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Paul Lockhart.
A Mathematician's Lament.
Bellevue Literary Press. 2009

rating : 2 : great stuff
review : 30 September 2013

Paul Lockhart, a brilliant research mathematician who has devoted his career to teaching kids, reveals math to be creative and beautiful and rejects standard anxiety-provoking teaching methods. Witty and accessible, his controversial approach will provoke spirited debate among educators and parents alike and alter the way we think about math forever.

For many years a 25 page essay written by Paul Lockhart has been circulating on the web. It came 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:

p10. It is, quite frankly, one of the best critiques of current K-12 mathematics education I have ever seen.

[“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.

p35.
SIMPLICIO: But don’t you think that if math class were made more like art class that a lot of kids just wouldn’t learn anything?
SALVIATI: They’re not learning anything now! Better to not have math classes at all than to do what is currently being done. At least some people might have a chance to discover something beautiful on their own.
SIMPLICIO: So you would remove mathematics from the school curriculum?
SALVIATI: The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas.

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 Worse still, in geometry the discovery process that leads to elegant proofs has been replaced by a regurgitation, of ugly proofs.

p68. All metaphor aside, geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. (Apparently I am incapable of putting all metaphor aside.)

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.

Paul Lockhart.
Measurement.
Harvard University Press. 2012

For seven years, Paul Lockhart’s A Mathematician’s Lament enjoyed a samizdat-style popularity in the mathematics underground, before demand prompted its 2009 publication to even wider applause and debate. An impassioned critique of K–12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living.

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.

Paul Lockhart.
Arithmetic.
Harvard University Press. 2017

Because we have ten fingers, grouping by ten seems natural, but twelve would be better for divisibility, and eight is well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages.

Paul Lockhart presents arithmetic not as rote manipulation of numbers—a practical if mundane branch of knowledge best suited for filling out tax forms—but as a fascinating, sometimes surprising intellectual craft that arises from our desire to add, divide, and multiply important things. Passionate and entertaining, Arithmetic invites us to experience the beauty of mathematics through the eyes of a beguiling teacher.