Help — I’m stuck in my ivory tower!

The UK Qualifications and Curriculum Authority is considering introducing a new A’level course (in Britain, A’level is the exam that is taken at the end of high school) called “Use of Mathematics”. As one might expect, this idea has not met with universal approval, and there is now a campaign to stop the idea in its tracks. (I should warn you that the preceding link is to a Word file rather than to a web page.)

The General Secretary of the National Association of Headteachers has this to say to the campaigners:

They should get down from their ivory towers. They should be out in the world where young people live and exist and they should be appreciative that young people have great skills in the use of technology and we have to latch on to that.

We cannot continue teaching an out dated 19th century curriculum. This is simply turning many children off education because it is completely not relevant to them at all.

Some sample papers for the new course have been made available, so let’s have a look at the up-to-date 21st-century curriculum that will enthuse a new generation of British schoolchildren. I’ll concentrate on one or two questions but if you want to see more, then the sample papers can be found at the bottom of this page. (Update: unfortunately, these sample papers have been taken down. I can’t help wondering why. Further update: at least some sample papers are now available at the bottom of this page.)

Before I discuss any of the actual questions, let’s imagine that we are sitting in a committee trying to devise a use-of-maths syllabus. What could be on it? Perhaps the most obvious place where mathematics is used is science, but that kind of use of mathematics is covered in mechanics questions and also in physics. Another place is statistics, but that too is on the existing mathematics syllabus. To help us, let us remember that we are looking for something that is relevant to schoolchildren. One might think that statistics was pretty relevant, but we had better suppress that thought and look for something else.

Here is an idea. Many children will one day find themselves taking out a mortgage. Perhaps we could devise a question that will help them think about how mortgages work. I’m not saying in advance that this will turn out to be a good idea, but let us at least try.

For later reference, I want to discuss an obvious problem about repayment mortgages and to do so in some detail. Suppose for simplicity that the interest rate for an interest-only mortgage would be 5% and that this rate never changes. If I take out a repayment mortgage of £50,000 and pay £500 a month, then roughly how long will it take me to pay off the mortgage?

Let me answer that in a way that comes naturally to me, and, I’m guessing, to most mathematicians. To start with, I would replace a discrete problem (payments once a month) by a continuous one (money leaking out of my bank account at a constant rate). Next, I would forget the numerical values, which obscure what is going on (for instance, they make it harder to keep track of units) and rephrase the problem like this: at time 0 I take out a loan of M_0, and thereafter the amount I owe, M(t) changes in two ways. On the one hand there is a constant-rate decrease of \lambda (because of my repayments) but superimposed on this is an increase (the interest I have to pay) that is proportional to the current amount I owe, which we can denote by \alpha M(t). In other words, M(t) satisfies the differential equation \frac{dM}{dt}=\alpha M-\lambda.

We can solve this by turning it upside down and getting \frac{dt}{dM}=\frac 1{\alpha M-\lambda}, which we can solve easily since all we have to do is integrate \frac 1{\alpha M-\lambda} with respect to M. From this we get t=\alpha^{-1}\log(\alpha M-\lambda)+C. Rearranging, we find that \alpha M-\lambda=\exp(\alpha(t-C)), so that M=\alpha^{-1}(\lambda+\exp(\alpha(t-C))).

When t=0 this is supposed to give us M_0. From that it follows that \alpha M_0-\lambda=\exp(-C), so M=\alpha^{-1}(\lambda+(\alpha M_0-\lambda)\exp(\alpha t)). Therefore, the amount of time it will take until M=0 is the value of t such that \lambda+(\alpha M_0-\lambda)\exp(\alpha t)=0, or \exp(\alpha t)=\frac \lambda{\lambda-\alpha M_0}, or t=\alpha^{-1}\log(1/(1-\alpha M_0/\lambda)).

From this expression we can see that I will never pay off the mortgage unless \lambda>\alpha M_0, though in fact it is more sensible to deduce that from the differential equation: if \lambda\leq\alpha M_0 then M will not decrease. (This is telling us the rate at which repayments must be made in order to keep up with interest payments.) Also, if we know a little about the shape of the exponential function, we can see that if \alpha M_0-\lambda is negative, then M will decrease rather slowly at first and much more quickly later on. This is a well-known phenomenon with repayment mortgages: initially most of the repayments are interest repayments (because the amount owing is large so the interest is large) but later on they are mostly capital repayments (because now the amount owing is small so the interest is small).

Of course, there is one final stage, which is to plug in some numbers. I won’t do it completely here, but I will point out that at least some thought is required if we want to work out what value of \alpha corresponds to an interest rate of 5% and what value of \lambda corresponds to a monthly repayment of £500. If we measure in years, then we need to choose \alpha such that e^\alpha=1.05 and we should take \lambda to be 6000. We are given that M_0=50000, and a reasonable approximation for \alpha is 0.05 (since e^x\approx 1+x for small x), so the time taken will be around 20\log(1/(1-c)), where c=50000/6000\times 20=1/2.4. So 1/(1-c)=2.4/1.4 which is slightly under 2, so we get a bit less than 20\log 2, and could get a better estimate with the help of a calculator (which is not just allowed but actually required in the use-of-maths exam).

Now what skills did I need in order to do that calculation? (Apologies if I’ve made a mistake in it — I have not checked it carefully.) There were basically two. The first was to transform the original real-world problem into a purely mathematical one. The second was to solve the mathematics problem, which involved solving a fairly simple differential equation and then doing some routine algebraic manipulations.

