In his autobiography Charles Darwin recalls intending to study mathematics at Cambridge only to discover that “the work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra” (presumably not helped by the “very dull man” who was his summer tutor). Winston Churchill, too, found mathematics a “heavy journey.” “When I plunged in, I was soon out of my depth,” he writes in My Early Life (1930). “We were arrived in an ‘Alice-in-Wonderland’ world, at the portals of which stood ‘A Quadratic Equation’”—that classic algebraic proving ground—beyond which were “further dim chambers lighted by sullen, sulphurous fires…reputed to contain a dragon called the ‘Differential Calculus.’” He credits a teacher at his boarding school with convincing him that mathematics “was not a hopeless bog of nonsense, and that there were meanings and rhythms behind the comical hieroglyphics; and that I was not incapable of catching glimpses of some of these.”

For many, the mention of algebra summons only unhappy memories of confused encounters with inscrutable equations. Such experiences underscore the challenge Paul Lockhart has posed for himself in TheMending of Broken Bones: A Modern Guide to Classical Algebra, a book that invites the general reader to gain an appreciation of—and perhaps even actively engage with—a centuries-old mathematical practice often called “symbol manipulation.” The title nods to the etymological origin of “algebra,” the Arabic word al-jabr, meaning “completion” or “mending”—as in the setting of broken bones. But Lockhart’s gesture is more than linguistic; he is well aware that there may also be plenty of emotional fractures in need of repair.

Lockhart, a mathematician who taught first at Brown University and UC Santa Cruz and then for many years at Saint Ann’s, a progressive private school in Brooklyn, argues that the injury is due to our ossified K–12 mathematics curriculum. That’s where we encounter algebra as the bread in a difficult-to-digest “algebra-geometry-algebra sandwich,” commonly delivered in grades eight through ten. Mending is Lockhart’s effort to show that it’s not just empty carbs.

TheMending of Broken Bones is Lockhart’s fourth book aimed at reeducating the public about mathematics. Throughout his writing it is clear that as much as he loves doing math, he may love even more getting others to love doing math. Indeed, “doing” is the operative word. Lockhart’s goal is to enliven appreciation through direct engagement with mathematical ideas. While many have a utilitarian view of the subject as a “tool for science and technology,” and competency has been in steady decline in the United States for more than two decades, quotidian concerns are not what have spurred him to his proselytizing. Rather, “the glory of [math] is its complete irrelevance to our lives.” Lockhart wants to make the case, urgently, loudly, and unapologetically, that “mathematics is the purest of the arts, as well as the most misunderstood.” Moreover,

there is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun.

These exhortations come fromLockhart’s first book, A Mathematician’s Lament (2009), which began as a manifesto that circulated samizdat-style around the mathematics and education communities in the early 2000s. Its thesis is that the standard pedagogy is a “nightmare” of “senseless, soul-crushing ideas” and that a “rich and fascinating adventure of the imagination has been reduced to a sterile set of facts to be memorized and procedures to be followed.” For Lockhart, math should involve “wondering, playing, amusing yourself with your imagination.” As he points out, children already know that “learning and playing are the same thing,” and when taught in this spirit, math can become a lifelong, joyful endeavor.

A Mathematician’s Lament is—in true manifesto form—a mixture of sarcasm, exhortation, irony, self-deprecation, poetry, and a little bit of anger. It also contains some opportunities to have fun with mathematical ideas. Lockhart gives new life to the well-worn formula for computing the area of a triangle, infusing it with the excitement of an epiphany. The fact that two odd numbers always add up to an even number is, he coaxes the reader to see, as simple as sliding one arrangement of pebbles into another:

You can almost hear the “Aha!” as they fit together.

