Mathematical impossibility


REVISITING THE LEAVING CERT:Armed with a degree in maths and a sense of superiority, Film Critic DONALD CLARKEattempted this year’s higher level Maths paper one. However, the last time he cracked open a maths book, ‘Top Gun’ was all the rage in the box office

WHENEVER THE opportunity presents itself, I will snatch a corner of The Irish Timesto bleat pompously about the average media pundit’s ignorance – proud ignorance at that – of mathematics, physics, chemistry and any other rigorous discipline that, unlike the humanities, requires genuine application to master. In university, when other budding journalists were reading the first and last lines of Middlemarchor learning the bare minimum about the Treaty of Utrecht, I was hunched over a contour integral or peering studiously at the business end of a Riemann zeta-function. Bow down before me, feeble arts graduates. I have a maths degree. (Actually, having taken a joint honours degree in English and maths at Trinity, I too pretended to read Middlemarch, but we won’t let that stand in the way of the cheap taunting.)

If evidence were required as to the rarity of a science education in media circles, it came when I entered The Irish Timesoffice and confirmed that, yes, I was about to sit the first paper of higher level Leaving Cert mathematics. Men who regularly write articles examining the unimaginably complex minutiae of Kyrgyzstani foreign policy behaved as if I had just unearthed the secrets of cold fusion (not that they’d know what cold fusion is). Women who write biographies of Schopenhauer in their idle hours seemed to feel I was about to offer a shorter, more elegant proof of Fermat’s Last Theorem (as if they could quote such a thing).

Hang on. What was I doing? To this point, I could point to my maths degree and allow colleagues to assume that I still had a close grasp of that demanding discipline. Of course, this was not remotely the case. It was way back in 1981 when I sat the Leaving Certificate in Villiers School, Limerick – John McEnroe was finally about to overcome Bjorn Borg and the Human League were turning from experimentalism to New Pop. It is a terrifying 23 years since I last attended a university mathematics lecture.

Now, English graduates will continue to read books (and speak English for that matter) long after they leave university. Those with history degrees will watch the occasional David Starkey documentary and plough through the odd Antony Beevor tome. But it is possible to live through an entire decade without, in either work or play, having to integrate a simple function or solve the most undemanding differential equation.

It was some consolation that I was about to fail miserably at a subject too much of the public think is really, really hard, but fail I undoubtedly would. Aware that it took two years of study to get a B last time around, I didn’t even bother to find out which fields of mathematics were on the first paper. Sauntering towards the improvised exam hall with my mathematical tables beneath my arm, I told the relevant editor that, in all likelihood, I would be no longer than 20 minutes. I just needed time to confirm the paper was impossible and dream up some self-deprecating gags.

Well, strange to report, 2.5 hours later I was still squinting at the paper and cursing myself for not having done a modicum of work. I was still fairly confident that I had failed, but, to my great surprise, I realised that, even for somebody who hadn’t cracked a book since the release of Top Gun, the paper was far from impossible.

Take section a) of question one, which said find the value of x/y when:

(2x+3y)/(x+6y) = 4/5

Is this a trick question? Surely anybody with even a smattering of algebra could have a crack at this. As you will undoubtedly be aware, you multiply the denominator on each side of the equation by the numerator on the other and then place the two resulting terms either side of an equals symbol. Simplify this and you discover that 2x=3y and, thus, that x/y = 3/2.

Or that’s what I should have done. When the paper was marked, it transpired that I made a mess of the algebraic endgame. How shaming. I have seen horses in travelling carnivals solve trickier sums by banging their hoof on the stable floor.

The second part was only slightly more difficult. Offered a quadratic equation, f(x), you had to feed (x+1) into the expression, divide f(x) by the result, and do some easy-peasy, lemon-squeezy factorisation. At this rate, I was going to get an A1 and, buoyed to glory by similar results in other subjects, head off to study veterinary science at UCD. (This joke is probably decades out of date, but, in 1981, that course required the greatest number of points.)

It hardly needs to be said that it was downhill from there. Attempting the third part of question one, I found that I had forgotten the basic rules for manipulating algebraic inequalities. More disgracefully, I was uncertain about the phrase “roots of the function” and – here, I hang my head in shame – could not remember the denominator in the formula for the solution of a quadratic equation. I did know it was 2a, 2b, or 2c, but felt that each looked equally likely.

On the upside, I knew where to look for De Moivre’s theorem in the mathematical tables (bizarrely, utterly unchanged since 1981) and, although the relevant answer ended up as a clutter of trigonometric terms and imaginary numbers, I demonstrated that I remembered what the term was used for. I also recalled the gorgeously ingenious principle of proof by induction and, by leaving out all the hard bits, offered some sort of incoherent response to that question.

Still, the scribbled mess that fouled my exam script would have appalled the 17-year-old Donald and caused contemporaneous authorities at the CAO to cast my application into the file marked “deranged optimists”. Yet I was taken aback by how approachable the first paper turned out to be. Indeed, if I’d learned a few relatively basic formulae – 2a! Of course! – and boned up on the manipulation of matrices, I reckon I could have passed the thing without breaking a sweat.

Of course, I would say that, wouldn’t I?

How Donald did


Presentation was fine. Presentation is not an issue with maths as long as it is legible.

Mistakes in basic algebra can be fatal at this level. The student only attempted five questions (he should have attempted six) and in all cases not all of the question was attempted. This is fatal. You must give the examiner enough material to correct in order to pass the exam.

Every Leaving Cert student taking this paper would have attempted the differential and integral calculus questions (Q6/Q7/Q8). The student did not attempt any of these questions. A teacher would have advised students to attempt these questions. The same material tends to come up every year.

The student seems to have been ill prepared and did not answer enough of each question or enough questions. He/she should have practised with past exam papers.


75/300 =25%


Marked by John Brennan, maths teacher at the Ballinteer Institute, Dublin. The marker was approached independently by The Irish Times, and the result has not been endorsed by the State Examinations Commission