathematical understanding can bring. However, I am also aware that many intelligent people thoroughly dislike mathematics, give it up at the age of 16, and have absolutely no regrets afterwards. Will two further years of mathematics really make a difference to such people, other than turning them off the subject even more?
One method that is sometimes proposed for making subjects more appealing is to make them ‘relevant’. In mathematics, this supposed relevance often takes the dismal form of ‘word problems’ such as this: two apples and three pears cost £1.80, while four apples and one pear cost £1.60. What do apples and pears cost each? To solve such a problem, the technique is to turn the words into equations and solve the equations. Here one might begin by saying, ‘Let A be the number of apples and P be the number of pears. Then 2A+3P=180 and 4A+P=160.’ Then, using standard techniques, one shows that A=30 and P=40, so apples are 30p each and pears 40p each.
But problems like that don’t feel relevant at all. This problem may pretend to be about a trip to the greengrocer’s, but we all know that it is really just a flimsy disguise for some equations. We also know that a question of this form would never arise at the greengrocer’s: if you want to know the price of apples, you look at the little sign that tells you the price of apples.
What is it that gives the stone-in-shoe question its appeal? Part of the answer is that one can imagine being in the situation described, or at least one can imagine that somebody might be in that situation. But that cannot be the whole story, because one can also imagine needing to know how much money to put away into a savings scheme in order to end up with a certain amount, and yet that question has no appeal at all. Another difference between the two questions is, I believe, more important: whereas the second question asks for a number, the first asks for a piece of advice. Many people, when asked to do a numerical calculation, switch off immediately, but almost nobody switches off when asked for advice: the natural reaction is to put oneself in the position of the person seeking the advice and to try to work out the best thing to do. The stone-in-shoe question exploits this instinct, at least initially, and it can then be answered without any calculations. (Just imagine how much less appealing the question would become if you were told the speed at which you walked and the speed of the moving walkways. Fortunately, you don’t need to know these.)
One might think that if calculation and solving equations were absent from a mathematics course, then there would be nothing left to teach. But that is quite wrong: there are plenty of things one could teach, many of them entertaining, important and useful in later life. Here are some examples.
We often need to make decisions based on incomplete data. Exact calculation is usually not possible in such situations, so it is very useful to be good at making rough estimates. For instance, will the benefits of building a high-speed rail line to Birmingham outweigh the costs? Even to begin to think about this question, one should have a rough idea of the number of journeys that would be made on the line each day. A useful trick for getting the right order of magnitude for quantities like this is to break the problem up into smaller parts. In this case we could estimate the number of hours per day that trains run on the line, the number of trains per hour, the number of carriages per train, the number of rows of seats per carriage, the number of seats per row and the proportion of seats that would typically be occupied. We would then need to multiply these numbers together. My own guesses, which I have made simple round numbers so that the multiplication will be easy, are 15, 4, 10, 20, 5 and 1. Multiplying those -together I get 60,000. Perhaps you would like to object to my assumption that the proportion of seats occupied is equal to 1. Of course I don’t actually believe that all seats would be occupied, but I think that most of them probably would be, and at this level of -accuracy rounding up a number like 0.8 to 1 is perfectly acceptable.
Another skill of genuine use is that of getting to the heart of a question by abstracting away irrelevant details. Consider the following dilemma faced by Alice, who has just been proposed to by her boyfriend Bob. Alice is very fond of Bob, who is a better match than any of her previous boyfriends, but she worries that whatever she does, she may end up with regrets. If she accepts his proposal, she risks going on to meet somebody she would much prefer to be married to, but if she refuses him, she risks never again meeting anybody as suitable.
Let us imagine that Alice is determined to be married by the age of 36, and that by that age she would expect to have had serious relationships with eight people, of whom Bob is the third, say. Then we can model Alice’s situation as follows. She is presented with a sequence of eight random numbers, one by one. At any time, she can say ‘stop’ and the number that has just been presented to her is the one that she must accept. What strategy will give her the best chance of accepting the largest number?
This purely mathematical problem encapsulates Alice’s difficulty and has a known solution. Given the numbers above, it can be shown that Alice’s best chance of avoiding later regrets is to turn down Bob and then go for the first person she meets who is better than Bob. However, the validity of this advice depends on a number of questionable assumptions — not least of which is that the ‘irrelevant details’ that were abstracted away really were irrelevant — so this question is a good example both of the power of mathematics and of its limitations.
