In which The Author fears that history might repeat itself
(With thanks to the great Douglas R. Hofstadter, whose title I have shamelessly ripped off!)
I’ve talked before about ‘dumbing down’ in education. Until now I’ve largely confined my observations to the woefully inadequate standards of literacy in the ‘higher education’ students I meet. Today I’m going look at the third of the Three Rs –’rithmetic.
This whole story dates back to 1983. As I’ve noted previously, I met my first girlfriend, Lisa W. because she’d failed her Maths O level, and Mother had suggested to her mother that Lisa could benefit from a bit of unpaid extra tuition, courtesy of yours truly.
Twenty years or so ago, when I was resting between engagements, Pam came into the Carpenters on a Friday afternoon, practically in tears. She’d enrolled on a BTEC course in Electronics Engineering at Aberdare College, and was regretting it after a morning of maths. While working in Hitachi she’d decided to try and get some relevant qualifications and aim for promotion. Now, a few weeks into term, it had all come crashing down around her ears, thanks to a couple of hours in the presence of a maths lecturer.
They’d been doing basic calculations about surface areas and volumes. As with Lisa, the mere sight of an equals sign had driven Pam to distraction. Like the little boy in the Gary Larson cartoon, her brain was full. I leapt into action immediately. We got into her car and drove up to Dad’s flat, where I gave her a crash course in symbolic manipulation.
It wasn’t as frightening as it sounds, by the way. Pam already had a calculator and writing materials. I got out a tape measure and I set her to work, working out the surface area and volume of a stereo speaker (a cuboid), a cardboard tube in which I kept my large-scale maps (a cylinder), and even a Toblerone packet (a triangular prism) – just to get her thinking about solids in three dimensions.
Over the next few months we spent most Sunday afternoons going through the outlines of fractions, algebra, geometry, trigonometry, very basic calculus, and logarithmic functions. We even had fun in the pub one afternoon; I did the empirical proof of Pythgoras’ Theorem, which I remembered from the appendix to Prof. Jakob Bronowski’s book The Ascent of Man. That had everyone transfixed. Even though they all remembered it from school, nobody had ever actually explained it to them before.
One day, Pam was panicking again because they were ‘doing pi’ (her words) and she’d never encountered anything other than the digits 0 to 9 and the occasional Roman letter before. I reassured her that π behaves like any other number – you can add things to it, multiply by it, raise it to a power, take its logarithm, and use it as a limit in integration. The only thing you can’t do is count up to it.
As she was still a bit adrift, I put three pens on the table. I put three glasses, three beer mats and three cigarettes beside them, and asked Pam to show me ‘the number three’. Naturally, she couldn’t – because ‘three’ was nothing but the number of elements in each set of similar objects. Pam liked this sort of Platonic approach to numbers, and it seemed that I’d stumbled upon something which she could grasp.
On another occasion, she was given a problem which involved a circular pipe, three quarters full of water, and she was required to find the cross-sectional area of the water. Pam, and I imagine most people, cut straight to the chase and said, ‘It’s three quarters of the total area.’
But it isn’t, of course. You have a half-pipe, and then a weird shape bounded by two parallel lines and two arcs, the area of which is more than half that of a half-pipe. In other words, the cross-section is more than three quarters of the circle. I showed Pam how to divide the weird shape into two sectors and an isosceles triangle, all with known sides or angles, measure each piece, and then put the whole thing back together again. Of course, we could have worked out the area of the segment and taken that from the whole, but this was more constructive and more realistic in her eyes. Slowly but surely she started to get the feel of visualising problems in space, and the lights were coming on.
One weekend she brought me a nice problem involving a piece of sheet metal, which was to be made into a cylinder. The class had to calculate the maximum volume that could be contained within that area of metal. It’s a fairly trivial problem in calculus, but Pam had gone up her own orifice for hours trying to solve it. I had a look and couldn’t figure it out myself. Eventually it dawned on me – the lecturer hadn’t told us whether the solid had one end, two ends, or no ends.
A hollow tube would have been easy enough. An open tin of beans would have been a bit more complicated, but not a major hurdle, and a closed tin of beans would have stretched us a bit. But we didn’t know which shape we were working on. Pam went back to the class the following week and sat smugly while the lecturer asked her colleagues who’d managed to finish their homework. Not surprisingly, nobody had. When he asked why, Pam put her hand up and told him that there wasn’t adequate information in the question. Nobody else had come up with the explanation – or at least, nobody admitted to it.
She was beginning to think critically about the content of the questions, and analyse them herself. That was the key. Suddenly the lights came on fully. Over the next few months Pam progressed to more computer-based mathematics and aspects of physics, which she was now able to handle with help from other friends who had done that part of the course. By the time the summer came, she was confident enough to tackle the exam – and was one of the two people in her group who achieved a distinction in that module. She progressed to study computing and got her degree a few years later.