I would guess that an average A’level student would find the first task quite hard, because it involves a bit of thought: it seems that most of the A’level syllabus nowadays consists of learning to do certain algorithmic tasks such as differentiating compositions of basic functions, and not much is devoted to solving problems or to solving the kind of modelling problem that constituted the first stage of the above calculation. But perhaps this is where the use-of-maths A’level would come into its own. One could do a bit less of the pure stuff, but by way of compensation one would learn how to take a real-world problem, transform it into mathematics, and solve the mathematics. I’m not sure I like that idea, but it could perhaps be justified, so let us now have a look at some questions in the sample papers.

Candidates are to be given something called a “data sheet”, which you might think consisted of tables of data that you then had to use your modelling skills to analyse and comment on mathematically. But actually the name “data sheet” is rather misleading: it’s more like a couple of pages with a few mathematical concepts explained. I think the idea is that the data sheet explains the mathematical principles and then your job as candidate is to use the principles.

For the question I want to talk about, which is number 2 on this paper, the data sheet is called “Waves as models”. I’ll let you read that if you want, but here’s the question.

2. The article states that, for the case of the simple pendulum, the angle, \theta, that the string makes with the vertical, t seconds after release, can be modelled by a function such as \theta=\frac\pi{12}\cos\left(\sqrt{\frac gL}t\right).

(a) What does this model suggest for the angle that the string makes with the vertical when it is first released?

(b) For this model, show that \frac {d^2\theta}{dt^2}=-\frac gL\theta.

Ah, so my guess was wrong. You aren’t asked to model anything. Instead, you are given the model! (The equivalent for my mortgage question would be to be told what the formula was for the amount owing at time t and to be asked to draw various conclusions from the formula.) So what exactly are the skills you need to solve the above question? Again, there are two.

The first is the ability to solve what in the US are called word problems: this means that instead of being asked to solve something like x+2=5 you are given an equivalent wordy problem such as “I have some apples in a bag, put two more in, count them, and find that I have five; how many did I have originally?” When you get used to these, they are rather easy: you just cut out all those stupid words and leave the maths. (However, interestingly, they were found very difficult when I taught them in the US. I was a PhD student at the time and the course was College Algebra 1021 at Lousiana State University, taken by people who would typically not go on to major in mathematics.)

What do we have to do for part (a) above? Well, the underlying maths problem is very simple indeed: what is \theta when t=0? To get to that problem, we had to interpret “when it is first released” as “at t=0” and we had to interpret “the angle that the string makes with the vertical” as \theta. The second of these tasks is trivial, since the question has just said that that is what \theta is, and the first is, well, not very challenging.

On to the second part. It’s telling us to differentiate \theta twice, but it just about qualifies as a word problem because it starts “For this model”. Luckily, we can turn this “word problem” into maths by simply ignoring the words “For this model”.

The setter of this question seems to have a touching belief that if the word “model” is splashed around a bit, then candidates are learning how to use mathematics. But they are doing nothing of the kind. They are learning how to solve word problems, and very easy ones at that. (For extreme examples of questions where the word model is used a lot, but plays absolutely no role in the questions, see the Calculus sample paper.) And the irony of it is that you learn how to do word problems in a conventional A’level syllabus, and you learn more mathematics in the process. Even worse, the problems in the use-of-maths sample papers aren’t real word problems: in a word problem you normally have to say something like, “Let’s represent this quantity by $x$.” Even that step seems to be regarded as too challenging on these papers.

There are in fact some questions about interest rates and the like on another paper, but the “data sheet” gives you formulae for everything, so that all you are required to do is take the values given to you in the question and plug them into the formula. (The data sheet, by the way, is made available before the exam.) Let’s just think for a moment about whether this is likely to be a valuable life skill.

Suppose, for instance, that you have a choice of two long-term savings accounts. One of them has a higher interest rate, but the other one has a bonus if you keep your money in for five years. Luckily you took use-of-maths A’level a few years ago, so you should be in a good position to decide which one to go for. Ah, but you’ve lost your data sheet, and in any case the data sheet didn’t tell you a formula for the amount you end up with when there is a bonus involved. What can you do? There isn’t an obvious internet search: the problem is that you have to think a bit, and unfortunately what you’ve been trained to do is plug numbers into formulae that are just given to you.

There are so many ironies to this. Those who propose the use-of-maths A’level will no doubt say that it is not a dumbing down of mathematics A’level (with its out-of-date nineteenth-century syllabus) but rather an equally rigorous exam that tests different skills. They will also say that the syllabus is more interesting and relevant. But it is blindingly obvious from the sample papers that it is not testing different skills (except perhaps the skill of understanding what the data sheets, which unfortunately don’t seem to be available in real life, are on about), and is deeply boring, and not even all that relevant to the people who are actually taking the exam, who should be enjoying their last few years of not having to think about mortgages, income tax returns and the like. (Does anyone seriously think that teenagers will be filled with enthusiasm by personal finance, when for adults, who are directly affected by it, it is an awful chore?) A conventional A’level student will do plenty of word problems and more mathematics, and will also solve modelling problems when they do statistics and mechanics. Who will end up better at solving mathematical problems that arise in the real world? You do the math.

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43 Responses to “Help — I’m stuck in my ivory tower!”

