Should Alice marry Bob?

Introducing a new ‘real world’ maths course, designed to engage every sort of pupil

3 November 2012

Two problems:

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.)

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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 (letters@spectator.co.uk).

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

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Show comments
  • Cogito Ergosum

    It is all very well to talk about the class discovering the solution, but this is a slow and erratic process. The phrase ‘random walk’ comes to mind. There simply is not sufficient time to cover the whole syllabus in this meandering way. One or two topics by this method, certainly; but most of the stuff has to be taught systematically.

    • http://davidwees.com David Wees

      School is not, fundamentally, about “covering a syllabus.” It is about teaching & learning. Unfortunately, sometimes our demands to “cover a syllabus” actually prevent good teaching and learning from occurring.

    • oldlearner

      That is one of the reasons online classes would be perfect for this type of “random walk”. students are liberated from the time constraints of the usual math class and can confer with others via the internet.

      • David Murphy

        Its true to a point, but you really need guidance from teachers, and also uncontrolled conversations on the net often degenerate into flame wars and macho swaggering by those who think they are great, and which can be off-putting to many of those you want to reach.

        I have been dong coursera courses lately and seen quite a lot of the macho swaggering type behaviour in the forums.

        So, yes, a great medium potentially but not ready for prime time yet.

    • Hector

      Agree, but the big issue is that in any case 99% of those following the “systematic” approach are unable to really use the learned stuff… Be honest, How many times have YOU solved a linear equation to get something of “value” in your real life. It is -in my opinion- much more important to motivate the student, and then he’ll be able to learn anything he needs “on the fly”.

      • Cogito Ergosum

        For example, the spreadsheet list of members for an organisation records 109 households and 162 members. From that, simple algebra yields 56 single members and 53 couples.

        Furthermore, one can only learn something ‘on the fly’ if there is already a good background knowledge into which the new stuff can be merged.

        In reply to David Wees, you must cover the syllabus in order to learn sufficient of what the teacher asserts is worth learning.

      • StatingDObvs

        “…but the big issue is that in any case 99% of those following the
        “systematic” approach are unable to really use the learned stuff… ”

        On what facts do you base this assertion?

        Seems evident to me the opposite is true. Learning to read and write is taught systematically. And the vast majority of those who learned it use it every day. It’s beyond your figure of 99%, and in the opposite direction you asserted.

  • Burbage

    I have used similar techniques – as I imagine most physics teachers will have done – where it’s important for students to see how laws can be deduced without relying too heavily on, often absent, mathematical skills. Though we tend to call them thought-experiments.

    However, the ‘discovery-led’ model of education can’t be relied on much, as the Nuffield Science experiment, a glorious monument to misguided optimism, easily demonstrated. Woolliness may be intrinsically appealing, but it’s rubbish for revision, and it’s much more difficult to find, let alone correct, failures in reasoning than failures in working.

    Critical thinking is important, but so is the knowledge of, and the ability to use, mathematical tools independent of context and, as an A’ level physics teacher, the need to additionally teach proper mathematics on a remedial basis isn’t always welcome.

    I wish you the best of luck, though I doubt, somehow, that your industrial correspondents will find the outcomes very different to those of the thousands of experimental approaches to teaching mathematics that have broadly failed in the past. But, you never know, there may be a book in it.

    • samar

      There is a frequent perception in the popular mind that school mathematics is good for little else other than getting into college. That’s not entirely true, of course. However, the ubiquity of computational devices coupled with the inability of most people to be able to think of a real life problem in terms of modeling the problem, robs school maths of its relevance. The challenge is in restoring relevance in school mathematics to an era when computation is not the issue but defining the problem in mathematical terms is.

      I think this article will go some way to doing that if it stimulates a re-think of school mathematics. The bigger challenge will be in getting experienced maths teachers to be able to align with it.

  • http://davidwees.com David Wees

    I’ve written about this exact same issue here: http://davidwees.com/content/fundamental-flaw-math-education

    I don’t think every example should come from the real world per say, but many of them should. If they aren’t obviously relevant, they should at least be interesting…

    • Alberto Godoy

      per se

      • http://davidwees.com David Wees

        Thank you for the edit. :)

  • skylark

    Paul Lockhart, who is high school math teacher with a research mathematics background, wrote a wonderful essay on this topic, later published as a book, called “Mathematician’s Lament”. His new book “Measurement” shows content and strategies for successfully teaching mathematics by these principles.

  • Baker Kawesa

    The first problem seemed rather irrelevant, but I liked the second one – perhaps because it’s about money. How to solve abstract problems that appear to have no practical value, no concepts, only boring formulas and pretend narratives – that’s school math for you, it’s no wonder students flee.