A third skill that is extremely useful is the ability to evaluate statistics, since we are continually bombarded with statistical arguments of widely varying degrees of soundness. For example, studies have shown that British vegetarians have, on average, higher IQs than the general population. Does this show that meat is bad for your brain? What other explanations might there be for an observation like this? How informative is an average anyway? Given some numerical data, what else can one usefully calculate from it besides the average? How large a random sample is needed if you want to be convinced that an observation is probably more than just a typical random fluctuation? One can get a feel for this kind of question without ever calculating an average or a standard deviation.
How should this kind of mathematics be taught? I strongly believe in two guiding principles. The first is to start with the real-world questions rather than with the mathematics. That is, rather than explaining mathematical ideas (about statistics, say) and then discussing how they can be applied to the real world, a teacher should instead start with a question that is interesting for non-mathematical reasons and keep a completely open mind about what mathematics has to contribute to the discussion.
The second is to make the discussion as Socratic as possible. Rather than asking the question and then explaining the answer, the teacher should just ask the question and leave the job of answering it to the pupils. The teacher’s role would be to guide the discussion, encouraging it when it moves in fruitful directions and making gentle interventions such as ‘Does everybody agree with that?’ when somebody says something wrong and is not corrected. This would be the opposite of the kind of spoonfeeding that goes on with GCSE and A-level.
Imagine if a teacher came into the classroom and said, ‘I’ve just read in the news that they are considering culling 70 per cent of badgers in certain areas of the country to halt the spread of TB in cattle. How on earth do they work out how many badgers there are in the first place? And how will they be able to tell whether the culling has worked?’ And imagine if the teacher admitted without any embarrassment to not knowing the answers. The aim would be to prompt a discussion in which the pupils were treated like adults and encouraged to think. The discussion would have many features that occur in real life: it would be open-ended, it would involve quantities that are hard to measure, it would be about estimates rather than exact calculations, and it would be responding to a non-mathematical need.
Can this possibly work? In February I was at a meeting about mathematics education at which Michael Gove was present, and at which I advocated this kind of course. The idea interested various people at the meeting, so in June I wrote a blog post about it, for which I compiled a list of over 50 questions that I thought could be the basis of interesting classroom discussions. One of those interested, Sir John Holman, arranged for me to visit Watford Grammar School for Boys, where I was given two hours with a class of about 25 sixth-formers, some from that school, some from the equivalent girls’ school, and some from a nearby comprehensive. Some were doing maths A-level and some were not. I discussed about half a dozen questions with them in the way I have been suggesting, and that left me convinced that it can be done.
Another person who was interested was Charlie Stripp, the chief executive of Mathematics in Education and Industry, an independent curriculum development body. He got in touch with me and said that MEI wanted to try to develop a course along these lines. Very recently, the government has agreed to provide the necessary funding, not just for developing the course, but for working out how best to assess it and for organising appropriate training for teachers, both of which will be essential, given how different this course will be from a traditional mathematics course. There is no guarantee that the course will be taken up by schools, and even if it is, it will not be suitable for everybody. But there is nothing to lose by making a course of this type available, and it is an experiment that is surely worth trying.
Some further questions for interested readers. The best answers will be published in next week’s letters page (email@example.com).
1. Roughly how often would you expect somebody in the UK to dream of the death of a loved one and that loved one to die the very next day?
2. You play a game in which when it is your turn, you can either add a point to your score and remove two points from your opponent’s score, or stop the game. You start with five points, and when someone stops the game you get £10 for every point you then have. Your opponent, whom you dislike, starts, choosing to add a point to his/her score and remove two points from yours. What should you do?
3. A divorcing couple are dividing up their possessions. The husband and wife agree about the financial values of these possessions but attach different sentimental values. Devise a good procedure for carrying out the division.
4. Roughly how many people could fit into the Isle of Wight?
You can find Timothy Gowers’s blogpost on teaching mathematics at specc.ie/howtoteach and his account of visiting Watford Grammar School for Boys at specc.ie/watfordgrammar