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All Things Being Equal

All Things Being Equal

Why Math Is the Key to a Better World
also available: Hardcover
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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.

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How to Age Better and Feel Better about Ageing
also available: Hardcover
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An Alphabet for Joanna

Many months before Joanna’s short escape, I stayed at a friend’s empty house in Buffalo, and visited my mother at the nursing home every day for a week. Each day I dropped down deeper inside her world. On my second visit, we retreated to her shared room. Her roommate wasn’t there. Beside Joanna’s single bed stood the last surviving piece of the bedroom set she’d inherited from her mother, an antique dresser topped with a bevelled-edged mirror in a curving walnut frame. I’d arranged for this to be used instead of the bland dresser issued with the room. There was a brass keyhole in the top drawer, but if there had ever been a corresponding key, it had been lost years ago.
Joanna sat on her bed and examined a wall-mounted fluorescent light fixture, running her fingertips over the bubbled texture of its yellowed plastic shade. She gestured toward it, invited me to appreciate its enigmatic power. “I like this,” she said.
I faced her in a chair I’d dragged in from the TV area across from the elevators. I’d positioned a meal tray so that it was between us, an improvised work space. I’d covered its surface with an array of coloured markers and two pages torn from a sketchbook.
When I’d visited the day before, there hadn’t been any photos on her side of the room, but now I noticed she had found and propped up three pictures on her dresser. There was a framed photo of my son, Levi, that I’d given her for Christmas four years earlier, when he was a newborn, and there were two loose snapshots. One was of the base of the Eiffel Tower, which Joanna had taken during our weekend trip to Paris together in 1994. The other photo was of me.
I picked up this last picture and studied it. I’m sitting by myself on the blue-and-green floral couch in our living room in suburban Detroit. I’m probably fifteen years old. I’m wearing a baggy acrylic sweater and my hair is pulled back against my head on the sides, a big puff of curled bangs clawing at my eyebrows. Everything in this photo now looked ugly to me: my clothes, the pink walls behind me, the purple calico print tablecloth on the side table, that awful couch, the ruffled muslin curtains my grandmother had made for us, just like the ones in her own house.
Joanna leaned forward, her soft, plump arm touching mine, and she too looked at the photo in my hands.
“You’re my beautiful baby,” she said. I kissed her cheek.
The room was dark. I crossed to the window over her roommate’s bed and pulled the fraying cord to raise the blinds. I sat down on the edge of the mattress and looked out through a tangle of bare tree branches to the street below.
Joanna had followed me over, and she stood behind me as we looked out the window silently for a moment. Then she pointed across the street and said, “You see that thing there . . . ”
“I’m not sure what you’re pointing at, the houses or the cars parked on the street?”
Her dark brows drew together. “I don’t know,” she said, and turned away from the window.
We settled back on her side of the room, sitting again with the meal tray I’d set up between us. I used the little speaker on my phone to stream the Beatles record she owned when I was little, back when we lived with my grandparents. It was the only record she had salvaged from the two years she lived in California before I was born.
“What colour marker do you want to use first?” I asked her. She hesitated and then pointed to purple, looking back up at me for reassurance. “Oh, that’s a great choice,” I said.
My mother held the marker awkwardly, looking down at it uncertainly.
“Here, I’m going to draw a circle on the page like this, and you can colour it in,” I said as I drew a wobbly round blob. “Oh, you’re doing such a good job,” she said.
After some hesitation, she slowly started to make short purple strokes along the inside of the circle. I continued to praise her as I drew a cluster of triangles and dots on my own page. After a while, she stopped moving her hand to watch mine.
“Yours is so beautiful,” she said.
“So is yours,” I told her, but I couldn’t redirect her attention back to her own page. “Here, we’ll draw this together,” I told her, putting my own drawing away in my bag.