  • Geoff.

    b. £200 per year then you keep the interest. Save yourself the bother of calculation.

    • John

      Depends on whether the reinvestment rate is changing or unchanging. The old “bird in the hand” issue. It also is silent on risk considerations

  • Bill Bonney

    I enjoyed your questions and I think they would engage many students. What I would add to this questioning technique would be some grounding of basic number theory. Many of my students lacked those foundational elements though, using sound teaching techniques, they realized mathematics was not beyond them. Frequently students need a little more time and sound teaching and they can master the fundamentals which gives them the ability to grapple with mathematical models.

  • rudy

    why not just admit some people like math, and most don’t :-)

    • http://www.learningnerd.com/ Liz Krane

      But it IS possible to learn to like math (or any other subject for that matter) if you have enough incentive, and it IS worth it to try. If nothing else, it’s at least a good challenge. :)

      • Sanjay

        ya the thing is there is a way for doing everything. Math needs to be felt and thats the challenge.

    • David Murphy

      Wh should we? Part of the problem is that maths is badly taught and taught better ti would mean more people would use it. We need many moe people educated in STEM subjects and maths is the underpinnig for all of them.

    • http://www.justarandomguy.com Akshay Bist

      While its true that some people don’t like math, and don’t appreciate it, that is no reason not to try and improve math education. If math is taught the right way, and made interesting, no reason why it wouldn’t fast become popular with the kids.

  • Franklin Vera Pacheco

    He is indeed a great mathematician but in this and other of his writings about math education he is pushing ideas I hope no one ever tries
    to implement or it will damage math education greatly and deeply. On one
    hand, almost paradoxically, everything he is saying is true. It is true
    that those real life problems are more compelling for a general a
    student. It is true that they are also quite important. Now, implement
    it and you will have in schools the same array of good, average and bad
    teachers presenting a class in this setting. The result will be that
    more people will get out of math classes a more educated intuition,
    better grasp of how to interpret statistical information. But. All, will
    lose, necessarily, getting to give to the mathematically inclined
    students good chunks of the technical parts of mathematics. For example,
    years ago you would in a basic Calculus book the whole method of
    integration of rational functions. Now not only a small part of it is
    taught but in the regular textbooks only part of it is written. It is
    true that most technical parts are not important for the whole majority
    of the students. But culture doesn’t work that way. Culture is built by
    individuals getting interested by some of the interests he finds around. If no
    one ever talks to no one about some small technical aspect of
    mathematics, only interesting for some, then you get a population in
    which that piece of culture starts getting lost. In my opinion, what is
    key is to make school more dynamic and personalized. Instead of having
    linear curricula to implement more ramified ones. Instead pushing
    students through courses talking about ‘interesting’ (in quotations
    because interest is always relative) things like this and many other
    good ideas, I think is better to force students a little bit, but once
    the level of failure reaches some limit to allow passing forward, but
    through different school curriculum channels. For example, and to stay
    with the topic of math let us give the example in math. Students can
    start by attending a math class with an average level. Let’s say a class
    teaching high school algebra as it was taught a few decades ago.
    Students that are finding this easy can be sent as soon as possible to
    more challenging classes. Either more advanced (in content) or more
    challenging (in difficulty, like those who train for math competitions).
    Those students struggling can be sent, if the marks are failing her/him
    to more washed out (washed out of the technical parts of mathematics,
    not dumbed down) math classes like these that Gowers is proposing. I think that only this way culture is preserved.

  • Vincenzo

    I like the challenges these questions present

  • http://www.learningnerd.com/ Liz Krane

    Making math more relevant will be different for each student — some people get interested in math via music or art or making video games. :) Most schools are just so focused on rigid syllabi, they don’t give students enough time to explore subjects like math in a way that actually leads to critical thinking skills and creativity. So there’s a LOT that needs to be changed before a course like this could really happen in a typical school.

    But I do love the ideas in this article and I hope it becomes a reality! It would certainly be better than the math classes I had to take.

  • P C Ray

    If every body follows the kind of discussion proposed and assuming that 75% may get adept in addressing non-quantitative decision making, they should pursue courses in Logic and Philosophy but not Mathematics, which they will not learn at all.

    Technical Mathematics also has great beauty, elegance, ability to handle very large and very small. It is not an exaggeration to say that it might lead one to an appreciation of the deepest spirituality and a feeling of the Almighty.

    – P C Ray ( 83 Yrs, and still studying mathematics)

  • Tay Moh Niat

    I believe this is a practical approach that encourage students’ engagement. It make learning meaningful through innovative teaching.

