Nothing comes easily to me.

I’m a mathematician, but I didn’t show much aptitude for math until I was thirty. I had no idea, in high school, why I had to turn a fraction upside down when I wanted to divide by it, or why, when I wrote a square root sign over a negative number, the number suddenly became “imaginary” (especially when I could see the number was still there). At university I almost failed my first calculus course. Fortunately I was saved by the bell curve, which brought my original mark up to a C minus.

I’m also a playwright. My plays have been performed in many countries, but I still won’t read a review unless someone tells me it’s safe to do so. Early in my career I made the mistake of checking the papers to see what two of the local critics thought of my first major production. It seems unlikely that they consulted each other before writing their reviews, but one headline read “Hopelessly Muddled” and the other “Muddled Mess.”

I often wish I was more like my literary and scientific heroes, who seemingly could produce perfect poems or solve intractable problems in a blinding flash of inspiration. Now that I’m a professional mathematician and writer, I console myself with the thought that my ongoing struggles to educate myself and the strenuous efforts that I needed to make to get to this point have produced an intense curiosity about how we achieve our potential.

A Slow Learner

From an early age I became obsessed with my intellectual capabilities and with the way I learn. When I started to teach in my twenties, first as a graduate student in philosophy and later as a math tutor, I also became fascinated with the way other people learn. Now, after teaching math and other subjects to thousands of students of all ages and after reading a great deal of educational and psychological research, I am convinced that our society vastly underestimates the intellectual potential of children and adults.

During my undergraduate studies, I showed as little promise in writing as I did in mathematics: I received a B plus in my creative writing class—the lowest mark in the class. One evening, in the first year of my graduate studies in philosophy, I began reading a book of letters by the poet Sylvia Plath, which I’d found on my sister’s bookshelf while babysitting her children. It appeared from Plath’s letters and early poems that she had taught herself to write by sheer determination. She had learned, as a teenager, everything she could about poetic metre and form. She wrote sonnets and sestinas, memorized the thesaurus and read mythology. She also produced dozens of imitations of poems she loved.

I knew that Plath was considered to be one of the most original poets of her time, so I was surprised to learn that she had taught herself to write by a process that seemed so mechanical and uninspired. I’d grown up thinking that if a person was born to be a writer or mathematician, then fully formed and profoundly important sentences or equations would simply pour out of them. I’d spent many hours sitting in front of blank pages waiting for something interesting to appear, but nothing ever did. After reading Plath’s letters, I began to hope that there might be a path I could follow to develop a voice of my own.

I imitated the work of Plath and other poets for several years before I moved on to writing plays. By that time, I’d taken a job at a tutoring agency to supplement my income from writing. The women who owned the agency hired me to tutor math because I’d taken a course in calculus at university (and I neglected to tell them about my marks). In my tutorials I had the opportunity to work through the same topics and problems again and again with my students, who ranged in age from six to sixteen. The concepts that had mystified me as a teenager (such as why does a negative times a negative equal a positive) gradually became clear, and my confidence grew as I found I could learn new material more quickly.

One of my first students was a shy eleven-year-old boy named Andrew, who struggled in math. In grade six, Andrew was placed in a remedial class. His new teacher warned his mother that she shouldn’t expect much from her son because he was too intellectually challenged to learn math in a regular math class. In the first two years of our tutorials, Andrew’s confidence grew steadily, and by grade eight, he had transferred to the academic stream in math. I tutored him until he was in grade twelve, but I lost touch with him until recently when he invited me to lunch. In the middle of our lunch, Andrew told me that he had just been granted full tenure as a professor of mathematics.

When I was growing up I would always compare myself to the students who did well on math competitions and who seemed to learn new concepts without effort. Watching these students race ahead of me at school made me think that I lacked the natural gift I needed to be good at the subject. But now, at the age of thirty, I was surprised to see how quickly I could learn the concepts I was teaching, and how easy it was for students like Andrew, who had never shown any signs of having a “gift” for math, to excel at the subject with patient teaching. I began to suspect that a root cause of many individuals’ troubles in math, and in other subjects as well, is the belief in natural talents and natural academic hierarchies.

As early as kindergarten, children start to compare themselves to their peers and to identify some as talented or “smart” in various subjects. Children who decide that they are not talented will often stop paying attention or making an effort to do well (as I did in school). This problem is likely to compound itself more quickly in math than in other subjects, because when you miss a step in math it is usually impossible to understand what comes next. The cycle is vicious: the more a person fails, the more their negative view of their abilities is reinforced, and the less efficiently they learn. I will argue that the belief in natural hierarchies is far more instrumental in causing people’s different levels of success in math and other subjects than are inborn or natural abilities.

In my early thirties, I returned to university to study mathematics (starting at an undergraduate level) and was eventually granted one of Canada’s highest post-doctoral fellowships for my research in the subject. In the meantime, I’d also received several national literary awards for my plays, including a Governor General’s Award. I don’t believe I will ever produce work that compares to that of my artistic and intellectual heroes, but my experience suggests that the methods I used to train myself as a writer and mathematician—which included deliberate practice, imitation, and various strategies for mastering complex concepts and enhancing the imagination—could help people improve their abilities in the arts and sciences.