Lockhart’s new book on algebra follows earlier offerings on arithmetic and geometry—both subjects that can feel, at least early on, like child’s play. Arithmetic (2017) frames the study of whole numbers and their basic operations (addition, subtraction, multiplication, division) as a natural investigation of the patterns that occur when arranging objects like the pebbles above, and of the various symbolic languages—from notched sticks to numerals—that have evolved for their representation and communication. The reader will gain a deeper appreciation for the utility of place value notation, as in our familiar decimal system, built on powers of ten, not to mention the landmark invention of the numeral “0,” both of Hindu-Arabic origins. The last chapter serves as an invitation to more elaborate kinds of counting, like the number of different possible poker hands or ways to seat guests around a dinner table, problems that may seem like smart-alecky brainteasers but actually form the foundation of modern probability.

Measurement (2012) is a gateway to a “universe where beautiful shapes and patterns float by and do curious and surprising things.” Intended as an antidote to the traditional geometry curriculum, it starts with triangles and ends with the geometry of space-time and connections between motion and shape. There are equations, but there are also plenty of hand-drawn illustrations. Sprinkled throughout are recreational rest stops where the reader is gently nudged to think (play) further on their own with a new idea or example. The area of a triangle relates to the area of the rectangle in which it is inscribed, but what about rectangles circumscribed by circles, or boxes within spheres? It’s best to keep a pencil and some paper handy.

These questions are intended as opportunities for exploration rather than assignments, and some may not admit the satisfaction of resolution. More and more—perhaps too much?—is asked of the reader as the book goes on. Lockhart is working to nudge the reader to think like a mathematician, and with that come the highs and lows of the creative process. “I’m not going to apologize for how hard it is,” he writes. “You will declare yourself a genius at breakfast and an idiot at lunch. We’ve all done it.” At times one gets the sense of a passionate teacher who lets his frustrations get the better of him. But Lockhart often does this in an effort to persuade the reader that the struggle of the mathematician is the same as that of any artist, and very much worth enduring.

So what about algebra? It is, says Lockhart, “the fine art of tangling and untangling abstract numerical information.” “Abstract” is the key word here. We’re no longer just solving the everyday problems of “the construction site and the counting house.” Instead, we are

driven by curiosity and aesthetics. We leave the noisy and complicated world of physical reality behind and move to a quieter, more peaceful realm of abstract pattern and ideal beauty—a place I like to call Mathematical Reality.

Well, eventually. Most of us are first introduced to algebra in the service of solving word problems. These generally take the form of “I’m thinking of a number such that when I do some stuff to it, it’s the same as if I do some other stuff to it. What is the number?” Sometimes we dress it up as a story. Maybe we’re asked to figure out the age of someone’s uncle or the amount of change in a friend’s pocket. In 1850 BCE, in a word problem written on a Babylonian cuneiform tablet, it was the weight of a mysterious stone:

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.¹ What was the original weight?

The discovery of general methods to solve such problems—to “reassemble” a number that has been “shattered into pieces,” as Lockhart puts it—was an extraordinary achievement. Algebraseems to have made its first appearance as a mathematical idea in one of the most important mathematics texts of all time, al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala, or The Compendious Book on Calculation by Completion and Balancing, put together around 813–833 by the Persian mathematician al-Khwarizmi. (Whispers of his name live on in the word “algorithm,” used to describe recipe-like computational processes that, like those in al-Khwarizmi’s book, are at times designed to unravel equations.) It was first translated into English in 1831 by the German Orientalist Friedrich Rosen, and it is full of rhetorical puzzles and explanations such as this:

If a person puts such a question to you as: “I have divided ten into two parts, and multiplying one of these by the other, the result was twenty-one;” then you know that one of the two parts is thing, and the other ten minus thing. Multiply, therefore, thing by ten minus thing; then you have ten things minus a square, which is equal to twenty-one.

