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 I take out a loan of , and thereafter the amount I owe, changes in two ways. On the one hand there is a constant-rate decrease of (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 . In other words, satisfies the differential equation .
We can solve this by turning it upside down and getting , which we can solve easily since all we have to do is integrate with respect to . From this we get . Rearranging, we find that , so that .
When this is supposed to give us . From that it follows that , so . Therefore, the amount of time it will take until is the value of such that , or , or .
From this expression we can see that I will never pay off the mortgage unless , though in fact it is more sensible to deduce that from the differential equation: if then 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 is negative, then 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 corresponds to an interest rate of 5% and what value of corresponds to a monthly repayment of £500. If we measure in years, then we need to choose such that and we should take to be 6000. We are given that , and a reasonable approximation for is 0.05 (since for small ), so the time taken will be around , where . So which is slightly under 2, so we get a bit less than , 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, , that the string makes with the vertical, seconds after release, can be modelled by a function such as .
(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 .
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 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 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 when ? To get to that problem, we had to interpret “when it is first released” as “at ” and we had to interpret “the angle that the string makes with the vertical” as . The second of these tasks is trivial, since the question has just said that that is what is, and the first is, well, not very challenging.
On to the second part. It’s telling us to differentiate 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.