Suppose you're on a game show and you reach the final stage, Select-a-Door. You have to choose one door out of three possible doors. Behind one is the jackpot prize (dinner with Michael Portillo, for example); behind the other two the booby prize (dinner with me and Michael Portillo). What's your chance of picking the winner? 1/3, right? Now, suppose you pick Door 3. The host - Noel Edmonds, of course - says to you OK, I'll help you. Behind Door 1 is one of the booby prizes. Do you want to stick with Door 3, or switch to Door 2? What should you do to maximise the chance of success? Switch, stick, or it makes no difference? (Answer at the end).
Most people not only answer this question wrongly, they feel so certain in their error that they react with anger to the correct solution. This is just one reason why I strongly welcome yesterday's announcement, that Carol Vorderman is to lead a taskforce for Michael Gove, to investigate ways to improve the teaching of maths in Britain's schools. Of course, given that I have a fondness for frayed corduroy trousers, prefer to cut my hair myself, and bought a very sensible anorak from Tesco to celebrate going to university, you might think well he would say that, wouldn't he? He's a maths geek.
But there are less esoteric reasons to welcome the announcement. Quite simply, an understanding of mathematical reasoning is, together with literacy and historical awareness, one of the most important gifts that a school can impart to its pupils. Why? I can see at least two main reasons: national scientific endeavour, and electoral literacy.
First, scientific endeavour. It's impossible to think of a single field of science or technology where an ability to have insight to higher mathematical reasoning is not vital. Telecommunications, the security services, neurosciences, all clinical research, all of psychology, economics, computing, industrial efficiency, engineering, atmospheric physics, all of physics really - supply your own list. My point is that if you accept that it is a Good Thing for Britain to be at the forefront of these activities, then we must produce university graduates with high capability in the mathematical sciences, or import those skills from competitor nations. If we don't get the maths syllabus optimised at school, then our university students begin their advanced education with one hand tied behind their back. There's plenty of evidence that this has become the case, with British Universities having to offer remedial courses to fresher science students. This is a modern phenomenon, and the only obvious inference is that something has gone wrong with the way maths is taught at schools.
OK, but you might think, who cares about the small proportion of science students, anyway, I don't hold with all that British Jobs For British Mathematicians malarkey, we can always attract top talent from China and India, what's the point of maths anyway, it's all rubbish and it's never done nothing for me. In a nutshell, this is the point of view of the esteemed Simon Jenkins, who wrote a depressing article along these lines last year. Well. Consider these propositions:
- There's a lot of evidence that road cameras decrease the chance of road traffic accidents, isn't there?
- New Labour has created 5 million new jobs, so it can't be accused of not improving the employment figures for Britons, can it?
- If a glass of wine a night doubles your risk of Disease X, that's frightening, and justifies acres of newsprint about my middle-class drink hell by Glenda Slagg.
- Since most terrorist attacks in Britain have been carried out by Muslims, it's correct to target Muslims at airport security checks.
These are just four examples off the top of my head. Now read this carefully: it's not my intention to convince you of the truth or falsehood of any of these points, so please do not read them as statements of my belief. It is my intention to convince you that without some ability to reason mathematically, Mr and Mrs Voter are completely in hock to the demagogues of the media and politics, and can be led to believe as true a proposition which, while it may sound intuitively reasonable, may quite easily be either false, or simply not sufficiently supported by the evidence. In the glass of wine example, it may well be true that your risk is doubled - from 1 in a billion to 2 in a billion. Is this excessive, versus the benefits your wine may bring? The road traffic argument is a perfect example of regression to the mean, which misapplication recently got Jacquie Smith in hot water over misinterpretation of knife crime statistics. The security check example requires proper application of conditional probabilities before a rational policy can be constructed - misapplication here is an example of the Prosecutor's fallacy, in which the probability of evidence, conditional on assumed guilt, is mixed up with the probability of guilt, given available evidence. (The probability of being a terrorist, if you are Muslim, is not the same thing as the probability of being Muslim, if you are a terrorist).
I think we can take a requirement for basic numeracy as given. (Incidentally, classical scholars, can you tell me why we call it arithmetic and not arithmetric? I've always wondered). But all that fussing you had to do with algebra was about more than 'solving for x'. It was a training to inculcate the notion that any deductive conclusion has antecedents which have to be combined sensibly in order to validate the conclusion. Sometimes you cannot deduce the conclusion, but in those cases the application of probabilistic thinking and inductive reasoning becomes vital. It's all maths.
Whatever the evidence about our inter-nation league-table slippage, we shouldn't despair. I have every faith in Carol Vorderman (a hero to every statistician) and in the determination of Michael Gove to bring real improvements to our school syllabus, and last August I got some first-hand experience that school pupils want to be stretched. A colleague and I spent a day with largely London-Bangladeshi sixth-form pupils in Tower Hamlets, doing a workshop (on regression to the mean), to try to convince them to stick with their maths and resist the lure of a career in the law. (I've nothing against lawyers! Some of my best friends ...). The feedback from the pupils and their teachers was very positive. So I believe that teachers themselves have a desire for constructive change, and I hope they can contribute to Ms Vorderman's taskforce. If the teaching of maths is never going to be a topic which sets the opinion columns of newspapers alight, I sincerely believe it to be a topic which a country cannot afford to get seriously wrong.
*
Footnote. Solution to Select-a-door. You should switch from Door 3 to Door 2. Most people don't see why switching makes sense, because most people don't understand how to think about conditional probabilities. Manipulation of conditional probability is the sine qua non of sensible political reasoning: given what we know, what should we do? I would feel more comfortable about our economic future if the Prime Minister felt more like Thomas Bayes and a bit less like Titian.
