WHEN I WAS QUITE YOUNG, I BELIEVE NO OLDER THAN 11, I came across two books that would determine how I thought about mathematics for the next 20 years, until, at the age of 31, I found the confidence to return to school and start a degree in the subject. One book was a collection of science fiction from the local library. It contained a story about two children who construct a Möbius strip that enables them, by a process I unfortunately can’t recall, to travel in time. The other book belonged to my older sister, who was studying psychology at the time. It was a thick book, full of charts and formulas, on giftedness in children.
Though I haven’t reread the short story since I was a child, I would be surprised if it was well written, and even more surprised if the mathematics behind it was sound. But the story awoke a greater sense of wonder than I have felt reading anything since: from it I gained the conviction that mathematics was a magical subject that would allow me, once I had mastered it, to transcend the everyday.
My sister’s book was less inspiring. I found the charts and formulas impossible to decipher, but I knew from the introduction what they implied: that I would become a mathematician only if I had inherited a gift for the subject.
The moment I learned that I couldn’t simply decide to become a mathematician, I began to reflect on my early childhood to see if I was lucky enough to have been born a prodigy. I read biographies of successful scientists and mathematicians so I could compare my development with theirs. I recall one book, The Mind, quite clearly. It contained a painting of two dozen geniuses, with an intelligence quotient printed beside each one. Gauss, a mathematician, had scored much higher than the rest, in part because he’d discovered a trick for summing the numbers from one to 100 when he was only eight.
It seemed clear, from everything I read, that a person born to do mathematics would never do badly on a test or struggle to learn a new concept. The thoughts and mental processes of a great scientist or mathematician were of an order entirely different from those of an ordinary person. As this belief sank in, I began to find math more difficult at school and my marks steadily declined. When I received a D (later belled to a C) in Calculus for the Life Sciences at university, I decided to drop the subject once and for all.
One evening, a year after I had finished my undergraduate studies, 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. (I still hadn’t learned how dangerous it was to read my sister’s books.) As a child, I loved writing as much as I loved mathematics, but in this, too, I had shown few signs of innate talent. Though I received a B in creative writing at university, it was the lowest mark given in the class.
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 meter and form. She wrote sonnets and sestinas, memorized the thesaurus, and read mythology. She also produced dozens of imitations of poems she admired.
Even with Plath’s example to guide me, I wasn’t entirely certain I could choose to develop a skill for something I loved. But I resolved to try to teach myself to write the way she had. Ten years later, I received a Governor General’s Award for my work as a playwright. Awards do not necessarily certify that an artist’s work is good, but they do at least indicate basic competence.
As I found it impossible to make a living as a playwright (the $10,000 award increased my cumulative earnings in theatre by about 300%), I supported myself with a series of part-time jobs. One day, when I was 28, I answered a notice at a placement centre for a math tutor. I had become interested in math again after helping a friend who was returning to school with a problem in calculus. Reading her textbook slowly and carefully, without the pressure of having to study for exams, I found the material much easier than I had at university. I convinced the woman who ran the tutorial agency that I could teach math because I had passed a course in calculus (she wasn’t aware of my mark) along with several courses in logic.
One of my first students was close to failing Grade 8 math. He had been told by his teacher that he wasn’t bright enough to do well in the subject. Having struggled with math myself, I decided to observe the boy carefully before I made any judgements about his ability. He proved to be an ideal student. He worked hard and soon developed an interest in mathematics. Because I hadn’t assumed that he lacked ability, and because I was lucky enough to work with him for a period of five years, I saw changes in his aptitude that few teachers ever see in their students. In Grade 13, my student did so well in the Sir Isaac Newton physics competition, he was offered a scholarship to Waterloo University, even though he hadn’t applied there. He is presently completing a doctorate in mathematics at another university.
