## Whatever happened to Euclid?

An interesting article on the BBC website about the innate nature of our understanding of geometry reminded me that I have been meaning to post something about the importance of geometry in mathematics education – and, more accurately, the damaging consequences of the lack of geometry in the modern curriculum.

When I was a lad – yes, it’s one of those tedious posts about how things were better in the old days – we grammar school kids spent a disproportionate amount of time learning geometry in pretty much the way it has been taught since the days of Euclid. In fact, I still have a copy of the classic Hall & Stevens textbook based on Euclid’s *Elements*, from which I scanned the proof shown below (after checking that it’s now out of copyright).

This, Proposition 5 of Book I of the *Elements*, is in fact quite a famous proof known as the *Pons Asinorum*:

The old-fashioned way we learned geometry required us to prove all kinds of bizarre theorems concerning the shapes and sizes of triangles and parallelograms, properties of chords intersecting circles, angles subtended by various things, tangents to circles, and so on and so forth. Although I still remember various interesting results I had to prove way back then – such as the fact that the angle subtended by a chord at the centre of a circle is twice that subtended at the circumference (Book III, Proposition 20) – I haven’t actually used many of them since. The one notable exception I can think of is Pythagoras’ Theorem (Book I, Proposition 47), which is of course extremely useful in many branches of physics.

The apparent irrelevance of most of the theorems one was required to prove is no doubt the reason why “modern” high school mathematics syllabuses have ditched this formal approach to geometry. I think this was a big mistake. The bottom line in a geometrical proof is not what’s important – it’s how you get there. In particular, it’s learning how to structure a mathematical argument.

That goes not only for proving theorems, but also for solving problems; many of Euclid’s propositions are problems rather than theorems, in fact. I remember well being taught to end the proof of a theorem with QED (*Quod Erat Demonstrandum*; “which was to be proved”) but end the solution of a problem with QEF (*Quod Erat Faciendum*; “which was to be done”).

You can see what I mean by looking at the *Pons Asinorum*, which is a very simple theorem to prove but which illustrates the general structure:

- GIVEN
- TO PROVE
- CONSTRUCTION
- PROOF

When you have completed many geometrical proofs this way it becomes second nature to confront any problem in mathematics (or physics) by first writing down what is given (or can be assumed), often including the drawing of a diagram. These are key ingredients of a successful problem solving strategy. Next you have to understand precisely what you need to prove, so write that down too. It seems trivial, but writing things down on paper really does help. Not all theorems require a “construction”, and that’s usually the bit where ingenuity comes in so is more difficult. However, the “proof” then follows as a series of logical deductions, with reference to earlier (proved) propositions given in the margin.

This structure carries over perfectly well to problems involving algebra or calculus (or even non-Euclidean geometry) but I think classical geometry provides the ideal context to learn it because it involves visual as well as symbolic logic – it’s not just abstract reasoning in that compasses, rulers and protractors can help you!

I don’t think it’s a particular problem for universites that relatively few students know how to prove the perpendicular bisector theorem, but it definitely is a problem that so many have no idea what a mathemetical proof should look like.

Come back Euclid, all is forgiven!

May 24, 2011 at 3:46 pm

What if the proofs came back, but the barrier to entry was lowered? We might be able to lose much of the jargon by using more common language, and not sacrifice the gist of it? I’d ditch all the latin. In “angle subtended by a chord at the centre of a circle is twice that subtended at the circumference”, we’d keep words like “angle”, but replace ‘subtended’. It’s not going to be easy. A dictionary reference for ‘subtend’ tends to look like this: “Geometry . to extend under or be opposite to: a chord subtending an arc.” – which is not very helpful to the student.

May 24, 2011 at 3:53 pm

Nothing wrong with “subtend” in my book. Do we have to assume that students are incapable of broadening their vocabulary? Aren’t we insulting their intelligence?

May 25, 2011 at 8:08 am

Are the students going to learn geometry on their own? My maths teacher taught me in class what ‘subtends’ means.

May 24, 2011 at 4:45 pm

As a private maths tutor I can only support you basic idea. Whether Euclidian geometry or something else the kids need to go back to learning how to formulate a logical argument or proof.

May 24, 2011 at 6:44 pm

Problem is you want to drive the maths curriculum with what is necessary for the 1% (or whatever) of children are going to end up doing a physics degree (or similar).

I would suggest just getting basic arithmetic (without a calculator) and things like working out areas and numbers of bricks needed to do a job are mo important to the other 99%

One you get to A-level, then I agree the emphasis should then shift to proofs etc. as the students then are heading for a science/maths type destination. Tying to teach such things earlier will just confirm to the majority that all maths is a waste of time and they wont even therefore apply themselves to the basic arithmetic and real-life maths they will need.

May 24, 2011 at 7:13 pm

Why do you assume that 99% of the population is incapable of learning such things. I agree that basic arithmetic is an essential pre-requisite for any higher mathematics, but this should much earlier, i.e. at age <11. If my generation could learn it then, so can today's. It's an insult to them to assume they can't cope. And education isn't just about utlility, it's about expanding your mind.

May 24, 2011 at 9:09 pm

It’s probably a very good idea that geometrical proofs should be taught to pupils so that this approach to problem-solving can be drilled in. Having said that, I didn’t spend a disproportionate amount of my maths education learning Euclid’s proofs but this approach was already second nature to me before starting my physics degree, through solving endless mechanics problems (inclined planes etc.), amongst other things, at A-Level.

