TWO OF THE FIRST JUMP STUDENTS, ANA AND MARGARET, FOUND it hard to concentrate for more than several minutes at a time when they started the program. Both girls were in Grade 6, but neither could consistently add a pair of single-digit numbers, even using their fingers. After several months of lessons they were making progress, adding and manipulating simple fractions. One day, while I was busy helping another JUMP student, Ana began leafing through a Grade 7 text she’d picked up from my table. When I turned to work with her, she handed me a sheet of paper on which she’d added some fractions. It was the answer to a word problem from the book. The problem was harder than anything I’d covered in my lessons thus far, and Ana had solved it without my help. Several months later, I gave Margaret a sheet of paired ratios with a term missing in each pair. Normally, I teach students to find the missing term in cases where they must multiply before I show them how to find the missing term by dividing. On Margaret’s sheet, I had mixed up the two kinds of questions. While I was busy thinking about how I would teach her to distinguish between the two cases, she answered the questions correctly without hesitation.
Even though Ana and Margaret had learned to work with fractions much more quickly than I had expected, these new displays of ability seemed almost uncanny. Both girls had been placed in classes for severely learning-disabled children: they were years behind even the students I had taught who were failing regular classes. With my help, they had learned how to manipulate fractions, but only by following a sequence of small, mechanical steps. Now, it seemed, they’d been able to teach themselves mathematical concepts without any assistance. New abilities had emerged, suddenly and mysteriously, from entirely mechanical work.
Not long ago, in the 1960s, mathematicians and scientists began to notice a property of natural systems that had been overlooked since the dawn of science: that tiny changes of condition, even in stable systems, can have dramatic and often unpredictable effects. From stock markets to storm fronts, systems of any significant degree of complexity exhibit non-linear or chaotic behaviour. For example, if one adds a reagent, one drop at a time, to a chemical solution, nothing may happen at all until, with the addition of a single drop, the whole mixture changes colour. And, as a saying made current by chaos theory goes, if a butterfly flaps its wings over the ocean, it can change the weather over New York.
As the brain is an immensely complicated organ, made up of billions of neurons, it would be surprising if it did not exhibit chaotic behaviour, even in its higher mental functions. Like the chemical solution that changes colour after one last drop of reagent, Ana and Margaret’s new abilities emerged suddenly and dramatically from a series of small conceptual advances. I have witnessed the same progression in dozens of students: a surprising leap forward, followed by a period where the student appears to have reached the limits of their abilities; then another tiny advance that precipitates another leap. One of my students, who was in a remedial Grade 5 class when he started JUMP, progressed so quickly that by Grade 7 he received a mark of 91% in a regular class (and his teacher told his mother he was now the smartest kid in the class). And Lisa, who couldn’t count by twos in Grade 6, now teaches herself new material from a difficult academic Grade 9 text.
If non-linear leaps in intelligence and ability are possible, as the results of the JUMP program suggest, why haven’t these effects been observed in our schools? I believe the answer lies in the profound inertia of human thought: when an entire society believes something is impossible, it suppresses, by its very way of life, the evidence that would contradict that belief. History provides ample proof of the blindness we display when observing ourselves. Only 50 years ago, everyone believed that women and people with dark skin were intellectually inferior to males with white skin. Unfortunately, this kind of prejudice still underlies most of our thinking about intellectual (and even artistic) ability.
Biologists tend to attribute differences in behaviour among animals of the same species to differences in material condition, and they test their hypotheses rigorously by varying those conditions. But ask the biologists themselves if they are gifted in mathematics and they will almost certainly reply, “I don’t have that kind of brain,” just as an artist might say, “I’m more of a right-brain person.” Twelve or 13 years in oversized classes, in a system predicated on the idea that only a few students will excel: these factors are not considered to be complicit in the problem. Failure in this system stands as irrefutable proof, even for the person failing, that one was born not to succeed.
Most people can recall a classmate in grade school who never seemed to work but always did well on tests. The fact that mathematical ability appears spontaneously in a gifted child is cited as evidence that ability is determined by genetics. But if the mind, like other complex systems, is subject to chaotic and non-linear effects, even siblings with the same genetic features, and who are offered the same opportunities, might develop entirely different abilities. Some small event in early childhood or at school might start an avalanche of learning in one child but not another. The fact that an avalanche occurs on one mountain and not another does not imply anything interesting about mountains. It does not prove that one mountain is more prone to avalanches or that an avalanche could never be started.
People who claim that they were born without mathematical ability will often admit that they were good at the subject until a certain grade, as though the gene for mathematics carried a definite expiry date. Most people will also recall an unusual coincidence: that the year their ability disappeared, they had a particularly bad teacher.
