# How proof dropped out of school maths

It is a very odd choice by Edexcel to have FP1 (Further Pure Mathematics 1) students study proof by induction, a particular (and rather subtle and paradoxical) form of proof, before they’re introduced to proof more generally. And it raises the question – just when and how proof disappeared from school maths. As far as I can make out, the story goes something like this.

I’m old enough that I was taught a lot of maths pretty much straight out of Euclid (though not actually from Euclid’s book). Students who did maths at all (beyond basic numeracy) had been taught that way for over 2000 years.

I vividly remember being taught the proof that there is no greatest prime number, at the age of nine: it was the most beautiful thing I’d ever seen.

I didn’t know it, but the end of that over-2000-year era was fast approaching.

Frank McCourt’s semi-autobiographical novel Angela’s Ashes contains a description of an Irish classroom scene in the early 1940s which tells a bit about what Euclid meant across that whole historical era.

Next day Brendan raises his hand. Dotty [the teacher] gives him the little smile. Sir, what use is Euclid and all the lines when the Germans are bombing everything that stands. The little smile is gone. Ah, Brendan. Ah, Quigley. Oh, boys, oh, boys. He lays his stick on the desk and stands on the platform with his eyes closed.

What use is Euclid? he says. Use? Without Euclid, the Messerschmidt could never have taken to the sky. Without Euclid the Spitfire could not dart from cloud to cloud. Euclid brings us grace and beauty and elegance. What does he bring us, boys?

Grace, sir.

And?

Beauty, sir.

And?

Elegance, sir.

In 1959, the year after the nine-year-old me learned the proof that there is no greatest prime number, at a conference on mathematical education in France, Jean Dieudonné, a member of the Bourbaki group, raised the slogan: “Down with Euclid! Death to triangles!”

Dieudonné’s idea was not at all to push proof out of the syllabus. The Bourbaki group’s general push was to make maths more formal, more rigorous, more abstract, less reliant on pictures. Dieudonné wanted to replace Euclidean geometry by more modern and abstract areas of mathematics, such as set theory and abstract algebra. (Abstract algebra is group theory, ring theory, field theory and so on, as distinct from the fiddling around with x’s and y’s we do at school).

According to a maths lecturer from Cyprus, writing on http://mathoverflow.net/questions/152352/is-euclid-dead, Dieudonné had a “success” which was eventually perverse from a Bourbaki point of view.

“Euclidean Geometry was gradually demoted in French secondary school education [and stripped of] the difficult and interesting proofs and the axiomatic foundation. Analogous demotion/abolition of Euclidean geometry took place in most European countries during the 70s and 80s, especially in the Western European ones.

“Together with Euclidean geometry there was a gradual disappearance of mathematical proofs from the high school syllabus, in most European countries; the trouble being (as I understand it) that most of the proofs and notions of modern mathematical areas which replaced Euclidean geometry either required maturity or were not sufficiently interesting to students, and gradually most of such proofs were abandoned”.

Marcus du Sautoy’s take on this (so he told me when he visited CoLA) is different. He thinks efforts like the Schools Mathematics Project (which tried to introduce more abstract maths into schools in the 1960s and 70s in England) failed not because the students couldn’t cope with the proofs, but because the teachers couldn’t: they hadn’t been adequately trained.

In any case, there was a backlash, and a demand that less abstract stuff and more calculation be reintroduced. It was reintroduced. And now the fact that much more varied maths was being used in science and engineering and economics and finance and statistics and computing and so on created a push for the new syllabuses to have a greater variety of content than the old Euclid-based ones. They were still stripped of proofs.

Part of the reason that proofs remained stripped out must have been pressure to include a greater variety of content in the syllabus. The syllabus comes to cover so many things that there’s no time to prove results or present them as more than black-box formulas to be applied (which means, of course, that they’re likely to be forgotten soon after the relevant exam is over).

I would guess that the tilting of the syllabus towards calculation and away from proof was driven also by the rise of credentialism and exam-obsession. Edexcel apparently allows markers two minutes to mark a paper. This is just about possible (though undesirable) with exams (and therefore syllabuses) based on calculation. It would be impossible with exams (and syllabuses) centred on proof. There is often more than one way to prove a result, so the marker would have to read the student’s work and think about it, rather than ticking everything which corresponds to what the mark scheme demands and then adding the ticks.

As the Cypriot lecturer notes: “I teach in a University… and we keep introducing new introductory courses for math majors, as our new students do not know what a proof is. Cf. the rise of university courses in the US that come under the heading ‘Introduction to Mathematical Proofs’ and the like”.

Talking with friends who have studied maths at university in recent years, they report a culture shock at the start. What they then encounter as maths, centred on proof, is very different from what they’ve done at school, centred on calculation.

Talking recently with friends who studied maths at university with me (1966-9), I remarked that our Further Maths students rarely use the word “if” (and pretty much never the words “if and only if”) in writing or talking about maths. They were shocked, and their shock made me realise just how much that important word is missing from school maths.

In his blog posts on the Cambridge introductory analysis course which he’s been teaching, Tim Gowers refers to “the ‘let’ move, which I’ve talked about several times in lectures”. (As in, e.g.: let the roots of a cubic be a, b, c; or whatever). I can’t offhand remember seeing the word “let” written by any Further Maths student this year (though, by ingrained habit, I suspect I use it a lot when talking to them and writing things out). Yet, as Gowers remarks, “the ‘let’ move” is a basic gambit in mathematical proof”. http://gowers.wordpress.com/2014/02/03/how-to-work-out-proofs-in-analysis-i/

So basic mathematical language is no longer covered in school maths. At a recent training event, one UCL lecturer who marks FP3 papers told me that he found that even the best students, even the ones who would get A grades, often knew nothing about how to set out and explain a mathematical conclusion.

They knew less about how to reason mathematically than my nine-year old classmates and I knew back in 1958. That they surely know more about hyperbolic functions, or how to do fiddly manipulations with trigonometric identities, is, I think, a poor exchange.