Does “our fixation with maths” add up?

(An almost “classic” 2011 parody video that was created in response to the actual question, “Should evolution be taught in schools?”, being asked in that year’s “Miss USA” competition. )

Does mathematics play an overly large role in school curriculae? At least for the country of Britain, this question was recently answered in the affirmative by journalist Simon Jenkins, who is apparently somewhat famous, in an article for “The Guardian”. After all, there is very little mathematics that the average person actually has to use in their lives, or even remembers. So why waste pupil’s time with linear equations and calculus, when that time could be used to instill in them “a knowledge of their history and geography, their environment, the working of their bodies, the upbringing of children, law, money, the economy and civil rights”, all of which seem more critically important?

Now, I studied maths, so I may be slightly biased. On the other hand, if Mr Jenkins’ proposals were widely implemented, it would significantly increase my own market value, so I hope you will indulge me when I explain why I think it’s his arguments that don’t add up.

One may immediately sense a slight contradiction in the inclusion of history on the above list of “important” skills – when was the last time you actually needed to apply your knowledge of the fall of the Roman Empire to the real world? And yet, I, too, have heard this point raised since I went to middle school in Germany: Classmates who rarely ever made such complaints about reading Goethe or Lessing (substitute two famous writers from your country if you like) asked, “What would I ever even need this for?”, right after getting a bad grade in a mathematical exam. But the immediate practical applicability of poetry or literature (and many or most other things taught in schools) seems actually much less clear than the one of math education.

Just pointing to the double standard between mathematics on one hand and history/literature/art/etc would be too cheap a counterargument, essentially an example of what I call a Hans Landa fallacy (after all, perhaps these subjects should be abolished or cut to size, too). Still, it is worth dwelling for a moment on how a defender of teaching the latter subjects to kids would justify it: They might argue that while their may not be an immediate practical application for knowledge of the origins of World War I, or for having read 1984, having engaged with these things teaches cognitive skills that are important in practice, such as political thinking and understanding complex texts.

Of course, the same argument can be put forward in favour of mathematics, so what is Mr Jenkins’ counter? Well,

Charge the maths lobby with the uselessness of its subject and the answer is a mix of chauvinism and vacuity. […] it falls back on primitivism, that maths “trains the mind”. So does learning the Qur’an and reciting Latin verbs.

Yes, but not all forms of “training the mind” are equivalent. Specifically, learning the Qur’an or Latin vocabulary exercises the faculty to retain specific facts in your memory. A good memory is an asset if you want to solve a mathematical problem, but it is not usually enough.

So what does mathematical education have to offer non-mathematicians? I believe a big part of the answer can be found by looking at some reasons why so many people, from tenth-graders to British journalists, raise the issue of uselessness for maths, but not for many other subjects that could very well be equally deserving. I believe there are three main factors at play here:

If Barbie ever taught us anything, it’s that math class is tough. and for most, though by no means all, students, it seems to be tougher than most other subjects. That usually leads to a higher rate of failure (in the form of bad grades), which in turn leads to an increased desire to downplay the importance of the subject where one has failed. Especially people who ace their other subjects, but not maths, may easily come to see it as a thorn in their side that they desparately would like to get rid off.

Second, one reason mathematics is hard to do for so many is that it is abstract, seemingly detached from our everyday experience even where it finds application to real life. That means it is (again, usually) difficult for our poor brain to find appreciation for or pleasure in maths, and that inhibits its ability to concentrate on the subject matter and process mathematical information.

Thirdly, mathematics is clear-cut. Even if you did get a bad grade for your essay interpreting a Shakespeare poem, you can claim that you just didn’t manage to guess the specific interpretation your teacher had in mind, and that you are an underappreciated literary genius. And it’s not impossible that you are right. On the other hand, your solution of a quadratic equation, or your calculation of a function’s critical points, is either correct or not, and you have no way to evade that fact. The Guardian piece even sort of acknowledges this point in the final paragraph, and with a very negative spin:

The reason [for the “obsession with maths” – PQI] is depressingly clear. Maths is merely an easy subject to measure, nationally and internationally. It thus facilitates the bureaucratic craving for targetry and control.