  1. Heinrich Says:

    I think the main problem of is that it is nigh impossible to teach the modeling part to disinterested pupils, simply because it does not involve a rote algorithm. (Not to mention that I doubt that most high school math teachers are themselves able to model the mortgage repayment like you just did!) It does seem to be possible to drill students to perform rote procedures to some extend, but it is futile to try to make someone understand what he does not want to understand. I know of know mathematician who become one against his will, to the contrary of course.

    On a side note, what exactly did the 19th century syllabus look like? Anyone know of old textbooks and class notes available on the internet? I wouldn’t be surprised if the math syllabus from the times of Klein and Poincaré is superior to the high school rote calculations of the 1990s.

    • Jason Dyer Says:

      I might recommend reading someone of Dan Meyer’s posts regarding getting the students “internal framework for processing the world before applying the textbook’s framework for processing mathematics,” like this one:

      http://blog.mrmeyer.com/?p=4018

      He also takes likes taking swipes at the pre-packaged textbook presentation as the post points out above.

  2. Phil H Says:

    Agree completely with the preceding comment.

    As for the repayment mortgage problem, I suggest that most people would simply use a spreadsheet. It’s a very simple matter to generate a month by month table showing repayments, interest charged and the outstanding balance. No calculus, or even summation of a geometric series – it involves nothing more than arithmetic. (It’s also easy to compare different packages, the effect of ad hoc additional repayments, and so forth.) This highlights the problem of devising a syllabus that provides “real world” problems that are both non-trivial and use mathematics that A-level students could cope with.

    Heinrich, Littlewood’s account of his education in his “Miscellany” may give some indication of the 19th century syllabus. He admittedly did a lot of private study, but you can probably still get a good idea of what was expected.

    P.S. Professor Gowers, could you please provide a link to the existing A-level syllabus?

    • gowers Says:

      It seems to be quite hard to find a straightforward link for that. The best I have managed to find so far is this.

      I basically agree with you about the mortgage problem — it shows that you either do some genuine modelling, which is almost certainly too hard for people taking a soft-option (not officially, but we all know that that is what it is) mathematics A’level, or you do something that you pretend is modelling but in fact is an artificial and stultifyingly dull exercise that will not help you use mathematics in later life.

      Having said that, a schoolchild who could generate a spreadsheet of the kind you describe would have had to learn something useful, which suggests to me that a much better idea than a use-of-mathematics A’level would be an A’level in how to do simple programming with a small amount of mathematical content.

  3. Américo Tavares Says:

    For your discrete problem about repayment mortgages I present a proposal for a direct solution. This solution, even if correct, is of course far less instructive that your reasoning!

    In general we have to find the value of constant payments A given the principal P during n monthly periods. The value of A in the period k is equivalent to the present value A/\left( 1+i\right) ^{k} monetary units, where i is the interest rate in each capitalization period. Summing
    in k, from 1 to n, we get the sum

    \displaystyle\sum_{k=1}^{n}\dfrac{A}{\left( 1+i\right) ^{k}}

    Now we have to sum a geometric progression with rate r=1/(1+i) and first term term u_{1}=A/\left( 1+i\right)

    \dfrac{A}{1+i}\dfrac{\left( \dfrac{1}{1+i}\right) ^{n}-1}{\dfrac{1}{1+i}-1}=A\dfrac{\left( 1+i\right) ^{n}-1}{i\left( 1+i\right) ^{n}}=P

    or

    A=P\dfrac{i\left( 1+i\right) ^{n}}{\left( 1+i\right) ^{n}-1}

    For the given problem, the payments will be made during n months, with i=5/12\%=\dfrac{5}{1200} and P=50\,000. Thus

    A=50\,000\dfrac{\dfrac{5}{1200}\left( 1+\dfrac{5}{1200}\right) ^{n}}{\left( 1+\dfrac{5}{1200}\right) ^{n}-1}=500

    or

    \dfrac{5}{12}\left( 1+\dfrac{5}{1200}\right) ^{n}=\left( 1+\dfrac{5}{1200}\right) ^{n}-1

    Solving for n, we get

    n=\dfrac{\ln 12-\ln 7}{\ln 241-\ln 240}\approx 129.63 months (10.802 years)

    and, as you proved

    20\ln 2\approx 13.863>10.802.

  4. Américo Tavares Says:

    Please let me only add one further interpretation of mine: the continuous effective interest rate can be derived from the 5% nominal interest rate, compound m times per year as follows:

    \underset{m\rightarrow \infty }{\lim }\left( 1+\dfrac{0.05}{m}\right) ^{m}-1=\underset{m\rightarrow \infty }{\lim }\left[ \left( 1+\dfrac{0.05}{m}\right) ^{m/0.05}\right] ^{0.05}-1 =e^{0.05}-1\approx 5.127 %

    (which approximates your formula e^{\alpha }=1.05 for \alpha =0.05), and in the general case of an annual nominal interest rate r as

    i_{E\;(m\rightarrow \infty )}=\underset{m\rightarrow \infty }{\lim }\left( 1+\dfrac{r}{m}\right) ^{m}-1 =\underset{m\rightarrow \infty }{\lim }\left[ \left( 1+\dfrac{r}{m}\right) ^{m/r}\right] ^{r}-1=e^{r}-1.

    From a point of view of a purely mathematical problem the model you are discussing in the context of the “use-of-maths A’ level ” is much more interesting and worthy.

