About the Author

John Mighton

John Mighton is a mathematician, author, playwright, and the founder of JUMP Math, a successful school program designed to tutor children who are having difficulties in math. John has written an inspirational book based on his experiences with JUMP called The Myth of Ability: Nurturing Mathematical Talent in Every Child, published by House of Anansi Press. He also released a follow-up book to The End of Ignorance. Mighton completed a PhD in mathematics at the University of Toronto and has lectured in philosophy at McMaster University. He held an NSERC postdoctoral fellowship for research in graph theory and knot theory at the Fields Institute and is currently an adjunct professor at the University of Toronto.

Mighton’s plays Scientific Americans, Possible Worlds, A Short History of Night, Body and Soul, The Little Years, and Half Life have been performed across Canada, as well as in Europe, Japan, and the United States, and have won several national awards including the Governor General’s Literary Award for Drama for Possible Worlds and A Short History of Night in 1992, and for Half Life in 2005. John was also the recipient of the 2005 Elinore & Lou Siminovitch Prize. Mighton’s play Possible Worlds was made into a full-length feature film directed by renowned director/playwright Robert Lepage. He is currently adapting Brian Greene’s book The Elegant Universe for the Lincoln Centre with Robert Lepage.

Books by this Author
All Things Being Equal

All Things Being Equal

Why Math Is the Key to a Better World
edition:Hardcover
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Excerpt

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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Jump at Home Grade 3 ed 2 /tp

Jump at Home Grade 3 ed 2 /tp

Worksheets for the JUMP Math Program
edition:Paperback
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Jump at Home Grade 4 ed 2 /tp

Jump at Home Grade 4 ed 2 /tp

Worksheets for the JUMP Math Program
edition:Paperback
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Jump at Home Grade 5 ed 2 /tp

Jump at Home Grade 5 ed 2 /tp

Worksheets for the JUMP Math Program
edition:Paperback
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Jump at Home Grade 6 ed 2 /tp

Jump at Home Grade 6 ed 2 /tp

Worksheets for the JUMP Math Program
edition:Paperback
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Myth of Ability /mm

Myth of Ability /mm

Nurturing Mathematical Talent in Every Child
edition:Paperback
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Possible Worlds

Possible Worlds

edition:Paperback
also available: Paperback
tagged : canadian
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The End of Ignorance

The End of Ignorance

Multiplying Our Human Potential
edition:Paperback
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Excerpt

Chapter One
The Waste Ethic
Twenty-five years ago, when I was studying philosophy at McMaster University, I wanted to write a book called "The Waste Ethic," which I hoped would be the first attempt in the history of the social sciences to accurately measure the amount of time people waste at work. I wasn’t interested in simply tracking the time wasted by people who hate their jobs or who are totally unqualified for their positions. I wanted to find out what proportion of our work goes into producing, marketing and disposing of the vast array of products that, before the advent of mass media, nobody knew they needed or wanted. I never did find time to write that book, but having spent the past twenty years teaching mathematics to thousands of children and teenagers – both gifted and challenged, and from both affluent and impoverished homes – I think I have a better idea of why we as a society are so efficient at wasting time.

It seems to me that two kinds of ignorance are always at work in our society, one extremely destructive and the other healthy. My career in theatre was initially shaped by the first kind of ignorance, in ways I am only beginning to understand. I came to writing plays rather late in life because I grew up thinking that to be an artist you needed to be born with a special gift. It wasn’t until I read Sylvia Plath’s letters to her mother and saw how as a teenager she had learned her craft in small, determined steps, dismantling poems like motors to see how they worked and writing imitations of the things she loved, that I began to believe there was a path I could follow to develop a voice of my own.

The destructive form of ignorance has divided many societies: it is the ignorance that says there are fundamental, inborn differences between people, between peasants and nobility or minorities and majorities. It is this ignorance that leads us, even in this affluent age, to neglect the majority of children by educating them in schools in which only a small minority are expected to naturally love or excel at learning.

