IN A TYPICAL ELEMENTARY CLASS, EVEN AMONG CHILDREN WHO are only eight years old, an enormous difference exists between the weakest and the strongest students. The most knowledgeable will be able to recite their multiplication tables to 12, while the most delayed have trouble counting by twos. The fastest will finish a page of work before the slowest have found their pencils. And the most eager will wave their hands to answer a question while the most distracted stare vacantly into space.
This gap in knowledge, ability, and motivation — which is already pronounced in Grade 3 and which grows steadily until, by Grade 9, students must be separated into streams — appears to make it impossible to teach mathematics in the classroom. A teacher working with the texts and resources now available for elementary students can expect at most one-third of their class to complete tests and assignments independently without making errors. The other two-thirds will not be able to read, write, add, multiply, reason, or concentrate well enough to carry out their work with a high degree of success.
Based on my work with the Grade 3 class described in Chapter 2, and on similar experiments conducted in five other classes ranging from Grade 3 to Grade 5, I am absolutely certain the gap I have described is an artefact of our system of education — an illusion that can be dispelled more quickly and with fewer resources than even the most optimistic educator might expect. In an elementary class, the gap can be eliminated, or closed to a point where it doesn’t affect the quality of the mathematics program, simply by using several volunteers over several months.
JUMP was not initially designed for the classroom. I decided to adapt it for that setting only after several teachers who were impressed with the results of the program invited me to test the method with their students. Though the material I had developed for the manual was intended for students in Grade 5 and above, I tried the fractions unit with a Grade 3 class. If it worked there, I reasoned, no teacher in Grade 5 could say their students were too young or too delayed to do the work.
During my first lesson with the Grade 3 students, I began to wonder if I had been too ambitious in my choice of grade level. I had previously tutored students who were 10 or older, in groups of two or three: now I was faced with 25 eight-year-olds, in an inner-city school where many spoke English as a second language. Most of the children in the class didn’t know their times tables, nor could they add or subtract readily in their heads. Several had been diagnosed as slow learners. Others clearly had trouble concentrating in a room full of children.
By the second week of lessons, the majority of the students, including one boy who had never answered out loud in class before, began to wave their hands whenever I asked a question. Sometimes a student who I thought was slower would finish the work ahead of one who was faster. Students who at first had needed constant encouragement to complete an assignment began to ask for extra work.
In the third week, the class asked if I could stay longer so they could do more math. Occasionally, I would have to stop the lesson so everyone could come to the board, one at a time, to show off their mastery of an operation. Even the weakest students were offended if they weren’t given a bonus question of greater difficulty than their regular work.
By the fourth week, it was hard to predict who would finish a worksheet first. Some students were still more likely to do so than others, but the extra time needed by the slowest students was usually negligible.
By the fifth week, the students had all scored over 90% on a Grade 6–7 fractions test. They went on to complete a unit on multiplication and division one or two years beyond their grade level, and a unit on solving word problems using logic and systematic search. None of the students has needed extra tutorials to keep up with the class.
Public-school teachers in five other classes, including one special-education class, have duplicated these results. In each of these classes a JUMP teacher (Maggie Licata, Katie Baldwin, or I) taught one demonstration lesson per week with several tutors assisting. The regular teachers, working from the JUMP manual and worksheets I had prepared, taught the rest of the lessons. In most of the classes the teacher was assisted twice a week by one or two JUMP tutors, and several students received occasional tutoring at recess. All of the teachers took more than five weeks to complete the fractions unit (the average time was about seven weeks); I believe this was because they had only just learned the method. In every class the students completed the fraction test with a final mark of A. Most scored over 90%.
In light of these results, there seems to be no excuse for the present state of mathematical education in our schools. Teachers can be trained in the JUMP method in several weeks, and high-school co-op students, university students, parents, and people with flexible work schedules can be recruited to assist in classrooms during the day. As elementary classes are usually 40 minutes long, a person who volunteers for two hours can work in two or three classes. At JUMP we have found that 20 volunteers are sufficient to establish an effective tutoring program in mathematics for Grades 3 through 6 in a mid-sized school.