Prior to the 20th century, in cultures where women were discouraged from studying mathematics, half the human population produced only a handful of mathematicians. No experiment could demonstrate more conclusively how a universally held belief becomes self-fulfilling.
Perhaps more than in any other subject, in mathematics it is easy to turn a good student into a bad one in a very short time. The myths surrounding the subject encourage children to give up the moment they encounter any difficulty. As well, mathematical knowledge is cumulative: a child who misses a step in the development of a concept cannot go on.
I conducted a rudimentary experiment with a Grade 3 class recently to see how much they could learn in a month if they were taught by the JUMP method, with adequate tutorial support. I gave four weeks of lessons (each lesson 40 minutes long) on fractions, followed by a week of review. Two volunteer tutors came into the class once a week, and the teacher assisted in most lessons. Five students, including three who were considered learning disabled or slow learners, received three extra tutorials in groups of two or three students. (The details of the experiment are given in Chapter 4.)
At the end of five weeks the class wrote a practice test, followed, the next day, by a 15-minute review, then a final. On both tests the children were expected to name fractions, add and subtract fractions, reduce fractions, change mixed fractions to improper fractions (and vice versa), add mixed fractions, compare fractions for size, and solve simple word problems involving fractions. Because at least half of the class didn’t know any times tables when I started the lessons, the denominators in most of the questions on the tests were divisible by two, three, four, or five (the children had learned how to multiply and divide by these numbers over the course of the month). Otherwise the tests were at a solid Grade 6–7 level. (The final test is reproduced on page 113.)
All of the students in the class scored over 80% on the practice test and over 90% on the final (with more than half of them scoring 100% on the final). By the time they wrote the tests, the weakest students in the class had shown remarkable improvements in concentration, memory, and numerical ability, so that there was much less of a gap between them and the strongest students. One boy, who had been recommended for a slow-learners class, finished his practice test ahead of half the class and scored 90%. (He took more time on his final, but still scored 92%.) The children were thrilled to be doing challenging mathematics because they knew they would not be allowed to fail. When I said to the class, “You all got As on the practice test — do you think I need to give you the final?” the students shouted in unison, “Yes!”
Based on my observations of hundreds of students, I predict that with proper teaching and minimal tutorial support, a Grade 3 class could easily reach a Grade 6 or 7 level in all areas of the mathematics curriculum without a single student being left behind. Imagine how far children might go (and how much they might enjoy learning) if they were offered this kind of support throughout their school years. It is possible, of course, that children who had a head start in early childhood might always remain ahead of their peers (although I believe that significant differences between children would tend to disappear if the children were all offered proper support in the lower grades). It is even possible, though I do not personally believe this, that some children might be genetically programmed to be more intelligent than others. But even if this were the case, it would not, I expect, be of any consequence in a society that educated its children. This is where the debate about intelligence misses the point. The results of JUMP suggest that we can raise the level of even the weakest students sufficiently to enable them to appreciate and master genuine mathematics. At this level, sheer intelligence is almost secondary. In the sciences, factors such as passion, confidence, creativity, diligence, luck, and artistic flair are as important as the speed and sharpness of one’s mind. Einstein was not a great mathematician technically, but he had a deep sense of beauty and a willingness to question conventional wisdom.
If a music teacher were to say, “Gifted children will simply pick up an instrument and play well; the rest will only become mediocre musicians,” we would take it as a sign of incompetence. Why, then, do we tolerate this view among teachers of mathematics? Why are our schools satisfied if only one-fifth of their students demonstrate a mastery of the curriculum? And why are most mathematics classes so large that only a few students could ever hope to ask a question and have it answered?
A simple analogy shows the extent to which people still believe that only a few children are born with intellectual ability. If children in any part of Canada were being starved to the point where they looked like famine victims, people would demand that they be fed. But children regularly graduate from our schools after reaching only a tiny fraction of their potential. Why do we tolerate this vast loss of potential, this great neglect of our children? It is not because we are inhuman. We must all believe, on some level, that these children are not being starved, they are simply incapable of eating.
Positive social change only occurs when enough people recognize something as unfair. Apathy alone does not stop people from acting: we tolerate suffering or injustice because we have failed to see something that seems obvious once it is understood (witness the treatment of people of colour in North America). What would happen if we devoted as much effort to teaching students as we do to assessing them and proving them different? The weakest students would likely surpass the standards now set for the strongest. As long as we insist that mathematicians are born and not made, we will tolerate poorly designed programs in our schools and classrooms in which children who have fallen behind cannot get the help they need to succeed.