  5. Américo Tavares Says:

    I hope to be able to make a more relevant comment this time. How can for instance an engineer (I take this example deliberately because I am a retired one) conceive NEW engineering solutions if he/she is only almost able to use formulae or tables? I think he cannot, unless he/she teaches him/herself to understand very well the maths he needs in the profession. And he/she may well be in a situation in which there is a need to find a new method in Applied Mathematics, in some particularly difficult engineering solutions or applications. Or someone, somewhere would have to do that part of the job. Of course other skills are also required but this one is important.

  6. chunter Says:

    I can understand why the change is requested, the issue is that the change being requested is wrong and won’t satisfy the problem.

    It has long been a frustration of mine that I was encouraged to take difficult mathematics classes (which except for Geometry, I wasn’t that good at it and after a time, I got seriously lost,) but none of these classes ever explained things like finance, loan amortization, investment yield, or even how to balance a checkbook or fill out a check. I suppose the classes carried the implication that I could figure all of that out, and as I recall, I did, but that is because I was good at those word problems you were describing.

    I enjoyed word problems, because despite having unsorted data to discover and organize, the underlying problem was indeed extremely easy to solve. I find that most of my modern uses of math are an exercise in reorganizing data, disposing of the unnecessary bits, and finding my answer. In my current customer service work, I sort through customers’ descriptions of problems, get rid of unnecessary information, and solve the customer’s problem. I suppose I found the right line of work.

    I discovered in Geometry class that people that had trouble turning words into a simple math puzzle also had trouble with pictures. My only frustration was being asked to provide proof that something is what you can tell it is at a glance, for example, “This triangle is right because it has three sides and a right angle.”

    This “Use of Math” business sounds like it is attempting to appeal to me, or to me as a teenager, and I agree with the problem it recognizes, but you cannot teach people how to turn language and real world problems into rote mathematics by showing them the formulae in advance. What we have is a problem of teaching our students how to research and study, how to take any problem and deduce a solution, without having to show it to someone saying, “This situation is in exception to the rules you’ve shown to me, I don’t know what to do.” Sadly, I don’t know how to teach real problem solving; although I solve problems for a living.

    I would not solve your mortgage problem by spelling it out into a large formula and solving it, I would try to subtract the principle and compound the interest repeatedly until I discovered the answer. Tedious, but I’m no mathematician.

  7. Mark Bennet Says:

    Tim

    I agree that science is a major field in which mathematics is allied, as is engineering. The dismissal of 19th century mathematics is crass – the mathematics of change – differential equations in three space dimensions plus time takes some getting hold of – and unless we put in the groundwork, our scientists and engineers will be floundering.

    But there is a great deal of practical maths in finance, too. For example, what accountants call ‘discounted cash flow’ – which is just a version of interest and geometric progression. If you combine this with some statistics you get analysis of financial risk – surely relevant and motivated at present. Then you could use population data to look at whether it is worth buying an insurance product or a pension: or you could look at business scenarios to see whether it was worth buying a machine or alternatively employing five more people. None of the maths is hard, but it takes some skill and understanding to apply the correct technique to each situation.

    Ah, but you get told the technique to apply, and you get given the mathematical content – as one of my colleagues used to say, “that’s not maths, it’s arithmetic”.

    The difficulty with teaching anyone how to interpret statistical data properly (I have this problem with school statistics, which don’t show confidence intervals) is that people would give the government a hard time if they understood some of the nonsense which purports to be proper interpretation of data.

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  10. Mark Bennet Says:

    Actually the question on the Catenary Arch in the paper you linked is a good illustration of where depth is lacking. Is this just a random aesthetically pleasing curve, or is there more to it than that.

    If I am in the business of aesthetics – where do I get a rich collection of aesthetically pleasing curves: this involves the study of functions and understanding what happens to functions when parameters are changed.

    If I am in the business of engineering – how do I understand the strengths and weaknesses of such a structure? How much will it cost to build? How does this compare with other possibilities?

    Part of the issue here is that the excitement and imagination has been taken out. The maths seems mundane and poorly motivated.

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  12. Ana Sá Says:

    Help, I am stuck in my ivory tower too.
    I simply do not understand how is it possible to get young people to be able to be good in modelling situations of reality and perform these tasks without the basic knowledges.
    For instance, if they never heard about differential equations, nor even in progressions, atrithmetic or geometric, and sums, how could they construct a model that woul lead to a solution to the problem above ?
    I don’t want to go so far. The above shown problem would leave University students in serious difficulties.
    The Portuguese Maths curriculum, specially at Secondary level has decreased along the last 14 years. Students no longer study Logic, for instance. They don’t learn how to structure an idea, they don’t have the faintest idea of what a demosntration is, they never heard of a demonstration process. So, everything is done – if done – without conscience nor any grade of centainity.
    If we think of the youngest (12 to 15 yrs old), the tasks they are supposed to perform are the ones that in 50’s and 60’s of XX century were performed by 6 to 10 yrs old in Basic School. I don’t understand how can the new techonologies may help to get children and young people to adquire the essential knowledg that would lead them, afterwards – only afterwards – to use the tecnology means to do things in a faster way.
    In my modest opinion, machines are marvellous means of not having to do some fastidious calculus and internet a very good mean to get information and comunicate. But this cannot replace the use of our capacities of thinking. If young people are not attracted to school because this one is a place of work and learning, is it a good solution to transform school in a an attractive place leaving behind those skills and building a fun playground full of computers ? Is it a good idea getting down to students and staying the whole Basic and Secondary down with them ? Or would it be a good idea to get down to students and , month by month, year by year , raise their level of learning and performing ? This is not valid just to Maths. All areas of knowledge are nowadays , in Portugal, being teached by minimum objectives with the excuse that otherwise young people will not have sucess and will leave school. But what kind of sucesse is this ? In my opinion, school is not helping to prepare young people to real life in any way, is not a vehicle of culture. It is a way of selecting exactly like it was in the past having or not gone to school : those who perform and get the graduation with only average results in this poor school, are not able to go far. They have a certification, a paper, that is all.
    I studied Sciences, so I only had 3 yrs of English in secondary school. That’s why I don’t write it very well, as you may see. But, at least I am able to understand the language and to write and speak it making myself understood (I hope…). Of course, I developed those 3 yrs reading in English all life throug. I don’t believe young people will do that nowadays: just hearing lyrics of songs or watching films doesn’t give them skills in the language. And I know what I’m talking about : I have 4 daughters and only the oldest one (32) speaks and talks English. None of the others do. They forgot the poor English they learnt in public school.
    Sorry I have written too much, but I am a Maths secondary school teacher and am in a serious difficulty : even if I don’t agree with curricula and methodologies, I have to teach what and how it is supposed nowadays. It is really hard for me to do that and , if I could, I myself would abandon school.
    Would someone want to comment these words of mine ?
    I may be wrong…