Tracy M, a lovely girl who used to work part-time and weekends in Blackwell’s in Treforest, had failed Maths GCSE. Her ambition was to go in for teaching, but without that qualification she stood no chance of being accepted. When it was just the two of us in the shop – most Saturdays – she and I would work through exercises and examples, interrupted only by the occasional overseas student wanting to buy a phonecard. After a few months of coaching, Tracy also got to know the short-cuts and the tricks of the trade. She passed on the resit.
When my cousin Aimee was fifteen, about ten years ago, she failed her Maths mock GCSE in spectacular fashion. Once again I was drafted in to help. I bought a study guide from the shop and took it home. She’d never even owned a geometry set, so I bought her one as well. I stocked up with graph paper, squared paper, and all sorts of useful visual aids to help her with her exams. However, before we could even start, her mother remarried and they moved to Cardiff. Somehow (more by luck than judgement, I expect) she came away from the exam with a B.
Then again, when one looks at the content of the study guide, it’s hard to see how anyone can fail. One question involved what looks like a Battleships grid of thirty or so squares, representing a very basic map of a village, with a school and a church and a railway station. The questions consist of things like ‘What is in square D3?’ and ‘What is north of the station?’ I think we were doing that sort of exercise when we were about nine years old.
Today, in Forensic Science, I stayed on for the second half of Gareth P.’s lecture. He told us last week that anyone who already had Maths A Level could leave early, as it would be a breeze for us, and that we’d be doing more advanced stuff today. I had a quick word with him after last week’s session, explaining that it had been a long time since I did my A levels, and wondering what sort of topics he’d be covering. He said it would be basic calculations (‘Like v = u + at?’ I asked, and he nodded), and that I needn’t stay in that case. But this week I did stay behind. After all, this week we were going to be doing more advanced stuff.
As I know from bitter experience, just the mention of the word ‘maths’ is enough to get most people running for the fire exits. That happened again today – we stopped for a break before Gareth carried on into the second half, and a fair number of people who’d signed in weren’t there when he read their names out. But what we were doing wasn’t mathematics. It was what my old teacher Geoff Abbott would have called ‘Sums’ – basic arithmetic and algebra, stuff we were handling when we were about twelve years old. Mathematics is the sort of insane things that John Nash works on in A Beautiful Mind. I had to try not to laugh when Number Numbness kicked in all around me.
Here’s one of Gareth’s examples (from memory):
The speed of sound is 343 m/s at sea level. If you see a flash of lightning, and three seconds later you hear the thunder, how far away is the storm?
I did a very quick calculation in my head and wrote down my answer. The gang in the row behind me were labouring hard over their arithmetic. On the end of the row diagonally in front of me, a young lad was stabbing frantically at his calculator. All around me I could hear the clicking of tongues, the scribbling of pens, the tapping of calculator keypads. I looked round at a sea of furrowed brows and smiled to myself. This isn’t Maths, I thought – this is sport!
Another one went like this:
A car travels for two hours at 100 km/h, two hours at 60 km/h, and one hour at 80 km/h. What is its average velocity?
After a few minutes the girl behind me nudged me.
‘Is that right?’ She showed me her result – 160 km/h, twice the correct answer.
I shook my head. ‘I think the police would probably have stopped him by then,’ I said.
Another question went something like this:
A farmer walks twenty-five kilometres in four hours. If he walks at 5 km/h for the first ten kilometres, what’s his average speed for the remainder of the journey?
Gareth wound the class up by telling them that it was a hard one. This one was a walk in the park (or maybe the fields) for me, but my colleague directly behind me was really struggling. Eventually she asked me for help. I told her we already know how long it’s taken him to go ten kilometres, so the rest is easy. After sweating blood she and her pals finally managed to sort it out between them.
At the end of the session, Gareth told us we were going to move on to acceleration next week. He explained that there are two types of acceleration: positive, or increasing speed; and negative. Gareth looked in our direction, and asked one of the lads behind me if he could think of another word for ‘negative acceleration’ and he suggested ‘de-acceleration.’ It was close, but I doubt if Suzie Dent would have allowed it on Countdown. I thought he was going to ask me at first, and I’d already psyched myself up to say ‘retardation.’ On the other hand, maybe I should keep that word to myself – I might be accused of taking the piss. Gareth wrote another formula on the board. It was the one I’d quoted him last week.
I think I’ll probably stay for next week’s session anyway, just to be on the safe side. As we were packing up, I gave my number to the girl behind me, just in case history repeats itself and I end up giving extra tuition again. You never know …
An Autobiography in Random Chapters 