Recently, in The Conversation, the Vice Chancellor of Monash University, wrote an article discussing MOOCs. He made some criticisms about the nature of assessment and grading that MOOCs offer. However, my attention was grabbed by two sentences:
The other major problem the MOOCs haven’t solved is assessment. They work very well for subjects like maths, which have objectively right and wrong answers, and can therefore be pretty easily marked by computers.
Now, here we have the Vice Chancellor of one of Australia’s leading universities – and indeed, one of the world’s leading universities (and incidently the University where I did both my Masters and my PhD) demonstrating an extraordinary lack of understanding about the fundamental nature of mathematics. He seems to think that mathematics is all about teaching students (in the fine words of John Power from Leeds University) about “finding ‘x'”. I suppose he thinks this is what mathematicians do: they “find ‘x'”.
But this seems to me to be symptomatic of how the world at large sees mathematics and mathematicians: that mathematics is, in essence, a cut-and-dry discipline, with objectively “correct answers”. And mathematicians are seen as dry creatures whose discipline and practice totally lacks the creativity that one might see, for example, in the arts. A poet, a painter, even a literary critic, must needs demonstrate creativity in their work, but not so for the mathematician.
Part of the problem is the way mathematics is taught at schools: from the earliest days of kindergarten onwards, mathematics is taught as if everybody is going to be an engineer. (Please don’t take this as a slight against engineers, for whom I have an almost infinite respect; this is merely a comment on the sort of mathematics that engineers are taught and use.) There is a careful gradation from counting, measurement, through to arithmetic, algebra and geometry, culminating in calculus. You’ll notice one thing missing: proofs!
All mathematicians know that it is the concept of proof which makes mathematics into an exciting, creative intellectual discipline. Here there is no “finding ‘x'”, but the deep, sustained, concentrated endeavour to provide a link between two apparently disparate abstract entities. Right from Pythagoras’ theorem (or, as one of my school teachers liked to say: Euclid Book 1 proposition 47) which magically joins algebra and geometry, right up to say, the classification of finite simple groups, proofs have been the excitement and glory of mathematics.
I know I’m biased, but I can’t think of any human activity which requires the same degree of creativity as does mathematics. There are no short cuts, there are no hiding places: a proof, like a sculpture, is exposed to the cold light of day. And this was also the view of Bertrand Russell:
Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture.
Another British mathematician, G. H. Hardy, put it this way:
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
It is this lack of exposure to the excitement of proofs (and there are plenty of proofs which require very little background, such as Euclid’s proof of the infinity of the primes, and some proofs of Pythagoras’ theorem) which is partly to blame for the low esteem in which mathematicians are held.
And of course this strikes at the very heart of my complaint of the quote from the Monash Vice Chancellor. Yes, you may well ask a computer to check if you have correctly “found ‘x'”, but that is hardly mathematics, or if it is, it’s only a very tiny corner of it. It’s as if “painting” meant not the glories of Rembrandt, Gainsborough, Jackson Pollock but the necessary work of the house painter. I haven’t yet seen an online system which will help in any way budding mathematicians work their way through proofs – their creation, and their evaluation.
One of the reasons I’m not a very good mathematician is that I don’t have enough of the level of creativity that good mathematics requires: I find it hard to see how a proof can be built. However, I have enough mathematical background to appreciate fine mathematics. And this is what I also try to do with my students: the curriculum may well require them to “find ‘x'”, but I try to help them build intellectual bridges between ideas. And of course, the excitement when they realize that they can indeed do this is possibly the greatest delight a teacher can know.
Please, fellow mathematicians and mathematics educators – don’t take ignorant slights against mathematics lying down. Over two thousand years of sustained human effort by some of the greatest minds in history deserves more than “pretty easily marked by computers”.