  13. Akhil Mathew Says:

    Indeed, calculators before high school used improperly have a track record of creating dependence and fostering unfamiliarity with basic arithmetic. For instance, I even once recall taking a precalculus course where students used their TI-89’s to compute derivatives and integrals for them.

  14. Anonymous Says:

    I have a slightly different take on the situation. Outside of my friends and colleagues in academia, I can’t say I know very many people that could do the mortgage calculation that you described above.

    Is this really a problem? I assume that many of these friends (like myself) would find an online “Mortgage calculator” to do the computation. In fact, this is probably a better solution then the by-hand computation, since it will be able to solve the discrete problem exactly, factor in nuances such as leap years, and output cool visuals with the numeric results.

    Of course, this doesn’t address the group of people who wouldn’t even care to investigate these issues before taking out a mortgage. However, I think that this is a failure of even the current education standards, and goes much more to issues of teaching effectiveness than curriculum.

    I would include some of the brightest people I know among the the subset of my friends who I wouldn’t expect to be able to do this calculation. In fact, the only friends of mine whose mathematical abilities present problems for in their work or daily life, are the mathematician.

    This said, let me now make a case for the stronger/theoretical math curriculum. First, I think most people would agree that the current curriculum / philosophy is almost useless. We spend 4 years of high school teaching students the (1) field operations of the real numbers (2) the quadratic formula (3) the geometry of the unit circle (aka trigonometry). Despite spending 4 years teaching what I can rigorously derive on two/three sheets of paper, we do it in a way that the people who care about the theory don’t get to see it, and the people who don’t care about the theory forget it a year out of high school.

    It is probably true that even in this (basic) form, the current setup adequately serves 95% of students. However, the issue is that if a student wants to go on to become an engineer or scientist then he needs a much better background in math. One approach to this is to simply offer multiple math programs of different rigour. The students who want to become engineers takes the “hard” one and the students who don’t take the easy one. For the most part this is the current system. However, I am slightly against it. First of all, I think there should be a bit of “eat your vegetables”-ness about your education. If a high school freshman comes to you and says he wants to be a ditch-digger and thus doesn’t need to be in high school, you don’t let him quite. So why should we let a 14 year-old decided to pass on math because he doesn’t think he will need it in the future?

    Another argument against the two option approach is that the student who takes the “easier” path is strictly worse off. High school isn’t like college where you don’t take class X so that you can take class Y. When there are two math options, the students who elect the easier path don’t get less math in exchange for more Y. They just get less math. Additionally if there is a choice between a theoretical math course and say a “real world math course.” I would make a significant wager that the top student in the theoretical math class would be able to ace the “real world math course exam.” Things like spreadsheet programing are important, but there is no reason they can’t be integrated into a theoretical math curriculum. (When I needed to use a spreadsheet for the first time, I figured it out quickly via the help menu and google. I suspect my ability to do this was rooted in the problem solving abilities I learned while training in mathematics.)

    A final (and only half-serious) argument against not having a rigorous (or only elective) math curriculum is based on the assumption that there is some innate component to math/scientific talent. If you assume that great scientists are randomly distributed in our population, then the quality of science drastically drops if a large portion of the population never gets introduced to the subject that they might be exceptional at.

    • Américo Tavares Says:

      I am not against the use of spread sheets. I myself have used a few, but only after being able to do the calculations by hand at least two or three times. I even programmed my pocket calculator for these and similar type of things.
      To me, the main point for an engineer is to guess the order of magnitude of the solution, so that he can detect faulty results due to data inputted incorrectly, which is something that may occur.
      How can one be better prepared to do that? I prefer the better understanding provided by hand made calculations, during the learning phase. But others may rather go directly to spreadsheets.

  15. gowers Says:

    Two quick points.

    1. I wasn’t suggesting that the mortgage question would be a suitable one for a use-of-maths A’level — it was more like the first question that popped into my head that might come up in real life and involve some mathematics that was less trivial than addition and multiplication, say. The real point, which was explicitly made by Phil H above, is that it is very hard to think of any real-world problem that involves both modelling and mathematics of the right level of difficulty. It seems to me that the sample papers are dishonest: they are pretending to have found appropriate questions, but are in fact not testing modelling skills at all and are making the mathematics very easy.