Two years ago, during a visit to the York Detention Centre, I saw the effects of this ignorance in its most devastating form. I had been asked to teach a lesson in mathematics to a group of teenagers who were awaiting trial and who were not thrilled to be spending their afternoon doing math. I told the students I had once struggled with mathematics myself and I promised to try to make the subject more interesting and easier than they might remember from school. I reassured them that if they didn’t understand something in my lesson it would be my fault for not explaining it properly, so they could ask me to explain it again. The teenagers responded to my promise exactly as I have seen young children respond: they raced through their worksheets and called for the tutors to give them extra work. One girl, whom I had heard complaining at the beginning of the lesson, made me put a check mark beside each of her answers. When I finished she said, "I've never had that in my life. I've only had this," and she wrote a large X across her page.

The letter X is a fitting symbol for our failure to care for those individuals who, like the girl at the detention centre, happen to struggle or fall behind in school or in life. The crossed lines evoke the barriers we place, out of ignorance and indifference, between the majority of children and their unrealized potential. But the letter X is also a universal sign for a different and potentially redeeming kind of ignorance: in the sciences and in mathematics, it is the letter most commonly used to stand for the unknown.

Einstein once wrote:

The most beautiful and deepest experience one can have is the sense of the mysterious . . . One who has never had this experience seems to me if not dead, then at least blind. To sense that behind anything that can be experienced there is something that our mind cannot grasp and whose beauty and sublimity reaches us only indirectly and as a feeble reflection, this is religiousness. In this sense I am religious.

The sense of the mysterious that Einstein described can define a person as much as their sense of courage or integrity or charity. People who experience the ineffable mystery of the world also tend to have a deep sense of humility, as Socrates demonstrated when he admitted how little he knew to the Athenians who put him on trial for his life. This sense of the mysterious drives such people to pull aside the veil or wipe away the fog that separates them from the mystery.

Every child who is well cared for naturally develops a sense of the mysterious. The feeling that behind every door another world is waiting can make a child’s world a paradise. But, once at school, children often begin to lose their sense of the hidden beauty of the world. By having to compete and be compared to their peers, many lose faith in their intelligence and their imagination; by having to struggle so hard to keep up, many come to believe that the world is beyond their understanding. The magical world that they once inhabited begins to recede until they can see no point in dreaming about or searching for anything beyond the world of their immediate needs and desires.

People are often surprised when I tell them that I am a mathematician as well as a playwright. Some people seem to believe that the brain can hold only one kind of information, or that when one side is working the other has to be left empty for storage. If they are lucky, students graduating from high school will likely believe that they have only one or two talents and that the majority of subjects offered at school are either uninteresting or beyond their grasp. As a society we are living under a vast spell or illusion. We have effectively hypnotized ourselves, but not in a single performance. It has taken twelve or thirteen years of school to put us in a suggestive state so that we all believe more in our limitations than in our potential, and it is difficult for anyone to snap their fingers to break the trance.

When I was a teenager I read a fair number of biographies of scientists and mathematicians. These biographies always gave me the impression that a person had to be born with a gift for mathematics and that someone who had this gift would never do badly on a test or struggle to learn a concept. This belief sank in very deeply; like many young people I would often give up on things because I was afraid of coming up against my limitations. It wasn’t until I was in my thirties that I had the courage to go back to school and study mathematics.

At university I had an enormous advantage over the other first-year students: I had been tutoring math for several years to supplement my income as a playwright, and I knew the high school material inside out. I had also learned some of the university material in advance. But occasionally I would do badly on a test and I would be paralyzed by insecurity. I remember lying in bed after I had failed one test, thinking that I had reached a threshold I couldn’t go beyond and that I would have to give up my ambitions. My work as a tutor had given me a great deal of perspective on the issue of ability, but I still couldn’t get over those fears. It took several years before I began to notice that the things I had found impossible on a particular test became trivial once I had a chance to practise and learn them properly. I began to wonder how many people have stopped themselves out of fear of failure from developing talents in things they found interesting when they were young.

