Hofstadter had always been fascinated with how the mind works. He set out to teach himself Euclidean geometry as an experiment, a way of observing the process of intellectual reasoning and discovery. He used the humble triangle as his guinea pig. “I was so naive at the outset of my many-year binge with geometry that I was caught off guard by the fact that the triangle had more than one centre,” says Hofstatder. “I had heard the terms ‘orthocentre,’ or ‘centroid,’ and ‘circumcentre.’ I sort of knew these things existed, but to tell the truth, when I found out that any triangle has several centres, I was really thrown. It seemed like a miracle. In point of fact, any triangle has an infinite number of centres, but they are not all equally interesting.”

“One time,” Hofstadter continues, “I made an analogy to the human body. Suppose I were to ask various people, ‘What is the most important part of your body?’ One person might reply, ‘It’s my brain.’ Somebody else might say, ‘No, it’s my stomach;’ somebody else might insist ‘my heart,’ ‘my sexual organs,’ or ‘my belly button’....I became inflamed and impassioned in trying to invent new geometrical notions. And I saw that the process of analogy-making lay at the core of all my discoveries. I started noticing beautiful patterns not only in triangles and among triangle centres, but there were exquisite patterns at the level of the ideas themselves.”

Following his triangle experiment, Hofstadter returned to math with the intention of reclaiming it. “I came back to group theory and Galois theory some thirty-five years later and I said, ‘By god, I’m going to make this hateful stuff lovable.’ ” He started a course called Group Theory and Galois Theory Visualized, though the running joke is that his approach is more like Pizza Theory. The visual metaphor Hofstadter uses to explain difficult group theory theorems such as Zassenhaus’ lemma is simple circles with slices and concentric circles within, which look a lot like pizzas. (It didn’t hurt either that he always brought pizzas to class.)

Hofstadter’s overriding conclusion from these intellectual experiments is that analogy-making plays a crucial role in doing quality mathematics—analogies are what allow the illustration and conceptualization and understanding of otherwise seemingly unfathomable ideas.

Of course, one could say math phobia has its societal upside. Like computer phobia or ignorance about the inner workings of an automobile engine, it has spawned expert classes—for example, accountants, who will happily do math for you. But if you disavow the paralyzing assumption that math is merely juggling numbers and puzzlingly hard, it is not so difficult to take charge of one’s repressed mathematical alter ego and have some fun.

John Mighton takes an open-minded approach with his Junior Undiscovered Math Prodigies (jump) teaching method. Mighton once struggled with math, then did a Ph.D. in it. He is currently a fellow at the The Fields Institute, a mathematical think tank in Toronto, and his book, The Myth of Ability, is attracting international recognition. jump fosters a can-do attitude by breaking down difficult material into manageable bites, and often tutors roam the classroom providing one-on-one attention and good old Pavlovian positive feedback: Be nice, never put an X on a page, and kids will beg for more. I was a jump tutor for a year and, while Mighton’s prescriptions remain controversial in some quarters, I witnessed students experiencing unparalleled personal mathematical successes. They were lapping it up, their hands shooting into the air to answer questions as if they were requesting an urgent bathroom visit.

Oxford’s Anne Watson holds that math is difficult but that it need not be fear-inducing. Says Watson, “I believe that every mind (except those which have specific damage of some kind) is capable of doing the kind of thinking that enables them to create mathematical ideas by making sense for themselves of a range of experiences, and I start from there. Thus good learning of mathematics includes arguments, discussions, creating ideas, testing things out, conjecturing what might or might not be true—in other words thinking hard about abstract concepts.”

Going a step further, John Conway, the widely respected John Von Neumann professor of mathematics at Princeton University, offers an irresistible and fun-inducing approach to mathematics. He’s part of a species of mathematicians more aptly characterized as “mathemagicians,” and one of his sleight-of-hand tricks involves spinning a penny on the tip of a coat hanger (whirling it over his head like a helicopter rotor) and bringing it to a graceful swooping stop—a trick he performs at kids’ math camps every summer. “It’s a mistake to assume that what mathematicians do is esoteric, deep, and difficult,” he told me, sitting in his “office,” the math department’s common room at Princeton. “All the great discoveries are very simple. Like Coxeter’s and Einstein’s.” Albert Einstein, another Princetonian, once stated in a lecture: “Describing the physical laws without reference to geometry is similar to describing our thoughts without words.”

Shortly after Conway set me straight on the beauty and simplicity of sleightof- hand mathematics, he was off to a reception of high-school-cum-university students who were keenly anticipating settling into classrooms this fall with math as their chosen path of higher learning.