    2. The result is that the use-of-maths A’level is teaching a strict subset of the conventional maths A’level. If I were teaching use-of-maths A’level and wanted my pupils to do as well as possible, I’d teach them the conventional A’level maths syllabus and perhaps give them a quick practice paper just before they took the exam (so that they could get used to the peculiarities of the data sheets — no other reason). And as Anonymous says, someone who takes a strictly easier course ends up strictly worse off. (There is another obvious possibility though, which is to have an easier course that you don’t call an A’level. But that already exists, and the point of the campaign against the new A’level is to keep things as they are.)

    • Américo Tavares Says:

      I apologize for having used the word “context” in “the model you are discussing in the context of the “use-of-maths A’ level ” “, because it may suggest that “the mortgage question would be a suitable one for a use-of-maths A’level”. It was a fault of mine and I did not mean that.
      Further, I am now aware that my first two comments, as formulated, are irrelevant, misleading and off the central issues of your topic.
      Please excuse me, Professor Gowers.

  16. Qiaochu Yuan Says:

    To me, the main point for an engineer is to guess the order of magnitude of the solution, so that he can detect faulty results due to data inputted incorrectly, which is something that may occur.

    It’s really quite amazing how many people have no sense of order of magnitude, even those who are reasonably capable of calculation. Perhaps we should be teaching Fermi calculations, dimensional analysis, and the use of extremal cases instead.

  17. Jason Dyer Says:

    The papers have been taken down from the Channel 4 site.

  18. Chris Stephens Says:

    This speaks to an issue that has been bothering me for some time now. I have not quite been able to pin down all the things about it that bother me, but I will do my best to verbalize what notions I have.

    1. We humans are not infallible when deciding what we dislike, particularly when we are young. For instance, when I was in university I would have declared without equivocation that I disliked potatoes and writing, and I would have been wrong on both counts. My mistake with respect to potatoes was due to ignorance — I simply wasn’t very familiar with their taste. Once I seriously tried them, I liked them. My mistake with respect to writing was more subtle. In retrospect, I have always liked writing in the sense of converting ideas into words (hopefully presentable, interesting words). What I disliked at the time were the particular writing *assignments* I was being asked to do. In my youth and inexperience I did not distinguish between the two, and the distinction is important.

    2. Nevertheless, let us stipulate that humans are at least reasonably good at declaring what they dislike. In particular, let us assume that a large portion of students genuinely dislike mathematics, as it has been taught them. Far more problematic is that we are very bad at declaring *why* we dislike something. In the U.S. (and, based on the blog-post, seemingly in Britain) we have latched onto this concept of “relevance” as the reason we dislike maths. I do not know where it originated, but educators and students alike use this idea as a primary reason for students’ lack of enthusiasm towards the subject. “I do not enjoy this, because I do not see how it is useful in my life.”

    I submit that it is FAR from proven that this is the reason for our dislike of maths. In fact, I have become convinced that it is decidedly NOT the reason.

    3. It seems to me that “relevance” and “academic study” do not mix well. If a topic is relevant to a person, will they then be inclined to study that topic in an abstract, academic way? I think they will not — UNLESS they are already given to abstraction and generalization by nature. Only someone who thinks like a mathematician will enjoy the abstract modeling of even an IMMEDIATELY relevant topic! I like the kind of calculation that Professor Gowers performed in the above post, not because I think it is useful, but because I think it is FUN. More useful (as has been pointed out) would be a spreadsheet.

    4. If we artificially force “relevance,” the students are not fooled. In my experience, all we do is propagate the myth that maths SHOULD have been relevant, while continuing to teach courses that are NOT relevant. And, in the meantime, we are no longer teaching the fundamentals.

    5. The DETAILS are not relevant in ANY course (mathematics or otherwise), unless one aims to be a professional in that field. Take, for instance, a history class in which one studies the causes of World War I. Are the specific details of the stories surrounding that topic relevant? Absolutely not! What IS important is the study of human behavior, on both the small scale and the large — THAT is why we study history. The DETAILS, though, are not relevant. One can make such analogies with any subject. Likewise, the *details* of mathematics are not relevant at all, for most students. We should not apologize for (or worse, deny) this fact. Forcing “relevance” in mathematics is doomed to fail, and misses the point besides.

    • Jason Dyer Says:

      Whoa there. Just because it is possible to handle real-life examples badly doesn’t mean they need to be jettisoned altogether.

      It it true that relevance does not equal interest. People (often inexperienced with actual classroom experience) will often reach for real-life examples as an easy panacea to classroom boredom, but just like anything else it is possible to have interesting examples and boring examples.

      Perhaps most important in interesting the students is that they *understand* the material. I found more than anything boredom comes from a feeling that students are performing an arbitrary series of manipulations; in particular they often attempt things in mathematics they have no natural intuition for. The students need “buy-in”; note the quote above, “internal framework for processing the world”. This does *not* necessarily mean turning 5 and 7 into 5 apples and 7 oranges, but it does mean nothing should appear to be arbitrary. Why is it that 4/5 * 2/3 is (4*2) / (5*3) but that 4/5 + 2/3 isn’t (4+2)/(5+3)? If your students can’t explain on some level, then it is an arbitrary rule.