I don’t mean, by advocating that schools recruit volunteers, to suggest that governments should be allowed to neglect their responsibilities; one full-time professional tutor in a school could replace two dozen volunteers. It will take a great deal of public pressure in these conservative times, however, to persuade politicians to provide money for tutors. I hope the results of JUMP will inspire parents and concerned citizens to hold governments — as well as publishers, school boards, and educators — accountable. No program in mathematics should be implemented in our schools before it is tested in classes of 25 to 30 students. If a teacher cannot be trained in several weeks to follow the program, and if a child cannot be expected, with minimal tutorial assistance, to complete assignments and tests with a high degree of success, then the program should not be allowed in the schools.
There are several principles on which any program that seeks to meet these standards must be founded. These principles, which I outline directly below, are not difficult to understand or implement. But I believe they have never been articulated or followed rigorously in our schools, if only because they would never have occurred to someone who has accepted the myth that mathematical ability is innate.
Any person who is committed to educating all children, not just the few who are initially more advanced than their peers, must begin by acknowledging that the gap I have described, though easy to close, exists among students in every school and at every grade level. Without acknowledging this, no amount of effort spent improving textbooks, training teachers, or developing sophisticated manipulatives and teaching aids will ever help the majority of students.
Before I began teaching the Grade 3 class, I prepared a series of worksheets of graduated difficulty. For most of the operations and concepts in the fractions unit, I made three worksheets labelled A, B, and C. On the A sheets, the denominators of the fractions were divisible by two, three, or five; on the B sheets by four and six; and on the C sheets by seven, eight, and nine. Some questions on the B or C sheets required several more steps to complete than those on the A sheets.
I made sure, before I introduced the worksheets, that every child in the class could count on their fingers (on one hand only) by twos, threes, and fives. Then I showed the class how to multiply by counting. To find the product of two and five, you count by fives until you have raised two fingers — the number you say when you raise your last finger is the answer. When I was certain everyone could multiply, either on their fingers or by memory, I introduced the worksheets. With each new concept or operation, I let the faster students work ahead on the B and C sheets while I helped the slower students on the A sheets. I would never move ahead until every child in the class had completed the A sheet for a particular concept. The weaker students were always allowed enough time to master a concept or operation completely, while the stronger students, who were given extra work, were never bored. After a month, as the weakest students began to catch up and work more efficiently, I needed fewer extra sheets. Several questions written on the board were usually enough to keep students who had finished their work ahead of the others busy.
I used graduated worksheets to teach the weaker and less knowledgeable students in the class exactly the same concepts and operations as their peers, albeit using lower times tables. When I tested the class, I used fractions that had appeared on the A sheets, so the weaker students could do as well as everyone else. After they had scored higher than 90% on their tests, these students were eager to learn new tables. On the A sheets for the next unit, I included questions that required multiplication by fours and sixes: a teacher could continue in this way until their students had memorized all of their tables.
I soon learned, in my lessons with the Grade 3 class, not to underestimate how hard it is to convey information efficiently to a group of 25 children. Even in private tutorials, a task as simple as copying a symbol correctly can be hard for a child. In the classroom, where there are countless distractions, and where the teacher cannot pay attention to every student, such tasks are even more difficult.
A teacher who wishes to ensure that all of their students will succeed in mathematics must start by introducing information in steps that are virtually impossible to misinterpret. When I taught the Grade 3 class to add fractions with denominators that were different, I spent the first five minutes simply making sure everyone was able to place the times signs in the correct positions. I was able to cover more steps at once as the children became more confident and attentive, and more adept at copying questions quickly from the board and carrying out complex operations. But if I saw that any students were struggling, I would only ask them to perform a step as simple as counting on their fingers or copying a symbol. (It is possible to teach all elementary mathematics in this way, as the units in Part 2 of this book illustrate.) By introducing new information in mechanical steps, and by allowing enough repetition, I was able to cover far more material than if I had omitted steps.
Just as a footnote: children enjoy discovering mathematical ideas for themselves, through experiments and open-ended activities. But this method of teaching should not, I believe, be used extensively in a large classroom until a teacher is certain the entire class has developed the numerical and logical skills, as well as the confidence and motivation, to do this kind of work. The enriched units in the JUMP manual are based on this style of teaching, but these units are introduced only after students have completed a number of more basic units in which they are guided in small steps.
I am quite willing to believe that a mathematics program very different from the one I advocate, even one based entirely on exploration and open-ended activities, might be superior to JUMP. But any such approach should be formulated in rigorous detail by its proponents and tested exhaustively in large classrooms with teachers who are not confident about math. If it doesn’t work for every student in a class, it shouldn’t be used.