Seeing how quickly several of my students were developing as mathematicians, I began to think about my own progress in the subject. Because I made an effort, in my tutorials, to reduce explanations to simple steps, many of the concepts and operations that had seemed mysterious in high school were becoming clear. It now seemed possible, by teaching myself mathematics the way I had taught myself to write, that I might realize my childhood dream of being a mathematician. I enrolled in two first-year mathematics courses at a local university.
When I received the results of my first test, I wasn’t surprised to see that I had failed. I knew the test wasn’t fair: some of the material had not been covered in class (it was taught several weeks later). As the professor handed back the tests, he read out the names of the few people who had passed and told the rest of us that we should consider dropping the course. Many students did exactly that; the ones I talked to believed that the professor had made an objective assessment of their prospects as mathematicians. I stayed in the course, but only because of the confidence I had gained as a tutor.
I failed other tests from time to time, in other courses, including two of three comprehensive exams for my doctorate. In the beginning, it was very hard to suppress the feeling that I had failed because I had finally reached a level of mathematics that was beyond me. Gradually I saw that questions I once found impossible on a particular test seemed trivial a month or a year later: there was no threshold that separated one level of mathematics from another. Sometimes I did badly because I was nervous (on my comprehensives I made mistakes that were equivalent, in higher mathematics, to 2 + 2 = 5) or because I hadn’t had time to digest the material properly. Sometimes the tests were unfair.
Unfortunately, few of the students who enrolled in mathematics the year I did survived long enough to gain the confidence and background they needed to do well and to overlook an occasional failure. Several of the professors seemed more intent on weeding people out than on teaching them: by fourth year, only eight of the hundred or so students who started were still in the department.
When I was growing up, my parents would billet students from Africa so they could go to our local university. My father often worked for free as a surgeon in hospitals in Africa, and my mother helped administer and raise funds for several Third World charities. By the time I had returned to university, I began to feel that I hadn’t really lived up to their example. Even after winning the Governor General’s Award, I wasn’t sure my writing amounted to much more than a self-indulgence. I searched several community papers until I found a volunteer program that was looking for tutors.
The program, which was listed under the Canadian Alliance of Black School Teachers, was run by two saintly teachers, Ken and Inez Johnson, who had spent years doing volunteer teaching and advocacy work for inner-city kids, and who didn’t seem to mind the fact that I wasn’t black. They assigned me to a Saturday morning homework club, where I tutored math, mainly to high-school students.
The Saturday sessions were very successful (I continued teaching there even after I started JUMP), but I was frustrated by one aspect of the classes. Some of the students would make spectacular progress for a lesson or two, then not show up again for a month. Though I was proud of the students who came every week and did well at school, I thought more often about the ones who were never there. I had the feeling, with those students, that I was bailing out a lifeboat. I knew the boat would sink eventually, and I could do nothing to prevent it.
For several years I thought about starting a program for elementary students. I assumed they would be less likely than high-school students to have given up on school. There were two spare rooms in my apartment, which, in addition to my kitchen, would make adequate classrooms. Inez had been vice-principal of a school a short walk from my apartment. She convinced the principal, Silvana Carletti, to select 15 children for tutoring. Several of my friends were brave enough to volunteer as tutors, even though they had never taught math. Half an hour before the first students arrived, I reminded the tutors how to add fractions. Beyond that, I wasn’t sure what we were going to teach the kids. I was, however, full of optimism: I had already decided to call the program Junior Undiscovered Math Prodigies — or JUMP.
After five minutes with my first student, I was certain the whole enterprise would be a terrible disaster. I had asked Silvana to send students in Grades 4 or 5, but the girl who sat down at my table looked older. When I asked what she had learned about fractions at school, she said, “Nothing.” I said she would find them very easy, but first I needed to know if there were any times tables she had trouble remembering. She looked at me with a blank expression. She had no idea what multiplication meant. Even the concept of counting by a number other than one was foreign to her: she was not able to count by twos. Silvana had assumed, when I asked for children who were struggling in math, that I’d wanted students in remedial classes. Lisa was in Grade 6, but knew less mathematics than a typical child in Grade 1. She was terrified by my questions and kept saying, when I mentioned the simplest concepts, “I don’t understand.” She also had trouble reading and told me she had never read a chapter book in her life.
