Recently all the sixteen year old school children in New Zealand have been taking their certification exams. A lot of controversy has arisen around the algebra paper. Apparently it was far too difficult for the students and many of them were reported to have left the exam room in tears. It would seem that the questions asked on the exam did not really reflect the curriculum as it had been taught during the year.
When the controversy first arose, the exam questions were published in the media so that we could all see for ourselves what the students had been struggling with. Out of curiosity I took a look at the questions and I even answered several of them. Despite the fact that I've not used algebra in any serious way for at least forty years, I found the exam to be quite straightforward and relatively simple (though some of the questions did seem oddly pointless). I must admit that I was rather surprised that the exam was aimed at sixteen year old students I would have been dealing with these kinds of exam questions round about the age of twelve or thirteen. By the time I reached sixteen, I was well into the study of calculus and I had left this kind of elementary algebra far behind.
At this point, I could play the old curmudgeon card and grumble that the youth of today have it far too easy. The study of mathematics (and presumably of other subjects as well) has been dumbed way down. In my day we really had it hard. (Cue the Four Yorkshiremen sketch...)
Of course, every generation thinks that way. Every generation is quite contemptuous of the education that the next generation receives. I don't really believe that standards have declined to the extent that some of the curmudgeons among us seem to think. All that has happened is that the subjects are just being taught at a different pace, and perhaps with a different perspective, than once they were. If today's students ever go on to study technical subjects as part of their tertiary education, I'm sure they'll be properly introduced to the mathematical tools they need at the appropriate time. I just learned those tools earlier than they will be learning them, which is really no big deal at all in the grand scheme of things.
There were two things about the exam controversy which worried and annoyed me.
Firstly it was reported that some of the maths teachers were also finding the exam questions difficult to answer. A photograph was published of a group of puzzled maths teachers staring at a whiteboard covered in scribbles which supposedly detailed their total failure to answer one particular question. This really is worrying. How can they possibly teach the subject effectively when they themselves are stumped by such elementary questions? What qualifications do they have to teach the subject? Are they, perhaps all English teachers who just teach maths on the side by keeping themselves the proverbial one chapter ahead of the students? Enquiring minds want to know...
The other worrying thing was the way the exam controversy was reported in the press and on radio and television. Without exception, the journalists who covered the story confessed themselves to be mathematical ignoramuses who found the whole thing quite bewildering. And what is more, one and all, they seemed to be quite proud of their ignorance.
In a particularly egregious example, one enterprising television journalist actually got a university mathematics lecturer to answer one of the questions in the studio. The lecturer launched in to a particularly lucid explanation of what he was doing and why he was doing it as he slowly worked his way through the problem. He reached a point where he noted that he had now derived one equation with two unknown values in it. In order to solve the problem, he said, he needed a second equation. At this point we cut back to the studio for the journalist to witter on about nothing very much for a while, then we returned to the lecturer who, as if by magic, now had the second equation that he needed. From that point on, the solution was trivial and he quickly reached an answer. Then we returned to the bewildered journalist who said he was sure that somebody in the audience might have understood that, but he himself had no idea what it all meant.
Obviously the decision had been made to broadcast this formal solution in small sections with linking material from the journalist, so as not to overload the attention span of the audience too much. But somewhere in the back and forth between the two, a vital step in the lecturer's argument ended up on the cutting room floor, rendering the whole explanation pointless and turning it into gibberish. But since the journalist clearly believed that the entire subject was gibberish anyway, he really couldn't see (and didn't care) that omitting a whole section, presumably for timing reasons, made a complete nonsense out of the whole business. After all, he clearly believed that no ordinary person could possibly understand such esoterica anyway, so what difference did it make?
Numeracy and literacy are two sides of exactly the same coin and without some skills in both those subjects it simply isn't possible to cope with the demands of modern society. Nobody expects the man on the Clapham omnibus to be able to apply tensor calculus to an analysis of stress in building materials or to be able to describe the significance of Anna Livia Plurabelle to the structure of James Joyce's Finnegan's Wake. These are specialist subjects which are best left to the experts. But we can legitimately expect the man to be able to perform simple calculations and to understand what he reads in a newspaper.
Illiteracy is generally felt to be shameful and people who cannot read will go to enormous lengths to conceal their lack of skill in this area. But no such stigma attaches to innumeracy and the man on the Clapham omnibus often appears to take a perverse pride in his ignorance. After all, what possible use are mathematical skills in everyday life?
When I was a child, I was very fond of the Jennings books by Anthony Buckeridge. Jennings was a pupil at an English boarding school. The books describe the scrapes and adventures that he had. They were very funny and I'd love to read them again, but they seem to have vanished from the world.
I remember one episode where Jennings has received a cake from his mother and he wants to divide it evenly among his friends. Recalling his geometry lessons, he uses his protractor to measure the appropriate angles so that everybody will have a slice of exactly the same size. Then he uses his ruler to draw the lines that define each slice. Picking up his knife, he begins to cut along the lines he has drawn. But the cake is very crumbly and the slices are very narrow. The cake disintegrates and Jennings and his friends are left with just a pile of crumbs. Jennings is disgusted and he says something along the lines of, "Huh! So much for maths. It's all very well in the classroom, but as soon as you apply it to real life everything just falls apart!"
I found this to be a delightfully satirical episode it's at least half a century since I last read a Jennings book, but that episode has stayed with me, fresh and clear, for all that time. It perfectly illustrates both the usefulness of mathematics and the dangers of applying it too literally.
I am also reminded of the joke about a mathematician who was employed to advise dairy farmers on how to improve the efficiency of their day to day operation. He spent many months following them around and taking copious notes. Then he retired to work on his thesis. Eventually it was complete and he called a big meeting of all the dairy farmers so that he could explain his ideas to them. He strode onto the stage in the full glare of the spotlight, and marched up to the whiteboard. He drew a circle and said, "Consider a spherical cow..."
But, more seriously, yes I do use simple mathematics in everyday life. In the supermarket I calculate unit prices so as to determine the best value for my money. In my car I calculate how long my journey will take, given the distance I have to travel and an estimate of my average speed. In the kitchen I adjust the quantities of the ingredients demanded by a recipe based on the number of people I am cooking for.
Numeracy matters and people who take pride in their lack of mathematical skills really need to take a long hard look at themselves. Maths phobia is a very real phenomenon, but it's nothing to boast about.