“Thing” more familiarly goes by the label x, which came into use centuries later, as did other symbolic shorthands like + and = and the superscript “2” to denote a square. In modern algebraic language, what al-Khwarizmi tells us is that x(10–x) = 21, but also x(10–x) = 10x–x2 (what you get when you multiply “thing” by “ten minus thing”), so that 10x–x2 = 21. In order to reassemble the “shattered” number in this equation—thereby “mending” it—algebra uses the idea of “balancing.” The most familiar balancing act, adding to both sides, is just one of many clever ways to rewrite equations—or tell stories anew—so as to bring the mysterious “thing” to light. When we identify “thing” as a number, we call it a “root.” In this example there are two roots, 3 and 7, meaning that if you replace x with either of those numbers (the infamous act of “plugging in”) and do the arithmetic you will find that the two sides of the original equation are equal—the number 21 has been “mended.” QED.

Until the nineteenth century algebraists were mainly arborists, pruning equations and extracting roots. Their most interesting discoveries were procedures that work in general cases, independent of the specifics of the story. Al-Khwarizmi came up with a few of them, the most famous of which is the quadratic formula, which could be used to get past Churchill’s gatekeeper and produced roots for equations involving squares. More elaborate formulas were discovered for cubics and quartics, or those “things” whose defining stories involve third and fourth powers.

In the sixteenth century finding such formulas was both a badge of honor and a source of income, as mathematicians sometimes engaged in public problem-solving competitions. There was a rule to this game: you were allowed to use only the numbers in the equations as well as a handful of operations in your formulas, the most complicated being the use of square roots, cube roots, and the like—also called “radicals” (a word related to “root”). A formula for roots constrained in this way produces a “solution by radicals.” Deriving such a process meant producing a formula of symbolic simplicity, but with very general applicability. As Lockhart says, “It’s hard to believe that this was not the ultimate goal of the ancients as well.” It is also an example of the minimalist aesthetic that guides mathematical artistry.

The great leap to modern algebra occurred when mathematicians began to suspect that a general solution by radicals for any fifth-degree equation (the quintic) was beyond reach. If true, this would show the limitations of the method and thus point to a need for new ideas. This is a theme common in the sciences. When classical physics couldn’t explain the results of certain experiments, mathematicians and physicists invented the language of quantum physics and searched for a new set of physical laws. In biology, the fact that inherited traits did not “blend” but combined suggested the need for discrete units of inheritance and led to the discovery of genes.

When it came to trying—and failing—to find a solution by radicals for the quintic, new algebraic ideas burst into existence. In 1830 the French mathematician Évariste Galois, freshly expelled from university and recently jailed for treason, announced his discovery that the key to solvability was the relationship between roots, something implicit in the fact that each was a solution to the same story—any of them could be the “thing.” He began to see the roots as equivalent to the story—the mathematical expression that defined them. As legend has it, he wrote down this insight and its ramifications in a furious burst of genius the night before he died in a duel, only twenty years of age.

By Galois’s era the notion of “number” had already evolved far from its natural number origins. The fossil record of this disruption is in the nomenclature: “natural” numbers give rise to “rational” numbers (fractions) and “irrational” numbers, which cannot be represented as fractions, such as the infamous square root of two—its discovery in ancient Greece was so upsetting that it was said to have caused violence among the philosophers. The rationals and the irrationals are grouped together as the “real” numbers, which are associated with measurements of material objects. They stand in contrast to “complex” numbers, which to the uninitiated might seem to make reference to confusion—but the name derives from the fact that these numbers are best thought of as a “complex,” or a grouping of a pair of real numbers.

The best known of the complex numbers is the square root of minus one, famously denoted as i, for “imaginary.” It was summoned into existence as “the thing that when squared is minus one”—surely a product of the imagination. The mathematician Barry Mazur, writing about the discovery of i in his little gem of a book Imagining Numbers (Particularly the Square Root of Minus Fifteen) (2003), frames this kind of leap as a granting of “permission.” He recounts a story of Gabriel García Márquez falling off his couch in shock after reading the first few sentences of Kafka’s The Metamorphosis. “I did not know that such a thing could be done!” he is said to have uttered. Such permissions extend across the arts (Arnold Schoenberg gave permission to make atonal music; Marcel Duchamp gave permission to treat everyday items like sacred relics), and also to mathematics. Galois and others did the same with symbolic expressions, which use letters or other symbols in place of numbers. With that, more mathematics was created, and other permissions were granted, both large and small.