My phone played one of the Beatles’ many hits from the year she went to see the band play Olympia Stadium in Detroit with her girlfriends. That was the year she turned fifteen. She’d told me about that show, how she couldn’t hear a single note they sang, the music drowned out by the sounds of the girls screaming around her. We sang along—“She loves you and you know that can’t be bad”—as our heads bent toward each other over the purple ring Joanna had made in the centre of the page. I added a black dot in the middle of the ring, then drew round petals around the perimeter, making a psychedelic flower with a cartoonish eye at its centre.
“Oh, that’s nice,” she said.
I continued to draw on her page, asking her to choose colours for me. “We’ll do this together,” I said again. “We’re collaborating.”
I adjusted my sense of time as we sang and I drew, listening to the whole double album with the door shut. We dropped out of time and space. We were the only two people in the world. We were alone at the bottom of an ocean.
Someone knocked and the door opened. It was John, a man I’d met the day before in front of the nurses’ station, beside the TV. He had told me he loved me as he shook my hand. He had the red face, bright-blue eyes and silver brush cut I associate with Midwestern football coaches. Like Joanna, he was wearing loose-fitting elastic-waist pants and fuzzy socks with no-slip strips. No shoes.
“Oh, hi!” he said, lighting up as he saw me. “What are you girls doing?”
“Oh, we’re just spending time together,” I said lightly, and looked back down at the drawing.
“I told you yesterday that I loved you,” he said.
This guy, I thought, and kept drawing.
“This is my baby,” my mother said, her hand on my arm. “Isn’t she beautiful?”
“Yes, I told you that yesterday,” John said. He turned his square body to me again. His eyes glinted in the flat fluorescent light. “Can I kiss you?” He was blocking the door.
“Nope,” I said with a big smile.
He smiled too. “Okay, I won’t do anything you don’t want. But I love to see you. Why don’t you visit more?”
I winced. “I live in Canada,” I told him.
“Oh,” he said, satisfied. “When are you coming back?”
“Soon,” I said. “I’ll be back soon.”
“Okay, you come see me. I love your mother, but I really love seeing you.” He winked. “I’ll leave you girls alone now.”
Joanna and I returned to our own zone again, but the spell was soon broken by screams on the other side of the door. My mother was not disturbed; she seemed to not hear the noise at all. I pretended I needed to go to the bathroom as an excuse to open the door and look out . . .
“You can just go to this one in here,” Joanna said, pointing to the bathroom between her room and the one next door.
“Um, okay, I just want to—” I poked my head out. I could hear a fight, but I couldn’t see a thing. “It’s okay, I’m fine,” I said, pulling my head in and shutting the door again.
When I opened the door an hour later to walk to the elevator with my mother, a woman of indeterminate age sat on the floor in front of me. This is a horror movie, I thought; this is the nineteenth century. The previous year, when Joanna had first moved from a nice assisted-living facility into this rundown nursing home, the only thing she’d said to me was, “There are a lot of people suffering here.” She had always been sensitive to the pain of those around her. Not long after this, she lost the ability to speak in complete, coherent sentences.
Her current home was the only place that would take her when the assisted-living facility pushed her out of their system—a system I’d chosen for their well-maintained buildings, for their reputation for keeping residents in their care after savings ran out and they transitioned to Medicaid. But my mother’s diagnosis of frontal-lobe dementia had made her an unattractive resident. Inappropriate was the word they used when they first approached me about finding her a new place to live. “She’s an inappropriate resident.”
I opened the door from our pocket of calm and saw a woman whose hands were wrapped around the doorknob of the room across the hall from us. The woman was hunched down on the floor with her knees pressed to her chest, the weight of her small, wiry body hanging from her grip on the doorknob. Our eyes locked and she began to sob. As my mother and I passed her, she stood up and followed us, asking for help, trying to get my attention. Joanna muttered, “Just ignore her,” as we inched sideways, gripping each other’s forearms, toward the elevator.