  • http://www.facebook.com/MostafaAlaaFCIH مصطفى علاء

    مقال رائع فعلا انا اللى كاتبه واحد من اكبر الرياضيين على مستوى العالم على حسب ما قرأت عنه

    • Cogito Ergosum

      What you write is, I hope, interesting. Unfortunately, it means as little to me as a sheet of printed music.

      And there is another subject which is badly taught. Not until I saw it clearly stated in a book written by a mathematician, which I read while at university, did I realise that there are twelve semitones in an octave. No music ‘teacher’ I ever met at school ever explained how music works: they just played it and hoped people would catch on.

      Entertaining people with music is one thing, teaching it is another. The same holds for any other subject: mathematics, English, history, …There is no substitute for systematic teaching.

  • ashraf abdalla mohamed manaa

    i like it

  • Pedro DC

    The answer to the second quesrion is £893

  • DT

    English teachers will remove all nuance. Politicians will tell the truth. History will stop repeating itself, and a math nerd will be elected the next PM. Imagine.

  • http://www.facebook.com/phil.bachmann.9 Phil Bachmann

    The answer to Mr Gowers’s concern is not to distort mathematics but to directly teach the perceptual thinking that makes mathematics useful: http://www.cortthinking.com (Edward de Bono’s CoRT Thinking).

  • Paul Robison

    I think he may have discovered engineering.

  • Bryan Meyer

    I loved this post! I teach in the US at a project-based high school which allows me to try these things with students. I know that, from the student perspective, results are positive. I always meet resistance here with national standards and public perception about schooling and definitions of what math is. I hope that those things continue to evolve and change so that your vision for math ed can exist in more places!

  • Eric I.

    Dan Meyer, a young high school math teacher, makes very similar points in his TED talk and discusses how he’s now approaching teaching. It’s a very compelling talk. http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html

  • D Short

    I gave up on reading this properly after the firs few paragraphs because it was so boring. But when I scanned to the end and saw question 4, I have heard that the entire world’s population coud fit ‘on to’ the Isle of Wight. I presume the writer meant ‘on to’ and not ‘into’ the Isle of Wight.

  • http://twitter.com/KeenKidsAtHome Lisa

    Despite being a reasonably bright person (member of Mensa), I came to the conclusion in high school that I didn’t like/wasn’t good at mathematics. In my last required course in mathematics I earned a squeak-through mark of 50% and I never looked at it (or regretted it) again. I never saw the point of it, and it was never presented as actually having a point.

    Fast-forward 25 years and I am now homeschooling my own bright children. One is naturally inclined to math (he gets that from his father) and has always had an innate joy about it, provided I don’t require much drill. He is several grades ahead, and gets excited about learning advanced materials. The other is, sadly, like me and is naturally math-averse. For him, I have spent lots of time and resources focussing on the beauty of math — the Fibonacci sequence in nature, fractals, stories of mathematicians in history, the history of math. We learned about Pythagoras and *why* he needed to develop his theorum (to solve a practical problem). No one had ever explained that to me before. I have become fascinated by math now, as has my son.

    As well, we started using a math curriculum called The Life of Fred (http://www.stanleyschmidt.com/FredGauss/11catofbooks.html) which not only introduces high-level math concepts from an early age, but is short on drill and long on fun and practical applications of math. We both can’t wait to sit down together and learn math. I really wish I could have enjoyed it this much when I was his age.

  • Sahil Gupta

    Nice thought and I believe with Franklin Vera Pacheco….It should be implemented dynamically and could be given as compulsory stllabi part as each student have to deal with rela life problems anyhow in the later stages of life.

  • Wills

    I feel I should pass on my Mum’s sure-fire solution to such dilemmas as the one in Question 3: “One cuts, the other picks.” Utterly infallible and prevented many disputes in childhood, and subsequently. Divisions are scrupulously fair as an unfair division penalises the divider.

  • Wills

    Beyond that, the secret to teaching more maths is to teach less of it. That is, a basic level of competence for daily life should be imparted, very slowly, to all at a speed they can understand. Though with a talent for it can be spotted and fast tracked at any point in their school career. 12-year-old PhDs? Fine. But even more important, every 18-year-old can understand the evil that is Wonga.com.

  • Joe Short

    I had a really good history teacher at my secondary school and a really poor maths teacher. The history teacher engaged us and was clearly in love with his subject. The maths teacher taught by text book and seemed totally disinterested in his subject. I have now just finished a coursera.org course on mathematics by Dr Devlin of Standford University, “Thinking mathematically” was a seven week university level maths course which was instructive, enjoyable and made mathematics an engaging subject to consider studying. I regret not having a Dr Devlin type of maths teacher when I was at school. His love of maths and what can be achieved came across from the very first lecture. Unless teachers are engaged in their subject and pass on that enthusiasm to their pupils changing how the subject is taught may not achieve anything.

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