The method of teaching in steps is efficient, in part because it avoids a phenomenon that might well be called “interference.” When a teacher introduces several pieces of information at the same time, students will often, in trying to comprehend the final item, lose all memory and understanding of the material that came before (even though they may have appeared to understand this material completely as it was being explained). With weaker students, it is always more efficient to introduce one piece of information at a time. When I teach rounding, for example, I start with numbers where the student has to round down, first to the nearest tens, then to the nearest hundreds, and so on. Then I repeat this process with numbers that round up. If I mix numbers that round up with ones that round down, as well as numbers that must be rounded to different place values, I will progress much more slowly than if I separate the various kinds of questions (this is particularly true in the classroom, where there are so many distractions).
This step-by-step method can serve to educate teachers as well as students, as it allows teachers to relearn math properly through teaching it. A Grade 4 teacher recently told me she had never understood the concept of probability, even though she had been required to teach it for a number of years. On two occasions, she had seen it demonstrated in her classroom, with sophisticated manipulatives, but she hadn’t been able to reproduce the lesson. After reading the JUMP manual, she was able to teach the subject confidently for the first time.
In fact, many elementary teachers will admit to being as terrified of mathematics as their students. We would see a vast improvement in the teaching of mathematics in our schools if texts and teaching materials were written in meticulous, well-formulated steps, where teachers were shown exactly how to proceed at every point.
In my lessons, I tried to give the third graders a perspective on mathematics (and on learning in general) different from the one they had been taught since kindergarten. The atmosphere I set out to create in the class (with the help of the teacher and the tutors) was as essential to the success of the experiment as the step-by-step method I’ve been describing.
In my first lesson, I told my students that many things in mathematics take practice and getting used to, so they should never assume they were stupid when they found something hard. I told them that as a child I had thought I was stupid myself, but that I eventually discovered new things in mathematics on my own. I assured them they were all smart enough to do well in the subject. If something I presented was unclear, it would be my fault for failing to explain it, not theirs for failing to understand. I asked them to tell me whenever they were lost, so I could make sure they received help.
To anyone who has not observed the results of JUMP in the classroom, these assurances might well seem misguided or unrealistic. A teacher accustomed to watching students struggle with math will almost certainly balk at telling an entire class they can excel in the subject. A former teacher who volunteered for JUMP initially refused to tell her JUMP students they were smart, because, as she put it, “Not all children can be smart.” And several teachers who tested JUMP with their classes only began to encourage their students sufficiently after children they thought were unteachable began to flourish.
When I agreed to test JUMP in classrooms, I hadn’t foreseen how quickly an entire class would respond to a simple promise that they would all do well. In the classes where JUMP was tested, it took one or two weeks for the most difficult students to participate enthusiastically in lessons. In hindsight, I might have anticipated such results. Older children are often too cynical to care about doing well at school, or too bored to believe that a subject like math could ever be interesting. The ones who fall behind develop sophisticated defences to cope with failure. But younger children, who haven’t been robbed of their natural curiosity or of their desire to be doted on by adults, will absorb knowledge without effort if they believe they can do it.
A program that allows children the luxury of success also allows teachers the luxury of giving them praise. This approach may take some getting used to for some teachers, especially those who have learned to rely on more traditional means (such as guilt, fear, or anger) to motivate their students, or who are afraid to encourage false hopes. Some styles of teaching are hard to abandon, even for teachers committed to the ideals of JUMP. Several months ago I watched a teacher who was learning the JUMP method help a boy with an operation from the fractions unit. She had followed the steps in the manual closely, but every time her student made a mistake she would say, with a hint of exasperation in her voice, “Why did you do it that way?” as though the boy had chosen to do the operation incorrectly. It took only a few minutes for the student to become flustered. The teacher pulled me aside and said, “I don’t understand why he doesn’t get it; he’s normally one of the fastest students.” When I told the boy that he had made a very good effort, but he had simply neglected to understand one thing, he was able to perform the operation right away.
I am not saying a teacher must always adopt the same tone with students: I can be quite firm with students who aren’t working or paying attention. As well, I sometimes pretend to believe students can’t do something so they can prove me wrong. But ultimately the students always know that I am impressed with them.
