It all adds up: Carol Vorderman in her "Countdown" days (PA Photos)
In a story published in 1918, H. G. Wells describes a boy being taught arithmetic. "Then Miss Mills taught Peter to add and subtract and multiply and divide. She had once heard some lectures on teaching arithmetic by graphic methods that had pleased her very much."
He goes on to explain how the teacher's explanation of someone else's ideas was confusing, but writes:
"Peter was rather good at arithmetic, in spite of Miss Mills's instruction. He got sums right. It was held to be a gift. [His sibling] was less fortunate. Like most people who have been badly taught, Miss Mills had one or two foggy places in her own arithmetical equipment. She was not clear about seven sevens and eight eights; she had a confused, irregular tendency to think that they might amount in either case to fifty-six, and also she had a trick of adding seven to nine as fifteen, although she always got from nine to seven correctly as sixteen."
Of course, we do it better these days. Or do we? Primary schools seem to manage well enough, and the teachers may be very clear that adding nine to seven is the same as adding seven to nine, and that seven eights is the same as eight sevens, but arithmetic does not stop there. Our children go on to learn algebra, but often with an inadequate background in arithmetic.
For instance, how would you do the sum: 3×17 + 7×17? I hope you would mentally reprocess it as (3 + 7)×17, giving 170. This is less work than doing the two products separately, and adding them at the end, but the point is that if you don't understand the short-cut using numbers, how can you understand it using letters? Replacing 3, 7 and 17 by a, b and c yields (a+b)×c = a×c + b×c. This is not a trick for the memory, like the principal parts of some irregular Greek verb, but a simple fact about numbers. Let's call it structural arithmetic. It's important — Carol Vorderman please note — because if you're not comfortable with numbers you won't be comfortable with elementary algebra, and you'll have a confused, irregular tendency to imagine (a+b)2 might be the same as a2 + b2.
When my son went to high school in America and took a year of calculus, the parents first met the teacher in his classroom. One anxious father asked what caused the kids the most trouble. "Algebra," boomed the teacher. And if algebra is the difficulty in calculus, then arithmetic is the difficulty in algebra. I know it's a cliché, but you have to master the basics, and that's where our secondary schools seem to be failing. Who do we blame? The teachers? Yes, some of them are not so wonderful, and need training to give them greater depth at arithmetic, particularly since they may have suffered abuse from calculators in their youth. But the real elephant in the room is testing. Testing, testing, testing. And for what? So the pupils can pass the tests? I hope not, because what is needed is understanding. I doubt if testing would show up the inadequate teaching mocked by H. G. Wells, and if the government wants to check how things are going, they can use sampling methods. No amount of testing will make a child learn what he or she is not taught, but it may hinder them from learning for themselves, and that, after all, is the only real way to learn.