The Sad Song of a Mathematician

Thoughts on teaching and learning

education teaching mathematics reviews Rashid Mhar

The sad song of a mathematician

Rashid Mhar

A link arrived in my inbox to a video discussing mathematics education. The video, a bit over an hour long, was a presentation by Mathew Crawford, an American educational entrepreneur and mathematician. My friend (Angus, no surprise to readers of Northern Edge) who sent the link was interested in my review of Crawford's ideas. Mathew Crawford is an educational entrepreneur who has established a successful business in the US offering supplementary maths classes to high school students and adults. His approach is to restore the story and ‘art’ to maths class and get away from the forced memorisation of formulas and the onerous task of solving as many problems as possible under exam conditions. He instead focuses on solving a few complicated problems by giving time to discovering solutions using creativity, exploring mathematics as journey with other students, and sharing the story and ideas of fellow maths explorers from bygone eras.

The video can be found in The Education Wars Part XV, and in it Mathew presents his arguments and calls on his results to advocate for his approach. It’s an interesting presentation, though perhaps if you don’t have an interest in mathematics or principles of educational practice it might be an hour too long. I know you are thinking this is a strange introduction to video review, he’s basically saying “muhh, s’okay if you got a spare hour+ then watch it.” And you’d be right I am not interested in saying anymore about Mathew’s video or more about his volume of writing, beyond he has some interesting pieces and hasn’t shied away from controversy. In this case, however, I’d rather like to point you to the story behind the story.

It’s quite fitting that Mathew in his 2022 video points to a 20 year old story from Saint Ann’s School, Brooklyn: ‘The Mathematician’s Lament’ by Paul Lockhart. Paul Lockhart was a young maths enthusiast who followed his love of the subject into teaching then into research and finally back into teaching. On that journey he found himself growing deeply dissatisfied with the way the K-12 curriculums acted as a highly effective weapon for the destruction of any young person’s love or even interest in the ‘art’ of mathematics.

He starts his polemic lament with a dystopian satire of a world where music is taught like mathematics:

“A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory… Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.”

Paul passionately develops his case. He uses many tools of argument and rhetoric to blend criticism with bold and emphatic claims that mathematics is a creative art and needs to be handled as such. However instead of quoting Paul’s arguments I’ll quote his quote from G.H. Hardy, an English mathematician who worked on Number Theory and Mathematical Analysis:

“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

Hardy, in 1940, published the swansong of his career “A Mathematician’s Apology”. An essay that developed into a book where Hardy wished for a new generation to take up the challenge of creating new mathematics as he faced the decline of his mental powers and health. In this example of soul searching humanity from an analytical atheist we see the antecedents that lead to Crawford's half a million students. Though we remain in a system with the apparent aim of using mathematics to take all that encompasses our hearts, souls and passions and summarise it into a succinct formula for an AI asset to use as a tool for behaviour management of the formerly free.

Lockhart puts forward many examples of his distaste, to put it mildly, of the curriculum and the way it is (it’s a 2002 piece so perhaps was?) delivered:

“To help your students memorize formulas for the area and circumference of a circle, for example, you might invent this whole story about “Mr. C,” who drives around “Mrs. A” and tells her how nice his “two pies are” (C = 2πr) and how her “pies are square” (A = πr2 ) or some such nonsense. But what about the real story? The one about mankind’s struggle with the problem of measuring curves; about Eudoxus and Archimedes and the method of exhaustion; about the transcendence of pi? Which is more interesting— measuring the rough dimensions of a circular piece of graph paper, using a formula that someone handed you without explanation (and made you memorize and practice over and over) or hearing the story of one of the most beautiful, fascinating problems, and one of the most brilliant and powerful ideas in human history? We’re killing people’s interest in circles for god’s sake!”

I shan’t try to summarise Paul’s arguments, nor will I attempt to argue in favour or against them. It’s fair to say Paul wrote what he wrote as that is what he honestly felt was the state of maths education. His piece eventually struck the small but significant world of US maths education. For my part, I enjoy the very romanticism of his lone stance as a man against a systemic world, but as practice, I prefer the permacultural principle of search for small, slow, steady changes tempered by observation as we work towards a solution to a presenting problem.

In March 2008 Paul Lockhart’s essay was discovered and published by Keith Devlin of the Mathematical Association of America (; suffice to say that it’s now hard to find on their modern website. It created a stir, which a good polemic is prone to do. Lockhart responded to the criticism in May 2008:

“..what I have written is a Lament, not a Proposal. I am not advocating any particular plan of action; I am merely describing the extremely sad and painful (and probably hopeless) state of affairs as I see it: mathematicians are not interested in teaching children, and teachers are not interested in doing mathematics. If I am advocating anything, it is only the obvious (and time-tested) idea of "learning by doing." If I have a method, it is only to convey my love for my subject honestly, and to help inspire my students to engage in a delightful and fascinating adventure… Have we really reached a point where one has to argue for teaching that "awakens and stimulates students' natural curiosity?" As opposed to what? I thought that was the definition of teaching! I find it a bit frustrating that I am put in the position of having to defend such a simple and natural idea as having students engage in the actual practice of mathematics. Shouldn't it rather be the proponents of the current regime who should have to defend their bizarre system, and explain why they have chosen to eliminate from the classroom the actual ideas of the subject?”

Paul Lockhart has very strong and controversial notions on what a teacher is, as do I, which I would say makes me feel a certain empathy towards him. I find a certain joy in that, 20 years later, Mathew Crawford has made a successful business by taking Lockhart’s words to heart and building an approach to teaching mathematics that stands outside the US system by being offered directly to students.

If you would be so kind as to allow me, I’d like to recommend Paul’s original essay for your perusal: A Mathematician’s Lament. You may also enjoy or prefer to watch Toby Hendy’s short video on the essay, which is on her YouTube channel Tibees, here presented as an alternative link Tibee’s A Mathematician’s Lament. However she only explores the earlier and more poetical arguments from Paul’s metaphorical contentions.

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