IMAGINE A SCHOOL WHERE THE FOLLOWING RITUAL IS OBSERVED. At the end of the year, after several days of coaching and preparation, the children are led to a cafeteria where tables have been set with plates of food, one for each child. A government official has inspected the plates; for a given grade each plate holds exactly the same foods, in the same proportions, at the same temperatures. To encourage a feeling of fair play and sportsmanship, the children have been instructed not to touch their knives or forks until everyone is comfortably seated. At a signal from a teacher, the children begin eating, madly trying to stuff as much food into their mouths as they can before a buzzer signals that the meal is over. Afterwards, the children are given a battery of tests to determine how well they are digesting their food.
Now imagine that only those children judged to be superior eaters are allowed to eat a full and balanced diet at school the following year. The teachers at the school, though well-meaning, believe only a few children are born with the capacity to digest food properly; the rest, depending on what kind of stomach they’ve inherited, can eat only one or two kinds of food, and even then only in small quantities. When challenged to defend this belief, the teachers point to the vast number of weak and unhealthy students at the school: even those singled out for special attention continue to complain of stomach disorders when placed on restricted diets.
One day people will look back on our present system of education as only slightly more rational or humane than this. A great deal of recent research in early childhood education has begun to show that, with very few exceptions, children are born capable of learning anything. Unfortunately, the existence of this research has done little to change the way children are being taught, at home or at school.
In 1998, when I was in the final year of a doctoral program in mathematics (a subject I had struggled with as a child), I persuaded several of my friends to start an educational charity called JUMP (Junior Undiscovered Math Prodigies). My goal at the time was rather modest. I knew, from my own experience, how easily children could become convinced they were incapable of doing well in mathematics. I wanted to give free, private tutoring in the subject to elementary students from working families in my neighbourhood.
Since its inception in my apartment, with 8 tutors and 15 students, JUMP has grown exponentially; it is now established in 12 inner-city schools in Toronto, with over 200 volunteers and 1,500 students. I expect the program will continue to grow at this pace, in part because the volunteers are not required to have any background in mathematics. Several of our best instructors dropped out of mathematics in high school. Working from a manual I developed for the program, tutors relearn the subject properly as they teach.
Six units from The JUMP Teaching Manual are included in Part 2 of this book. These units illustrate fully the teaching method used in JUMP. The method is easily learned and becomes automatic with practice. Teachers who work through these units with students should see very quickly how they can adapt the approach to teaching any kind of mathematics.
In many of the units in the manual (as illustrated in the fractions unit, Chapter 6), new concepts and operations are introduced in extremely mechanical steps that a student cannot fail to grasp. While the steps are simple, the goal of the JUMP method is not to produce students who can do math only by rote. In some units, students are taught to solve problems requiring careful reasoning and systematic search; in others, topics normally covered in high school or university are introduced. (Two enriched units are included in Chapters 10 and 11 of this book.)
The manual assumes that young children are capable of understanding advanced mathematics, but it does not ask students who have fallen behind to struggle with open-ended problems without guidance, as do many of the texts now used in schools. Even in the most advanced units of the manual, students are taught how to find solutions by first working on simplified models of a problem.
Over the past four years, I have observed a great many remarkable leaps in intelligence and ability in students taught mathematics using the method described in this book. My first student, who was in a remedial class in Grade 6 and couldn’t count by twos, is now in a Grade 10 academic program a year ahead of her grade level. And in several elementary classes where JUMP was tested, all of the students, including many who were thought to be slow learners, incapable of concentrating or learning advanced mathematics, scored over 80% on a Grade 6–7 fractions test after less than two months of instruction. To my knowledge, results of this sort have not been documented in our schools.
There are several reasons why such dramatic improvements in mathematical ability, particularly among remedial students, have not been observed in the school system. Most people who are good at mathematics develop a talent for the subject when they are quite young. As adults, they are scarcely conscious of the steps they follow in solving problems. Consequently, they may find it hard to isolate or describe those steps and will often blame students for their own failings as teachers.
Because I had struggled with mathematics myself, I was inclined when I started JUMP to observe my students carefully to see why they were failing. I didn’t have to look far. I examined a book of sample problems in mathematics used in Ontario’s schools. Though the book was intended for teachers, there was almost no discussion of how to guide a student, step by step, to understand and solve the problems. The book was a catalogue of failure: it described, in meticulous detail, all of the incomplete and erroneous answers a teacher might expect from students and suggested a mark for each answer. This approach did not strike me as unusual: I’ve never seen a text among the many teachers’ guides I’ve read that consistently introduces mathematical concepts in an order any student could grasp, or that lays out the steps of an explanation in a way that any teacher could communicate. I have seen guides filled with excellent exercises and activities, but none aimed at closing the gap between the weakest and strongest students. Books that purport to teach teachers often seem more concerned with classifying students: one learns how to label children according to learning styles or disabilities, but not how to deliver a lesson that will be understood by every child in a class of 25.
Apart from the lack of effective texts (and inadequate training for teachers), there is, I believe, a more fundamental reason why dramatic improvements in mathematical ability have not been observed in our schools. Until recently, a theory that even allowed for these improvements did not exist. Most models of learning assume that intelligence and mathematical ability are fixed: by reducing explanations to trivial steps, one can add only tiny increments to a student’s knowledge. Slower children will become a little better at math, but only by parroting what they have learned by rote. The results of JUMP appear to contradict this expectation. In Chapter 2 of this book, I will argue that a new branch of mathematics, chaos theory, may account for the non-linear leaps in ability that have been observed in JUMP students. I will also call into question a number of universally held beliefs about mathematical ability.
Based on my work with elementary students, I am now convinced that all children, except possibly those who are so severely disabled that they would not be enrolled in a regular public school, can be led to think mathematically. (I say “possibly” because I have not worked with children who are outside the regular school system: it wouldn’t surprise me if these children were capable of more than people expect.) Even if I am wrong, the results of JUMP suggest that it is worth suspending judgement in individual cases. A teacher who expects a student to fail is almost certain to produce a failure. The method of teaching outlined in this book (or any method, for that matter) is more likely to succeed if it is applied with patience and an open mind.