When a class works as a body, the students are carried forward by a common excitement. To maintain a sense of momentum in the class, it helps to have an assistant in the room several times a week, especially when new concepts are introduced. It also helps, in the first few months of the program, to have several extra tutors on hand once a week, so that students who are extremely delayed, unmotivated, or inattentive can be given lessons individually, or in small groups. Three or four students in each class needed this extra help in the classes where JUMP was tested. With most of these students, the extra tutoring was only required for a few months: a girl in Grade 4 who had received no formal schooling and who could barely count when she started the fractions unit needed only two months of weekly 40-minute tutorials to become one of the top students in the class (she received 98% on her fractions test). A boy in the same class who couldn’t concentrate well enough to take part in lessons or complete assignments now always waves his hand to answer questions.
A teacher who kept their weakest students in at recess or after school several times a week might be able to match the results of a program with tutors. I think it is unreasonable, however, to expect this extra work of teachers: from what I have observed, they are already overtaxed managing 25 to 30 students. If we cared about educating children, our teachers would be given more support in the classroom, particularly in the early grades.
I was glad, in my first lessons with the Grade 3 students, to have an assistant in the class to help with those who were delayed or unmotivated. Whenever I demonstrated a step, the teacher and I would walk from desk to desk, making sure these students could repeat the step. After several weeks, however, I found I needed an assistant for an entirely different reason. As all of the students were beginning to complete their work without errors, the ones who had needed help would race to finish their assignments so they could be given a mark. They were thrilled to be able to call an adult to their desks to write, “Wow,” or “Excellent,” or “Perfect” in their books. For many, it was the first time they had received such praise at school.
I will never forget the moment I saw the aspect of the educational system that most impedes learning. I was standing in a room full of excited children who were all boisterously calling for the tutors to come and mark their work. It struck me how different the scene would be if, instead of As and A-pluses, the tutors were writing Cs and Ds in half of the children’s books. With infants, the capacity to absorb knowledge cannot be separated from the capacity to be adored. It seems likely, from the way the most difficult students flourish in JUMP, that this is still true of children who are 10 and 11 years old. Otherwise, why would children with such different needs and impairments, ranging from severe psychological and behavioural problems to dyslexia and attention deficits, respond so quickly and in such a similar manner to the prospect of learning mathematics?
If children need to feel admired in order to learn, it follows that a system of education that measures or rewards their progress by assigning them a rank will never be as productive as one in which they are all expected to meet (roughly) the same standard. When only a fraction of a class receives a mark that is considered good, the majority will inevitably convince themselves that the subjects they did poorly in are difficult or boring.
I’m not advocating that children be given inflated marks to make them feel good. The marks the Grade 3 students received on my fractions test were an accurate measure of their ability to manipulate fractions. Indeed, the results of JUMP show that schools could set a standard, much higher than the one they have adopted, that every child could be expected to meet.
In a court of law, only adults are held accountable for their actions. In schools, however, children who are lazy or uncooperative are often treated as if they have made a free and informed decision to receive a grade of D or F. If children are more like infants than adults, then adults, not children, should be held responsible when a child fails. Adults should provide encouragement and rewards, as well as the attention that drives healthy children to learn from the day they are born.
Tests would not be used as threats or as measures of rank in a system that assumed education was the right of every child; they would serve as a way to guarantee that a child was not being neglected and as certificates and rewards for a child’s work. In the Grade 3 class I taught, children who were absent for the fraction test nagged me until I allowed them to write it: for them, the test was a chance to show what they had learned.
Schools cannot be improved simply by compelling students to write standardized tests, as many politicians seem to believe. Tests accomplish nothing if teachers aren’t trained or provided with the means to prepare students to do well on those tests.
After seeing how children flourish with even a modest amount of attention, I have come to believe that when a child fails a test it should be regarded as a failure of our system of education. And when millions of children, year after year, fail tests they could easily pass, it should be regarded as the failure of an entire society to care for its young.
The costs of neglecting so many children are mounting, especially in neighbourhoods where parents cannot afford to hire tutors for their children. Hundreds of children would have fallen behind if not for the volunteers of JUMP alone. For want of classroom assistants, effective texts, and trained teachers, millions of other children will become convinced that they are bad at math before they graduate from school. It is disheartening to think of the potential wasted in our schools, and of the myths that will excuse this waste, until the day children are granted their right to be educated, and there is no mark assigned for failure.