The Mending of Broken Bones by Paul Lockhart
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The Mending of Broken Bones:
A Modern Guide to Classical Algebra
By Paul Lockhart
The Belknap Press of Harvard University Press 2025
Paul Lockhart, a long-time mathematics instructor and for decades an algebra teacher at Saint Anne’s School in Brooklyn, opens his new book, The Mending of Broken Bones with an old problem – a very old problem, from a Babylonian stone tablet dating to roughly 1850 BC:
I found a stone but did not weigh it.
After I added one-seventh of its weight,
And one-eleventh of this new weight,
The total was one ma-na.
What was the original weight?
Readers breaking out in cold sweats at the mere sight of such a problem are probably flashing back to their school days, and they’ll take precious little consolation from the discovery that schoolchildren have been tormented in this way for millennia. Lockhart is quick to strike the Good Cop pose, sidling up to those trembling readers and telling them that he, too, finds the whole problem faintly silly:
The first thing I want to point out is the extreme artificiality of this question. At first glance, it might seem to be a math problem of the most mundane and practical sort, dealing as it does with stones and weights and all … But on closer consideration, we realized that the scenario depicted here is actually quite absurd. How would anyone come by this ridiculous information in the first place? You mean I somehow have the ability to determine the combined weights but not the weight of the stone I started with?
See? We’re all just laughing together over how silly that old problem is. But don’t be fooled: Lockhart is still a math teacher, perhaps the ur-math teacher, and that silly old Babylonian problem is still going to be on the final exam. And as Lockhart gently implies throughout his book but never condemns, the huge majority of adults in the world today wouldn’t have the faintest idea how to solve it.
The way to solve it is algebra, which Lockhart describes as “the fine art of tangling and untangling abstract numerical information." He posits his entire book (and his earlier book, A Mathematician’s Lament) on his belief in that woolly old academic fib, that math in all its variety is a language, a fun exercise, a “fine art.” Humans process and appreciate language in the left temporal lobe of their brain, whereas math and spatial issues tend to concentrate on the inferior parietal lobe. There’s some messy overlap, of course, but neurological science has nonetheless clarified what humans have known for thousands of years: mathematical aptitude, or the lack thereof, seems to a very large extent to be hard-wired in some humans and not in others. That should long since have put to silence this perennial nonsense about math being a language or a fan dance or a double feature at the Vista View over in Coralville, or whatever – and yet, the nonsense continues.
It would be flatly unjust to characterize Lockhart’s book as nonsense. He’s so clearly, almost painfully a true believer that this universal language can be learned, and for nearly 400 pages, he tries to teach it. Readers who’ve been feeling guilty for years about just how quickly they abandoned all those grueling algebra problems the instant they got out of school, how eagerly they fed those algebra textbooks to their beagles and went on to live lives of untroubled innumeracy, will probably never have a more congenial, patient teacher than the one they find in these pages.
But that teacher never really strays far from ancient Babylon. He proposes a hypothetical situation, explores it a bit, then chuckles and agrees it’s all a bit silly:
Here we can imagine, if we like, that w is walking down the street minding its own business, and all of a sudden it got multiplied by 7/4 and is now 10. How can we figure out who it was before this happened? The idea is to simply undo what was done. We started with w, did this thing to it, and now it’s 10. So the plan is to start with with 10 and then undo this thing, and that will get us back to w itself. To undo multiplication by 7/4, we simply multiply by its reciprocal, 4/7. So we find that w must be 4/7 x 10, otherwise known as 40/7. (If you like, we can rewrite this as 5 5/7, but it doesn’t really do all that much for you.) Just to be sure, let’s check that this number really does work: half of 40/7 is 20/7, and one-quarter of it is 10/7. Adding these together, we get 70/7, which does indeed equal 10.
“Of course, no algebraist then or now would care about this particular problem or its solution,” he’s quick to add. “What we do care about is the technique, the way we untied the knot.” And although readers might be tempted to quip, “If w is able to walk, isn’t w able to talk? Couldn’t we just ask w what happened?” they should concentrate and bear down. After all, it might be on the final.
Steve Donoghue is a founding editor of Open Letters Monthly. His book criticism has appeared in The Washington Post, The American Conservative, The Spectator, The Wall Street Journal, The National, and the Daily Star. He has written regularly for The Boston Globe, the Vineyard Gazette, and the Christian Science Monitor and is the Books editor of Georgia’s Big Canoe News