A county council gathers data about where road accidents take place, identifies an accident blackspot, places a speed camera there, and notices that the frequency of accidents decreases. Moreover, this is not a one-off: many other councils have observed similar reductions. Does this show that speed cameras improve road safety?
The answer isn’t as straightforward as one might imagine – and the way to understand this is through mathematics. Unfortunately, the way the subject is taught often leaves people with a narrow and misleading view of what maths is. No wonder Simon Jenkins attacked the subject in Thursday’s Guardian.
Mathematics should be a tool for increasing one’s thinking power but for many children it is just a set of rather pointless rules for manipulating symbols. The problem becomes clear if one asks children a question such as the following: a number 35 bus pulls up at a bus stop and eight passengers get on; what is the age of the bus driver? A large percentage of children, their minds numbed by years of symbol manipulation, will give the answer 43. This is a tragedy: rather than being trained to think, these children have been trained to do the opposite.
To return to the speed cameras, the evidence initially seems conclusive. However, the correct answer is neither a clear yes nor a clear no, but rather that more research is needed. Consider what would happen if the locations of accidents were completely random. Then, just by chance, some places would have noticeably more accidents than others while the data was being collected and these places would be identified as blackspots. But since they would not in fact be more dangerous than anywhere else, the later accident rates at these “blackspots” would tend to decrease to more like the average, whether one installed speed cameras or planted apple trees. This phenomenon is known as regression to the mean. Further investigation is needed to determine whether speed cameras make a difference over and above the difference one would have expected anyway. (It turns out that they do.)
Regression to the mean is one of several statistical phenomena that are counterintuitive until you understand them. But once you do, you become better at making decisions. This is important for individuals – whether we like it or not, we all have to take major decisions based on statistical evidence – and it is even more important for people in positions of authority, whose decisions affect other people.
It is therefore good for the health of a country if its population has high standards of mathematical literacy: without it, people are swayed by incorrect arguments, make bad decisions and are happy to vote for politicians who make bad decisions on their behalf.
So how might mathematics education be different? The way it is often taught, children are asked to take a huge leap of faith: that the symbol manipulation that seems pointless now will one day be useful to them. But this is true for only a small minority of children, who enjoy the symbol manipulation for its own sake and later find themselves drawn towards Stem subjects, where it is indeed very useful. The rest know perfectly well that they will never reach this promised land. What can be done for them?
An indication is given by the speed-camera example. It shows that regression to the mean is an important mathematical phenomenon that can be explained without the need for any calculation or symbol manipulation. Why not use examples like this to bring statistics alive? That way we could explain the point of means and standard deviations rather than just asking people to calculate them.
Of course, some proficiency in calculation and symbol manipulation is important – and it improves one’s conceptual understanding – but it should not be all that is taught. We could also ask children open-ended questions, such as whether it is more dangerous to travel by car or by aeroplane. A question like is not explicitly mathematical, so it is less likely to trigger the brain’s off switch. And if it doesn’t, the ensuing discussion will convey why we should care about multiplication, division, averages and probabilities, what we can say about them when we do not have exact numbers handed to us on a plate, and how to frame mathematical questions to help make decisions that are of practical interest.
I am not suggesting that all maths should be introduced this way. But until our mathematics classes encourage people to think, rather than merely play games with marks on paper, the Simon Jenkinses of this world will continue to confuse mathematics with mindless symbol manipulation, attacking the subject itself when their real target should be today’s curriculum.