Some time in 1971 I bought a book which infuriated me: Why Tommy Isn’t Learning by Stuart Froome. This book is an indictment of the (British) educational system which, at that time, was deeply into child-centred learning. Froome argued for the teaching of basic arithmetic and, to make a point, compares the questions on a 1970s teacher-training examination and a 1924 eleven-plus examination—the examination that 11 year old children used to take in the UK to divide them into the sheep (going to Grammar School and an academic education) and the goats (going to a technical school for vocational training).
Here’re a couple of the questions from the 11+ examination (I suspect that, even today, there are many 11 year olds who wouldn’t be able to do them, even were the imperial units converted into metres and even were a calculator to hand).
- A man cycling at 10 miles per hour passes a man walking at 4 miles per hour. Twelve minutes later the bicyclist has a puncture. How far ahead of the walker is he when this happens? If the cyclist takes 20 minutes to mend the puncture, will the man walking pass him while he is mending it or not? How far apart will the two men be when the bicyclist is ready to start again?
- A bicycle wheel travels 7 feet 4 inches for every revolution of the back wheel. If the wheel revolves twice in every second, at what rate in miles per hour is the bicycle travelling?
Readers in North America should note that these were not multiple-choice questions.
At the time I bought this book I was an undergraduate in mathematics and therefore knew everything. It’s interesting how my knowledge has decreased over the years. Anyway, I was so infuriated by Froome’s book that I actually wrote to him to say that the ability to do arithmetic, in particular mental arithmetic, was not a trick needed by a modern child.
I work for Nortel, a company which has supported the development of the $100 laptop as part of the One Laptop Per Child programme of the Massachusetts Institute of Technology (a fairly young (1860s) but prestigious university in the USA). The first pictures of this have emerged this week and raised again the question of whether children should be using computers to do their arithmetic.
One of the WIKI pages on the One Laptop Per Child project argues strongly, and more fluently than I, the case I made in my letter to Froome all those years ago.
“Some people might object that it is useless to have children understand algebra when they don’t know how to multiply 78543 by 17629. I have four answers. …. First, algebra captures a way of thinking that is far more important in life (both practical life and intellectual life) than multiplying big numbers. Second, a calculator multiplies better. Third, if it is necessary to know how to multiply this can be learned far more quickly and far less painfully after learning how to think mathematically. The fourth answer, and perhaps the most important, is expressed in the words of my mentor, the late Bob Davis who used to say: “School math teaches the art of getting the right answer without thinking.” Intensive practice of mechanical skills before one knows enough to fully understand them is a great inhibitor of creative thinking.”
Having helped many high-school and university students with their mathematics homeworks I must admit that I have always been totally bemused by what is taught in mathematics courses. I have recently been working with a business student struggling over Eigenvalues and Eigenvectors (even from the point of being able to pronounce them). I know nothing of business and am sure that these are valuable somewhere in there but the textbook certainly gave no examples. I’ve seen high-school students struggling over quadratic equations which I suspect that they will never ever need in their later life and I’ve realised how strongly our mathematics education is determined by chance—we teach quadratics because they’re soluble analytically and quintics aren’t; we get kids to jump through hoops to integrate the most obscure functions, not because those are functions that would ever be met in real life but because they happen to be integrable!
I think I’m going to rework most of the high-school mathematics curriculum and make room for a lot more mathematical philosophy.