Towards the end of the 1960s, and during the 1970s, a decade when some people felt Britain was going to hell in a handcart, a group of ambitious intellectuals introduced the "new maths" in schools across the country. If traditional mathematics, numbers and equations were a bit hard for many people, why not replace them with something more up to date that everyone could understand? And as the rate of inflation went up and the winter of discontent set in, perhaps it was better to forget about numbers and let this bold cultural revolution in mathematics create a new generation of proletarian intellectuals able to take their place at the vanguard of a new understanding of mathematics. What happened?
At the time, my mother asked me, her son with a degree in mathematics: "Do you understand the new maths?" I was nonplussed. What new maths? Of course mathematics is developing new ideas and new methods all the time, but you can't learn and understand them until you've learned the basics. Was this a new method of teaching the basics? Apparently not - they were learning about sets (very roughly speaking, collections of things), rather than numbers and algebra. When I recently asked a colleague why, he said it was meant to teach logic. Logic? Now that's important. Logic at an elementary level deals with such things as syllogisms: all men are mortal; Socrates is a man; hence Socrates is mortal. And contrapositives: if rain implies clouds, then no clouds implies no rain. But it doesn't imply that if there's no rain, then there are no clouds. Common sense? Yes, but there is a logical structure here and the way to learn it is to relate it to things we understand, rather than abstractions that most people don't understand terribly well.
Logical reasoning is important and it used to be part of a good mathematics course up to the age of 16. It was done under the heading of geometry, a venerable subject that has used careful rational argument for over two thousand years, ever since the Greek mathematician, Euclid of Alexandria, wrote the definitive text on the subject. Starting with simple axioms, such as the idea that between any two points there is a line, Euclid developed the main results of plane geometry. Assumptions were clearly stated, and the results, which we call theorems, were carefully proved so that no unspoken assumptions were made and no gaps appeared in the logic.