Galois realized that the important thing was to focus on the relations among roots and not on whether you could actually express them in terms of known quantities. He saw that the roots of an equation had a kind of symmetry. For example, i could just as well be -i, since any number and its negative both square to the same number. Inventing i implicitly brings -i with it, an algebraically indistinguishable mirror image—akin to flipping your mattress, a transformation that leaves your bed and bedroom unchanged. Symmetries in nature could also be encoded as symmetries in systems of equations, and with them, new natural features emerged as solutions—equations describing the electron, for instance, suggested the existence of a positron before it was experimentally observed.

The study of symmetries would by the late nineteenth century turn into what we now think of as modern group theory, which quickly dispensed with the concerns of numbers for a more general approach. One could, for example, consider the points that make up the surface of a cue ball, which can be rotated—or rearranged under the symmetry of rotation—while leaving the cue ball intact. Abstract group theory is about symbols that interrelate according to a small number of terse axioms. The extraordinary complexity that evolves from this simple framework is astounding.² Lockhart tells us that despite its long track record of scientific utility (especially in physics and chemistry), the development of group theory, as with other kinds of pure mathematics, was driven more by aesthetics than applicability:

It’s not so much that there are rules…as there are creative decisions that either do or do not please us as mathematical artists. If you can create something lively and interesting—or even (Lord help us) useful—then your work will be appreciated and valued.

Groups are just a start. Toss in multiplication and you get “rings,” toss in division and you get “fields,” and each of these structures has its own complexities. More generally, the study of abstract algebra focuses on the nature of relationships in made-up worlds of symbols that passed tests of depth and elegance. And so while the study of groups, rings, fields, and their variations might seem like just working through the implications manifest in some simple rule-based symbol manipulation, that’s as reductionist as saying that poetry, painting, or music is just moving around words, dabs of color, or fragments of sound.

From pedestrian beginnings, algebra rises to dizzying heights, and admittedly it can get complicated. The struggle ensues. But for those willing to take it on, there is much to be appreciated in The Mending of Broken Bones. A mathematician myself, even I learned something new from some of Lockhart’s detailed examples, including a deep technical digression in his discussion of elimination theory—the beautiful but complicated mathematics created to uncover simple representational forms of certain kinds of algebraic expressions. A math-curious and forgiving reader might skim those sections rather than close the book; for those who have some background, even if they are just at the beginning of wondering where math can go, The Mending of Broken Bones will be a worthwhile and energizing read. The partially initiated may find themselves thinking about a life of math. The professional may be reminded of what drew them to the subject in the first place.

I’d encourage those who are heavy on the curiosity and perhaps lighter on the background to start with Arithmetic, work their way up through Measurement,and then move on to Mending. (A Mathematician’s Lament might serve only to rile you up.) If you’ve ever had a love of playing with blocks or drawing shapes or collecting pretty much anything, there is a latent love of mathematics inside you waiting to be kindled or rekindled. Even Darwin was able to concede that his early “impatience” was “foolish” and that it cost him access to the “extra sense” possessed by mathematicians.

Lockhart believes to his core that each of us is capable of that experience of appreciation, and like Churchill’s anonymous teacher, he wants each of us to be able to catch “a glimpse of something beautiful and pure, a harmless and joyful activity that has brought untold delight to many people for thousands of years.” As machines do more and more of the grunt work in our age of artificial intelligence, a deeper appreciation of mathematics’ intellectual artistry, and the opportunity to try it out, might be all we have left. Lockhart offers us a road to that end. Let the mending begin.