Nine years ago I was looking for a way to give something back to my local community. It occurred to me that I should try to help kids who needed help with math. Mathematicians don’t always make the best teachers because mathematics has become obvious to them; they can have trouble seeing why their students are having trouble. But because I had struggled with math myself, I wasn't inclined to blame my students if they couldn’t move forward. If a student didn’t understand my explanation I assumed there was something wrong with the explanation, not with the student.

Many of my closest friends are actors. Actors often have a good deal of time on their hands and they will do anything for attention, so I was able to convince a group of my friends, even the ones who had dropped out of math, that they too could tutor math. We started an after-school tutoring program in my apartment called JUMP (Junior Undiscovered Math Prodigies).

The JUMP program was founded on a very lucky accident. I had asked the principal of a local school to send some kids to the program who were having trouble with math. She misunderstood me and delivered the most challenged kids in the school, including a number of kids in special education classes who were performing far below grade level.

Working with these children, and with the thousands of children I have taught since JUMP started, has been one of the most inspiring things I have done. I am now convinced that the brain can develop new abilities more readily than traditional theories of intelligence allow, and that children who face challenges at school are capable of far more than we expect of them. Many of the students from the first few years of JUMP had moved into academic classes by the time they entered high school. And thanks to the work of hundreds of teachers and volunteers, JUMP is now an in­class program that delivers the complete curriculum in mathematics from Grades 1 to 8. JUMP materials and methods are being used with encouraging results in several hundred schools in Canada, the United States, Australia, South Africa and Britain (which is home to the world’s largest single JUMP program, with five thousand children).

In this book I will describe the methods and results of the JUMP program, but this is not primarily a book about JUMP, or even about mathematics. The examples I give illustrate broader points about the way we educate children and about the expectations we have of weaker students. It is possible to apply the methods I describe to various subjects. Deep psychological principles, which are as important as the curriculum or the content of the course, must be taken into account in the teaching of any subject: including the way children perceive themselves in groups and the way cognitive abilities can be developed by playing with subtle variations on patterns. I argue that, because we have ignored these principles, our educational system is far from optimal and the means we have developed to educate children are extremely inefficient.

The philosopher Wittgenstein said that people get themselves into trouble when they fail to pay attention to the precise meanings of words. I argue that many inefficiencies in education stem from confusions over the meaning of some words used in educational theory. These confusions are not of academic or philosophical interest alone: they have had a profound effect in our schools.

It is not easy to describe the things I have seen in the educational system, or the ideas behind JUMP, without over-simplifying. I often return to the same idea several times to provide a different angle or perspective. It is almost impossible to say anything in education that is not both true and false at the same time. The things that children respond to are often surprising and counterintuitive, and words and categories can fail to capture the complexity of a child’s behaviour. It is easy to attach a label to a particular style or philosophy of teaching, but I hope that educators will wait until they have read the whole book before they try to classify or name the methods I describe in this book. One of the greatest threats to education is educational theory that does not take account of the extraordinary complexity of the mind or of the subtleties in what children find interesting and meaningful and in what motivates them to learn.

When I was growing up I often dreamed about becoming a good mathematician or a good writer, but never about becoming a good teacher. Given a choice, I prefer sitting alone in my office doing mathematics to teaching, giving talks about mathematics or writing workbooks for children. But over the past ten years I have effectively given up my careers in math and the theatre to work forty to fifty hours a week as a volunteer for JUMP. And there are several reasons why I have set aside my other work in order to teach and develop educational materials.

I think it is fair to say that there has been no great improvement in the state of education in the past thirty years. And over the same period, despite the substantial efforts of governments and charities, there has been no significant improvement in the condition of the world’s poor. At the same time, a succession of breakthroughs in productivity and in technology has transformed our society and created vast fortunes around the world. When so much has changed in contemporary society, how have the state of education and the level of poverty remained the same? I believe our lack of progress in these two areas is connected.

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The Myth of Ability

The Myth of Ability

Nurturing Mathematical Talent in Every Child
edition:Paperback
also available: eBook
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