For many years a 25 page essay written by Paul Lockhart has been circulating on the web. It cam to prominence in 2008 when Keith Devlin featured it in his monthly column at MAA online. Devlin said there, and repeats in the foreword to this book:
[“K-12” means primary and secondary education in the US.]
I read the essay. It is a true lament, that is, a “passionate expression of grief”, of how an intrinsically beautiful subject has been so badly mistaught.
It has been dulled down to mere regurgitation of formulae, with no understanding of the essential artistic process of discovery that leads to those formulae. Worse still, in geometry the discovery process that leads to elegant proofs has been replaced by a regurgitation, of ugly proofs.
Having learned that there was an extended book version, I immediately went and bought it. The book version has two parts. Part I, Lament, is the original essay. Part II, Exultation, is a constructive account of how mathematics might be taught to bring out the beauty, the artistry, of the subject.
I was one of the lucky ones who was good at maths at school. But I didn't become passionate about it until the age of about 14, when a new teacher arrived, who was passionate about the subject himself. I think he was the first teacher that I had ever had who was actually passionate about his subject. He loaned me a "popular maths" book, and from then on, I was riveted (very many thanks, Colin Chapman). That book gave me a first glimpse of the artistry and beauty behind the school formulas. I was also lucky enough to study the School Mathematics Project (SMP) curriculum, which emphasised understanding over regurgitation and meaningless manipulations (which some current curricula have still not incorporated, disappointingly). At the time we were exasperated by some of its heavy-handedness, but in retrospect, I realise that it was an excellent introduction to mathematical thinking. Even so, it was nothing like what Lockhart is advocating.
His critique is savage, passionate, devastating, thought-provoking. Read it.
In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.
Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience first-hand the playful excitement of mathematical work.