When I was taking my degree in math, I often wondered how my life would have gone if I’d selected a different book from my sister’s bookshelf the night I discovered Plath’s letters. I felt lucky to have regained the passion I’d had as a child for creating and discovering new things and lucky to have been encouraged to follow my passion by my parents and family. Watching my students become more engaged and successful in math, I began to feel that I should do something to help people who’d lost faith in their abilities, so they could regain their confidence and keep their sense of wonder and curiosity alive.

In the final year of my doctoral program, I persuaded some of my friends to start a free, after-school tutoring program called JUMP (Junior Undiscovered Math Prodigies) Math in my apartment. Twenty years later, 200,000 students and educators in North America use JUMP as their main math instruction resource, and the program is spreading into Europe and South America. Its methodologies have been developed in consultation with and guided by the work of distinguished cognitive scientists, psychologists and educational researchers, many of whom you will meet in this book. These methods are easy to understand and apply, and they reinforce confidence in your abilities rather than assigning you to a particular skill level. They can be used by adults who want to help children learn any subject more efficiently or who want to educate themselves and pursue a new path in life, the way I did.

Before I describe these methods and the research that supports them, I will look more closely at some myths about intelligence and talent that prevent us from fully developing our intellectual abilities and that create an extraordinary range of problems for our society. Because people have so much trouble imagining that they could be good at subjects they struggled to learn in school, they also have trouble imagining what the average brain can accomplish or understanding the magnitude of the losses our society incurs when we fail to educate people according to their potential. This failure of the imagination creates a self-fulfilling cycle of frustration and lost opportunities for many people; to escape from this cycle we need to re-examine our most basic beliefs about what it means for people to be “equal” or to have equal opportunities in life.

Invisible Problems

Every society is plagued by invisible problems that are particularly hard to solve—for no reason other than because they are invisible. Sometimes a society has to collapse before the problems that stopped it from progressing can be seen. And sometimes this process can take centuries.

The ancient Greeks were remarkable innovators. They established the first democracies and produced a staggering number of mathematical and scientific breakthroughs. But this great, progressive society was hobbled by an insidious problem they could not see. Even the most enlightened thinkers of 400 BC were convinced that women were inferior to men and that slavery was as good for slaves as it was for their owners. Aristotle wrote, rather chillingly, that some people are born to be masters, while others are only fit to be “living tools.” The Greeks couldn’t begin to solve the most serious problems of their time because they couldn’t conceive of a more equitable society.

Over the past three hundred years, the idea that every person is born with the same inalienable rights and privileges, regardless of their race, gender or social status, has slowly taken hold across the world. In theory, in most nations, we have all been granted these same rights.

In practice, however, these rights are not always upheld in the same way for every person. And in many parts of the world, the impact of these rights on people’s quality of life is still rather limited. Even in Western democracies, people who are born with the same inalienable right to vote don’t necessarily enjoy the same social or economic opportunities.

Half of the world’s wealth is owned by 1 percent of the population, and tens of millions of people still don’t have enough to eat or proper access to health care or sanitation, even in the developed world. We are confronted by an array of threats—including economic instability, climate change, sectarian violence and political corruption—that have a greater impact on the poorest and most disadvantaged people of the world. In such a world, it’s hard to imagine a society in which people are born “equal” in any material sense or in which they can exercise their basic legal and political rights in the same way.

The laws and constitutions created to give everyone a fair chance in life have only partially succeeded in levelling the playing field. That’s because the most serious disparities in our society are not simply the result of legal or political inequalities but are also caused by a more subtle and pervasive form of inequality that is difficult to see. This kind of inequality might seem to be a by-product of social and political forces or of the deficiencies of capitalism, but I believe it is primarily caused by our ignorance about human potential. In the developed world, this inequality can affect the children of the rich as much as it does the children of the poor (although wealth does help mitigate its consequences). In many ways, it is the root cause of other inequalities. I call this kind of inequality “intellectual inequality.” And I will argue that it can easily be eradicated, particularly in the sciences and mathematics.

In this book I will sometimes use examples from JUMP Math to illustrate various principles of learning and teaching. But this is not a book about JUMP. My claims about human potential and the methods of teaching and learning that can unlock that potential are backed by a large body of research in cognitive science and psychology that is independent of JUMP. One day this research will be more widely known, and we will all be compelled to set much higher expectations in mathematics and other subjects for ourselves and our children, whether or not we use any particular math learning program. When we have understood and absorbed the full meaning of this research, our present beliefs about our intellectual abilities will seem as antiquated and as harmful as the belief that some people are born to be slaves and others masters. And the problems we have struggled to overcome since antiquity—which originate in our failure to foster intellectual equality—may finally be addressed.