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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.

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John Ibberson
April 9th, 2010
1:04 AM
Rene Thom's ariticle '"Modern Mathematics": An Educational and Philosophic Error?' appeared in AMERICAN SCIENTIST vol. 59, Nov./Dec. 1971 pp.695-99. Jean Dieudonne's reply 'Should We Teach "Modern" Mathematics?' appeared in AM. SC. vol.61, Jan./Feb. 1973 pp. 16-19.

November 19th, 2008
4:11 PM
Just 3 points... (1) Years ago that finest of teachers George Pólya (in "How to Solve it", I think it was, can't find my old copy) stated simply that anyone who has been through school, without experiencing some moderately serious, systematic and rigorous Euclidean geometry and theorem proving, could justifiably accuse his educators of selling him short. But then the exhilarating mathematical explorations that Pólya brought to teaching are impossible to imagine in any but the most exceptional schools today. (2) Back in the early 1970's none less than René Thom issued an impassioned plea to schools to ditch the empty, barren abstractions of set theory that were meaningless to most pupils, trivial to the best, and utterly stifling the development of real mathematical insight and imagination - in favour of a return to the inexhaustible intuitive riches of Euclidean geometry. ( I wish I could find the original essay.) (3) In the late 1980’s, before I made a career change out of teaching, I recall my head of a school mathematics department asking Oxbridge don’s what were the most noticeable changes they had seen in recent undergraduate intake. Answer 1: a marked falling-off in ability to think spatially at all, let alone in three dimensions. (A rather dire impediment to work in virtually any branch of science or engineering, one would have thought, let alone to any work in higher mathematics.) Answer 2: A woeful lack of any conception of systematic proof from clearly identified assumptions or axioms leading to an ever growing structure of theorems and results. Speaks for itself, doesn’t it? The fads and fashions and easy options of the semi-educated will always win out with the politicians, bureaucrats and ideologues of the “new”. Very sad.

October 31st, 2008
3:10 AM
I couldn't agree more. In fact, it perplexes me that what seems to have been substituted in many universities for the training in logical argument (however philosophically inadequate) that Euclid used to provide in schools is an elementary course in abstract logic (some propositional calculus and a little predicate calculus) that is presented without any motivation at all. I bore my students occasionally with John Aubrey's anecdote about Hobbes from the Brief Lives -- a perfect illustration of the power of Euclid's exposition, and of the appeal his theorems can have for people willing to read them carefully.

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