      There was an article in Science last year which travelled the blogwaves which ostensibly demonstrated the advantage of abstract examples in learning mathematics. Unfortunately, the example chosen was horribly confusing and actually worked *against* natural intuition. If the experimenters had instead used a clock example rather than the water pouring one (see the link above) it would go with the inuition of the learners rather than against it.

      Avoiding the sense of being arbitrary is sometimes easier to illustrate with the abstract, sometimes with the concrete. In both cases in education they are rhetorical devices and must be treated with the same pragmatism as rhetorical points in a speech; whether they work may depend on context, audience, sequencing, culture, and unfortunately for educators random cosmic rays. Most teachers are familiar with a lesson that comes off as crystal clear in one period, but repeating the same lesson again for a different group somehow turns into mud.

      Note also that examples as quoted from the paper above are hardly the only use of models and relevance. One may approach a particular model scientifically, gathering data, and then trying out various models of best fit. One may simply use a model as a demonstration of a concept, like having students build a reflective parabola out of aluminum foil and file folders to illustrate the focus. One can approach a problem as an engineer: we need to build this thing, what mathematical tools do we need? — in some cases this kind of question can be used to have the students themselves discover new mathematics.

      One last point is that “fixing” a problem from boring to interesting can hinge on things that might seem trivial to a mathematician but are essential to the psychology of a child. I can guarantee none of my students would be interested in the house example, but a great many are interested in a (roughly equivalent) example involving buying a car. Buying a house is an activity way off in adult-land, whereas buying a car is something they can relate to. Of course, note the caveat above on context; perhaps in New York (where less people own cars) students can relate more to the woes of apartment buying.

    • Chris Stephens Says:

      Probably I am overstating my case in my above post. To clarify, I am not opposed to real-life examples. What I am opposed to is the philosophy that teaching of mathematics should involve “relevance.”

      It is also possible that I am taking too narrow a definition of “relevant.” As I read the term “relevant,” Gowers’s mortgage derivation above is certainly a real-life example, but is not at all relevant to anyone’s life.

      Mathematics as a modeling tool is a wonderful topic, and should be included in the curriculum at all levels; but I believe it is a mistake to claim the models are directly “relevant” to the students. [Are we modeling the physics of the local bridge they cross every day? Well (they say), who cares? The bridge is already built, and even those who designed and built it probably didn't do the calculations we're doing!]

      I agree with you that understanding is key in interesting the students. I also think surprise and mystery are powerful tools (of course, they can only be used once in a while).

      In my experience the students are comfortable with *some* arbitrary rules. If too many manipulations are arbitrary, they certainly disengage (quickly!). However, it seems that most of them also do not wish for *everything* to be explained. Some of the more ruthlessly pragmatic students, in fact, want to distill everything down to an algorithm, and want to understand the underlying reasons only insofar as it helps them to execute the algorithm correctly.

      The right balance between when to ask them to understand and when to ask them to accept is a constant challenge, I think.

      By the way, I believe your water-pouring link is broken.

  19. Bourbakism & the queen bee syndrome | neverendingbooks Says:

    [...] a debilitating level under the pretense of ‘usability’. Tim Gowers has an interesting Ivory tower post on [...]

  20. Jason Dyer Says:

    One more try at the water pouring link.

    Algorithms are a funny case, in that students believe they understand a topic if they know an algorithm (and sometimes it may be good enough). The problem is when algorithms start to pile on algorithms and they hit a topic which requires an actual understanding of prior material — say, logrithms — and then their world crumbles around them. (Since we’re talking solely of motivation, having them believe they understand is enough.)

    The earliest place this occurs is fractions. Legions of students get the two fraction cases I mention above mixed up, causing them to hit the “stupid arbitrary rule” threshold. This is not helped by some teachers at that level who believe the same thing.

  21. gowers Says:

    Adding fractions is an interesting example, because there’s a delicate boundary (which will vary from child to child) between questions that provoke nonsense and questions that provoke sensible thought. For instance, if you ask people what half a cake plus quarter of a cake makes, then they will probably visualize a cake with a right-angled piece removed and say three quarters of a cake. Most such people, I would guess, could even do without the cake and see without much difficulty that \frac 12+\frac 14=\frac 34. And I think very few people would be tempted to say that \frac 12+\frac 14=\frac 26. But change the question to \frac 12+\frac 13, and my guess is that some people would at that stage switch to formal mode and say \frac 25, some would switch to formal mode but use the right method, and a small proportion might actually visualize a cake cut into six and see clearly why the answer is \frac 56. When it comes to letters instead of numbers, the proportion who add numerators and denominators will be higher. I think this backs up the principle of doing examples first (if you’ve added a lot of fractions by actually thinking, then you will know from your first-hand experience that it isn’t as simple as adding the numerators and adding the denominators), or at the very least of having the principle drummed into you that if you make a general statement then you should look at a few instances of it to check that it isn’t ridiculous (e.g. pretty well any choice of a,b,c,d that you substitute into \frac ab+\frac cd=\frac{a+c}{b+d} will show that it’s false, and similarly for (x+y)^2=x^2+y^2 and other favourite mistakes).

  22. Elizabeth Truss Says:

    Reform has put the Use of Mathemetics pilot papers on its site:

    http://reform.co.uk/EfR/EducatorsforReform/tabid/131/Default.aspx

  23. Hans Says:

    I wonder how many university graduates with good math grades, who are now not involved in math for a living, could solve a real-life math problem that is more complicated than simple algebra. Even if they did well in school math, I will guess that beginning about 6 months after their last semester, most people will never be able to do any math more complex than solving a simple equation for a single unknown again in their lives. My experience is that most people lose their math skills before ever having an opportunity to use them in real life.