However, I think it’s optimistic, at best, to think that this kind of maths can be successfully taught to pupils of all ability levels in secondary schools. The abstract nature of the problems and the vocabulary involved do present real problems in teaching this material, especially given pupil behaviour and concentration levels. I think it needs to be targeted to pupils of the right ability levels who can actually benefit from it and have their minds expanded.

You mentioned that you did these problems at grammar school, which obviously didn’t teach pupils of all ability levels. Did 99% of school children in your generation, in all types of schools, really study Euclid’s proofs, gain a good understanding of them and benefit from learning this approach to solving problems? (With the emphasis on understanding them, and not just dutifully copying them out).

May 24, 2011 at 9:19 pm

Point taken. I did go to a selective school. Euclidean geometry was part of the O-level mathematics syllabus which was the forerunner of GCSEs, but I don’t know what fraction of school pupils took O-levels.

However, my point is really that nobody learns this stuff at school now, really, not even the most gifted…however they are identified or defined.

May 24, 2011 at 9:39 pm

I agree with the main point of your post.

I was in secondary school during 1998-2005. I remember applying the rules of Euclidean geometry to solve problems… but not proving them, which perhaps should have been the case.

May 24, 2011 at 9:46 pm

I did my O-levels in 1979 and A-levels in 1981. We all spoke Latin then.

May 25, 2011 at 8:40 am

Euclid’s Elements lasted longer as a textbook than any other, some 23 centuries! I do not agree that it is ‘geometry’ cf ‘algebra’; the proofs in it *are* algebra. The history of mathematics is at least partly the history of better notations (eg, Arabic vs Roman numeral system), and things like

Mod (exp(i theta) ) = 1

subsume Pythagoras (since it expands to cos^2 + sin^2 = 1).

May 25, 2011 at 11:31 am

O levels were taken by roughly the top 25% of the population. I was one of the last years to do them in 1986 and certainly their formal approach was well suited to myself, as it probably would be to the majority of scientists. However most people are not going to become scientists, and for them an emphasis on proof rather than application is generally a turn off in my experience. Pythagoras is very useful as a way of working out the length of the side of a triangle, and everyone should leave school knowing how to use it, but the formal proof involved in constructing squares on the sides of a triangle is just confusing to most.

The problem is we want those who turn up in our lecture theatres to have done the proofs without having disadvantaged everyone else by trying to teach them irrelevances. The old O-Level / CSE distinction helped to do this, and getting rid of that was probably the mistake.

May 25, 2011 at 12:56 pm

I agree with your last point.

May 25, 2011 at 1:28 pm

The argument against the O-level/CSE split was of course that a large number of bright kids who were more than capable of dealing with O-level demands were shunted off in the direction of CSE owing to classification errors or just plain thoughtlessness on the part of their teachers. I had my eyes opened to this some years back when a friend lacking in O-levels started doing maths at night school. He got an A at A-level easily – so somehow the school system had completely let him down. The anger against that sort of cockup was and remains a powerful argument against a simple binary system.

The solution is to have optional extensions, which you decide to access late on. So I not only did O-level maths aged 16, I did further maths O-level at the same age. This was a more rigorous course including calculus. The sad thing is that (I think) no such option now exists. As a result, I find myself lecturing 1st-year maths methods to 19-year-olds, covering material which has a substantial overlap with the old FM O-level. Many of the kids struggle: this is partly because grasping these things is easier the younger you start, and also because it’s the first time they’ve probably been challenged to be in any way rigorous. Bringing back FM at 16 would give a foundation for a more rigorous A-level and would cure many of the problems we all bemoan in the maths capabilities of modern undergraduates.

May 25, 2011 at 1:38 pm

’tis true.

I still have the O-level mathematics papers I took in 1979. They’re significantly harder than modern A-level mathematics, and a struggle for most first-year physics students.

May 26, 2011 at 11:08 pm

Well, I went to a comprehensive school (the first to be established in Britain, or at least the first across both Wales and England, I understand) and I studied Euclidean proofs at the age of 12 or 13 years. That was in a streamed group of the top 50% in ability of that year.

The examination board whose age 16+ papers we sat provided three mathematics papers of different degrees of difficulty. All candidates were entered for two of these. Those weaker candidates, who would conventionally have been entered for a CSE qualification, would be entered for the more basic two of the three papers, and would have been taught for these more basic papers. Those stronger candidates, who would conventionally have been entered for O-Level, would sit the more demanding two of the three papers. Performance in these examinations would determine whether candidates were awarded an O-Level or the more basic CSE. A candidate who sat the more basic two papers but performed very strongly in them could be awarded an O-level pass. (This system existed only in mathematics, although some dual O-level/CSE system existed in a few other subjects in a slightly different form.)

(I believe all candidates were given an O-level and a CSE certificate. I therefore have an O-level in mathematics and a grade 1 CSE, but I have never referred to the CSE when listed my qualifications as it is irrelevant.)

I also studied for a separate O-Level examination in Calculus and Coordinate Geometry, although the grading was only pass or fail (not the conventional A to F).

August 26, 2011 at 11:06 am

[…] I’d be interested in any comments you might have, especially if you’ve actually done GCSE Mathematics (recently or a long time ago). I suspect the most obvious difference is that in my day we did much more geometry… […]