I had promised a lesson in fractions, and as I had no idea what else to do, I began counting slowly on my fingers by twos, asking Lisa if she could do the same. I wasn’t certain she would ever understand the concept of fractions but wanted at least to see if she could carry out basic operations, changing the denominators of simple fractions by multiplying on her fingers. Lisa made several attempts to count to 10 by twos but couldn’t remember the correct sequence of numbers. As she was clearly growing more nervous with each failure, I told her she was brilliant, even though she could only repeat the sequence up to six. The encouragement helped her focus, and by the end of the lesson she had made more progress than I expected. The next day, her mother told me Lisa had had a nightmare that she wouldn’t be allowed to return to tutorials. I was the first teacher who had ever told her she was smart.
Lisa has been in JUMP for three years now. I find it hard to believe that she ever had trouble mastering simple mathematics. Her rate of learning seems to double by the week: lately she has even started teaching herself new material from a textbook when I’m not available to answer her questions. Lisa recently moved from a remedial class (where she was still being taught the most basic math) into a regular Grade 9 academic class. Several weeks ago I also found out that she’d enrolled, of her own volition, in Grade 10 math for the next semester. Soon Lisa will be a year ahead of her grade level.
Though I had asked Lisa’s mother to place her in an academic math program, I began to have misgivings during the first week of class. There were still large gaps in Lisa’s background I felt would be difficult to fill in, particularly in a semester course. A teacher at the high school called Lisa’s mother several times to ask why Lisa had been placed in an academic class. She even told Lisa directly that she should move to a basic class. According to the school’s records, Lisa was still at an elementary level; on top of that she had failed her first test. But Lisa, who previously had been terrified of teachers, refused to leave the class. When she received a 90% on her second test, I breathed a sigh of relief. Since then, Lisa has learned an enormous amount of new material, including much that her peers have been working on for years. Though there are still large gaps in her background (her marks oscillate wildly between 40% and 100%, with most falling in the C to B range), I am certain she will pass her course.
Lisa is not an isolated case. Three of the four oldest students in JUMP moved from remedial classes to academic Grade 9 classes recently. (The mother of the fourth student was convinced by a teacher that her daughter would not be able to handle academic math.) As well, many of the younger students in JUMP have made remarkable progress, which I describe in more detail in Chapters 2 and 4.
Lisa advanced from a Grade 1 to a Grade 9 level after only a hundred hours of tutorials (fewer lessons than she would have received in a single year of school). I am certain she would have progressed much further if I had been less quick to judge her and more adept at presenting advanced material. The students in JUMP have taught me how little I knew about teaching. The main principles of the program were thrust upon me, in embryonic form, even in the first lessons. The students Silvana had selected were so delayed that I was forced, as I had never been with previous students, to reduce mathematics to steps no one could fail to grasp. Just to keep them going, I was compelled to tell students who appeared unteachable that they were smart, only to realize later what I’d said in fact was true.
It can take an hour to convince a child he or she is intelligent; with an adult, sadly, it could take forever. Recently I told a teacher of enriched mathematics about the remarkable changes I’d observed in children enrolled in JUMP. I was responding to his claim that real mathematical ability can never be nurtured, because it is genetically determined. He said that though I was smarter than he (because I had gone further in mathematics), he had more experience with kids. I never questioned his experience (though it had been solely with teenagers who had already suffered at least nine years in the public school system), but I found his lack of confidence telling: he easily might have gone further himself if he hadn’t been prevented by his beliefs.
I sometimes wish I could travel back to my childhood to regain the time I wasted in self-doubt (and not just in mathematics). But I’m grateful to have found some faith in my abilities. One day, I hope that same faith will be considered the right of all children, and they will not have to wait for some accident, like a book of letters, or a tutor assigned by mistake, to reveal the vast potential that might otherwise have been lost.