(Of course, deficiencies in how mathematics is taught play a huge role, too. However, Simon Jenkins does not argue for changing the pedagogics, he argues for abolishing most of mathematical teaching, with only basic arithmetics and some basic statistics exempted. Therefore, I find only the relevant features of the subject itself to be of importance here.)

But the exact reasons that people have for disliking maths are also reasons why they might need to be instructed in it nevertheless: Because it presents us with tough problems that get us to our intellectual limits faster than other subjects, because it introduces us to a high level of abstract thinking whose power is demonstrated by all of modern science and technology, because it is an exercise in concentrating on material that we don’t at first find it natural to concentrate on, because it confronts us with clear-cut feedback on whether we are right or wrong about something, because of all these things, mathematics indeed does train cognitive skills that other subjects don’t usually engage to the same extent.

And other subjects, of course, engage cognitive skills that mathematics could never develop to the same extent. Problems in real life can be messy, and frequently do not have the kind of uniquely determined solution of mathematical ones. They require evaluation of arguments, critical thinking, understanding of texts, research abilities, memory, and so on. But precisely because definitive solutions often can’t be found in such situations, it seems important to me that people have exercised their mind in a context where they can. Would you go free-climbing in the mountains without experience under the controlled conditions of a climbing wall?

The late Austrian mathematician Karl Menger (not to be confused with his father, the economist with a C) probably put best what I am trying to get at here:

Not that, if insight into the method of mathematics was more widespread, people would necessarily say more prudent things, but they would surely say fewer imprudent things.

While I view the above as sufficient reason to keep including significant amounts of mathematics in peoples’ education it is by far not the only problem I see with Mr Jenkins’ argument:

For one thing, his academic background (the Philosophy, Politics and Economics course in Oxford, according to Wikipedia) may make him underestimate the range of professions that actually do need mathematical knowledge in practice. It is my understanding that the precision mechanic may need to apply Pythagoras’ theorem, and the chemical laboratory worker may have to solve the occasional equation (both examples lifted from this German brochure). And because of the broad range of applied mathematics, it can help improve things even where it is not absolutely necessary, including for the “executives, accountants, salesmen, designers and creative thinkers” Jenkins states are more important than mathematicians.

Moreover, when we do acknowledge there is good reason for society to keep some maths people around (which Jenkins does), we must also admit that we cannot already predict with certainty which students can or will be them when they are only fourth-graders, IQ tests nonwithstanding. But starting from scratch and teaching higher mathematics to an undergraduate who has never learned the mathematical basics that pupils are usually exposed to before entering university will prove veritably impossible. (Unless that person has made an effort to learn them later in life, but that is going to be much harder and consume time.) As a former tutor for first-semesters, I can assure you that even how well someone has been taught calculus in the two years before enrollment makes a noticable difference.

So, even if you buy into the notion of only teaching pupils what they directly need in their future jobs, as long as a human being’s aspirations and dreams are subject to change, you may find it necessary to teach to everyone what only a few will actually need.

And lastly, you may advocate the teaching of a subject like e.g. history not only for its direct usefulness or for the thinking skills it promotes, but also because you share the idealistic goal that people should at least have a trace of understanding for the world and the culture they live in. But mathematics is the foundation for understanding science and engineering at a technical level, and has been since the scientific revolution in the 17th century. The mathematical ideas of Archimedes and Galileo are no less foundational to the modern world than the liberal democratic ideas of Solon and John Locke (and even if you don’t know their names, you probably have heard of liberal democracy). Not everyone needs to have a deep understanding of the former, just like most people don’t actually have a deep understanding of political philosophy, but they should have received some instruction in it.

At the risk of making myself unpopular, I will go even further: If you, to take an example from the article, do not know how to calculate the area of a circle (yes, it’s Pi Day, and yes, the last link is actually about circumference, complementing the second one, which explains why the formula is true), there is nothing wrong with you, in the sense that you are not stupid, or a bad person, for such a reason. But you do have a gap in your knowledge of the world that is at least similarly large as not knowing who Picasso was, or never having heard a Beatles song.

All of us, including myself, probably have more such gaps than we care to admit to ourselves, maybe even in our own fields of expertise. Nevertheless, we can try to learn continuously throughout life. And for that, it is very advantageous to have gotten at least a rudimentary foundation in many fields of knowledge early in life. Including the field of mathematics.