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In a short story by Jorge Luis Borges, a European woman asks a pro­fessor from Bogotá what it means to be Colombian. The man hesi­tates before replying, “I don’t know. It is an act of faith.” Colombia is like that. Nothing is as expected. Magical realism, celebrated as Colombia’s gift to Latin American literature, is within the country simply journalism. Gabriel García Márquez wrote of what he saw. He was an observer, a practicing journalist for most of his life, who just happened to live in a land where heaven and earth converge on a regular basis to reveal glimpses of the divine.
Only in Colombia can a traveler wash ashore in a coastal desert, follow waterways through wetlands as wide as the sky, ascend narrow tracks through dense tropical forests, and reach in a week Andean valleys as gently verdant as the softest temperate landscapes of the Old World. No place in Colombia is more than a day removed from every natural habitat to be found on earth. Cities as cultured as any in the Americas were for most of their history linked one to another by trails traveled only by mules. Over time, the wild and impos­sible geography found its perfect coefficient in the topography of the Colombian spirit: restive, potent, at times placid and calm, in moments tortured and twisted, like a mountain that shakes, crum­bles, and slips to the sea. Magic becomes the antidote to fear and uncertainty. Reality comes into focus through the reassuring lens of the phantasmagoric. A god that has given so much to a nation, as Colombians never fail to acknowledge, always gets his piece on the back end.
Certainly there was some kind of magic at work in the genesis of this new book, which celebrates the Río Magdalena, Colombia’s river of life. In 2014, I was invited to Bogotá by Héctor Rincón and Ana Cano, both acclaimed journalists from Medellín, to help pro-mote the Amazon volume of their series Savia Botánica. With the backing of Grupo Argos, one of Colombia’s most prominent corporate citizens, they had assembled teams of botanists, photographers, and journalists to survey the five major regions of Colombia with the goal of producing an elegant illustrated book on each—the Llanos, Amazonas, Chocó, the Caribbean coast, and the Andean Cordilleras. These Savia Botánica volumes were not to be sold, but gifted as complete sets to every library in the country, all with the goal of sending a message to a new generation of young Colombians that theirs was not a land of violence and drugs, but rather a place of unparalleled natural wealth and beauty, home to, among many wonders, more species of birds than any other country in the world.
One day, as we wrapped up a discussion of the latest Savia Botánica volume, I casually mentioned that, having focused on the Colombian landscape, perhaps it was time to pay attention to the rivers. I proposed, half in jest, that we do a book on the Río Magdalena, the Mississippi of Colombia, the vital artery of commerce and culture that runs a thousand miles south to north, traversing the entire length of the nation. To my surprise and delight, my new friends embraced the idea without hesitation, as indeed did Grupo Argos, which immediately offered its unconditional support for the project. That whimsical remark turned out to be a defining moment, for the research and writing of this book would in the end consume nearly five years.
Colombians think of the Magdalena as having three sections—Alto, Medio, and Bajo—divisions with overlapping and even shifting boundaries that nevertheless reflect geographical, historical, and cultural distinctions far more profound than the simple terms high, middle, and low would imply. Thanks to the generosity of Grupo Argos, I was able to explore the Magdalena in all its dimensions, from source to mouth, in all months of the year, with every shift of the seasons, from the uplands of the Macizo Colombiano to the sand and stones of the Caribbean shore. Altogether, I made five extended forays to the river: two with the Savia team, led by Héctor Rincón and Ana Cano, surveys that covered the entire drainage, and two subsequent explorations that concentrated on the Medio Magda­lena and the musical traditions of the lower river and the Caribbean coastal plain. The fifth brought me back to the Arhuaco mamos, old friends from my time in the Sierra Nevada, as we returned to Bocas de Ceniza to make ritual payments at the mouth of the river, even as the streets of Barranquilla erupted all around us with the magic and joy of Carnaval.
The Río Magdalena is not just the country’s main artery; it’s the reason Colombia exists as a nation. It is the lifeline that allowed Colombians to settle a mountainous land that geographically may well be the most challenging place on the planet. Within the Mag­dalena drainage live four of every five Colombians. It is the source of 80 percent of the nation’s economic wealth, the engine that drives the economy, the river that powers the lights of the great cities. Like the Mississippi, its shadow to the north, the Magdalena is both a corridor of commerce and a fountain of culture, the wellspring of Colombian music, literature, poetry, and prayer. In dark times, it has served as the graveyard of the nation, a slurry of the shapeless dead. And yet always, it returns as a river of life. Through all the years of the worst of the violence, the Magdalena never abandoned the people. It always flowed. Perhaps, as this book suggests, it may finally be time to give back to the river, allowing the Magdalena to be cleansed of all that has soiled its waters. Colombia as a nation is the gift of the river. The Magdalena is the story of Colombia.

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On Fire

On Fire

The Burning Case for a Green New Deal
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