    The people that seem to retain math skills had some interest in math and took the time to think and learn and to try it out in real life scenarios. They had (and probably still have) an interest in mathematics beyond the need for good marks, beyond the goal of satisfying some academic requirements.

    This is one point that ivory tower types do need to remember when debating a math curriculum. An interested person will learn and retain math, even if they start with a high school curriculum that isn’t perfect. Enlarging the base of people who become interested in math is a very noble goal. Nothing would increase the level of math skill in the population like a broad interest in math.

    Maybe this updated test is a good way to interest more kids and maybe it isn’t, but you have to admit that it would be great if it did actually manage to interest more kids in math. So in addition to knocking it down, maybe it would be good to come up with ideas for interesting kids while keeping the kinds of skill training that you find important.

    Also, anyone with a graduate degree in math (such as myself) must take care to remember that they do not really understand how most people see math. Most people approach math in the way that a mathematician may approach running or learning to dance or mastering the art of the salesman – these are things that give great benefit to those who master them, and yet hold zero interest to the average mathematician. So we must remember that we are talking about the masses here – the kids who start loving math in high school will not be hindered by this test, they will excel anyway. The proponents of this test are talking about taking some of those kids who don’t love math in high school and maybe helping them see why they should get to know it a little better in college. That’s a noble goal, even if they may not be approaching it the right way.

  24. disquisitionesmathematicae Says:

    What we need is an inquisitive mind- then and only then will mathematics flow on its own.

  25. Niall MacKay Says:

    I use this problem in my first-year teaching (of maths at a UK uni). It fits nicely into the context of difference equations: it’s simple to write down the general solution of a first-order difference equation, specialize it to the mortgage, and compare the continuous limit with the solution of the differential equation (and then remind students of the specialization to compound interest which they already know and how it connects with various definitions of the exponential function).

    An anecdote demonstrates the perils of using a spreadsheet instead. A friend at a US investment bank contacted me with a ‘tricky problem’, namely solving this (in the context of bonds), then computing the yield (interest rate) from the coupon (repayments). Their software was taking appreciable time to do this, because it was solving the latter problem iteratively — it was iterating an iteration.

    I sent him the solution by return of email and he responded that it gave _nearly_ the same result as his numerical solution. I waited a day or so and in his next email he reported finding his numerical error…. They ended up being very pleased with the speed with which their new software put answers on the dealers’ screens.

    Anyway, I agree with Tim about the UoM sample papers, which are quite dreadful. However, the _syllabus_ is not bad. If you had to construct a UoM A-level you might well do what they have done: take all the calculus from the A-level core (C1-C4), right up to first-order separable DEs, and then apply it, to modelling problems of which this would be a perfect example.

    My (personal) perfect framework for A-level maths qualifications would drop decision maths [modules] entirely, keep the statistics, revise mechanics to emphasize modelling and physical thinking (especially via dimensional analysis and scaling arguments), and apply calculus in two new ‘Modelling’ modules. If there is to be a UoM A-level then I would tweak the syllabus to include some dimensional analysis and scaling, and some discrete equations (following Robert May’s advice) — and replace the crap models with some decent ones. There’s so much you could do with this syllabus – finance (as here), population growth and logistic limits, reaction kinetics, epidemiology,…

    In the end I think the argument is for giving mathematicians from HE a greater role in producing the papers (and perhaps the syllabuses). These really need a leavening of mathematical intelligence, experience and good ideas.

  26. Use of Mathematics and Hyperscopes « Maxwell’s Demon Says:

    [...] this change.  The proposal at the moment is too much about fitting numbers into equations.  (See Tim Gowers’ analysis). One way to think about the new A-level is that it could play a role similar to “Classical [...]

  27. Бывают странные сближенья … « liidia-22008 Says:

    [...] towers: I and II. Метки: and, ivory, towers Комментариев нет Дневник [...]

  28. Use of mathematics II « Gowers's Weblog Says:

    [...] answer is that a few months ago I wrote a post in which I criticized a proposal for an A-level in Use of Mathematics. Amongst the responses I got to that post was an email by Joseph Malkevitch, from the mathematics [...]

  29. Proof? What do you mean, proof? « episodic thoughts Says:

    [...] misuse of the word proof in daily life by many people. Surely that should be something which any math curriculum should tend to correct at an early [...]

  30. Jennie Golding Says:

    I follow the arguments with interest: they’re fine for university mathematicians. But what about 16-year-olds with a weak grade B in GCSE? (I would suspect none of the above was in that position, ie about the 4th quartile of mathematical ability, but please correct me if I’m wrong) The pilot study has shown they *do* find the syllabus interesting and make good progress mathematically; they have chosen to continue with mathematics when most of their peers have dropped it altogether, and they give very positive feedback (while mostly getting weakish AS grades, broadly comparable with their other subjects). These are not ‘the clever core’ who opt to do AL maths: they are additional students. I agree the assessments need some attention (though there’s a good profile for the use of spreadsheets): perhaps teh contributors above would like to bend their attention to improving what’s offered to this additional group of young people.

  31. How should mathematics be taught to non-mathematicians? « Gowers's Weblog Says:

    [...] 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 [...]

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

    [...] esta otra, aunque un poco más antigua también está [...]

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