A HUNDRED YEARS AGO, RESEARCHERS IN LOGIC DISCOVERED that virtually all of the concepts used by working mathematicians could be reduced to one of two extremely basic operations, namely, the operation of counting or the operation of grouping objects into sets. Most people are able to perform both of these operations before they enter kindergarten. It is surprising, therefore, that schools have managed to make mathematics a mystery to so many students.
The following chapters contain excerpts from The JUMP Teaching Manual. The units selected for this book are only a small part of the JUMP program and do not cover the full elementary curriculum. They do, however, give a fairly complete picture of the method of teaching used in JUMP. The units are intended as models or templates: a teacher or parent can easily adapt the techniques introduced here to teach any topic in elementary mathematics.
The units in Chapters 6 and 7, which contain material on fractions and on multiplication and division, are among the most basic units in the JUMP manual. They demonstrate how a teacher can reduce mathematics to simple operations, such as counting and grouping objects into sets, that any student can perform. The unit in Chapter 9 on ratios and percents shows how this approach can be extended to more advanced material. (The units from the manual in Chapters 6, 7, and 9 are not complete: they should be supplemented with conceptual exercises of the sort introduced at the end of Chapter 6.)
It is not the intention of the JUMP program to produce students who can only do mathematics by rote, or who experience mathematics as an endless series of mechanical drills. Even in the most basic units, students are expected to explain how operations work and generalize rules to deal with new cases by themselves. A student who has completed these units should have acquired the skills and the motivation to begin more independent work in mathematics. The units in Chapters 8, 10, and 11 show how a student can be led, through the systematic use of games, puzzles, and other activities, to solve problems and discover mathematical ideas on their own.
In Chapter 8, the notion of a coordinate system is introduced by means of a simple card trick. By figuring out how to perform the trick, students will learn (and likely remember) how the position of an object relative to an observer can be represented by a pair of numbers. And they will discover, by modifying the trick, several deep ideas about mathematics and space.
In Chapter 10, a method of teaching students to answer more open-ended mathematical questions is introduced. Mathematicians and scientists are often faced with problems that have more than one solution, or problems in which a solution can only be found by searching among many possibilities. In school, students are usually introduced to problems of this sort in a manner that is very unstructured. Problems are presented in stories or real-world situations in which different concepts have been mixed. Usually the particular skills required to solve the problem have not been isolated. Often the language of the problem is a barrier to children who have trouble reading English. In Chapter 10, all of the terms a student will need to answer a question are introduced in preparatory exercises. As well, students are shown how to solve more advanced problems by first working on “toy” models of a problem.
Chapter 11 introduces the notion of a finite state automaton, a simple model of a computer that can be played like a board game. By designing and writing a code for this model, a student will learn how a working computer can be represented by a string of letters. And by feeding data into their finite state automata, they will discover how a computer performs basic functions, such as identifying patterns in numbers and codes.
In teaching mathematics I often use simple diagrams or concrete materials. A finite state automaton can be “built” using a penny, a piece of paper, and a pencil. The notion of fractional equivalence can be taught using coloured blocks (Chapter 6), and objects in sets may be represented by lines inside boxes (Chapter 7). In abstract mathematics, the ability to draw a picture or create a model in which only the essential features of a problem are represented is an essential skill. Teachers may use more sophisticated manipulatives to supplement the units if they wish. With a delayed student, however, teachers should follow the mechanical procedures of the fractions unit very strictly. In classrooms where we have tested the JUMP method, students who had been struggling showed marked improvements in motivation and ability in a matter of weeks. They found it easier to work with concrete materials after they had finished the unit.
Students in Ontario must demonstrate a solid understanding of mathematical concepts to meet the requirements of the elementary curriculum: they must be capable of explaining how they found solutions to problems and of applying their knowledge in novel situations. Why, then, in the fractions unit of The Jump Teaching Manual, are operations introduced in entirely mechanical steps (and why at a level beyond current expectations for elementary students)? I have described already the enthusiasm that spreads contagiously through a class when students believe they are capable of learning material beyond their grade level. A teacher will find it hard to generate this kind of enthusiasm if they allow even a third of their class to feel they have no aptitude for mathematics. The fractions unit is designed specifically to help a teacher capture the attention of their weakest students and to build the basic mathematical skills those students need to progress to more advanced conceptual work. (See Chapters 3 and 6 for an account of these skills.)
None of the exercises in the first 15 sections of the fractions unit requires a mastery of English or of higher-order concepts. Students who speak English as a second language (a growing part of the population in many schools), students who are delayed in reading or writing, and students who have fallen behind academically are allowed the same chance to succeed as their peers. To do well on the Advanced Fractions Test (see page 113), students must simply perform the various operations introduced in the unit correctly. (The units that follow the fractions unit in the JUMP manual gradually introduce more conceptual work.)
Often the teachers who have seen their weakest students succeed on the fractions test are changed by the experience: they become more conscientious about breaking explanations into steps and are less likely to underestimate their students. The fractions unit is as much a test of a teacher’s ability to teach the weakest students in their class as it is of a student’s mastery of operations involving fractions.
Teachers who try the fractions unit but find they are leaving students behind should ask: Can all of my students multiply without hesitation by twos, threes, and fives (as taught in section F-1 of the unit)? Am I being careful to isolate the most basic steps in the operations? Am I teaching one step at a time and allowing enough repetition? Am I following the hints given in the unit for teaching weaker students? Am I expecting my students to learn or remember things that aren’t relevant to the steps I’m trying to teach? Am I building momentum in my lessons by allowing all of my students to succeed? Am I giving sufficient encouragement? Am I spending extra time with my weaker students and giving them extra practice? Am I excited about my students’ progress? Do I believe that all of my students (even those diagnosed with severe learning disabilities) can learn mathematics?
Some educators believe that teachers can actually arrest or delay the intellectual development of their students by teaching them how to perform an operation (such as adding fractions) before they have learned the concepts underlying the operation. However, while it is true that children who understand mathematics should ultimately be able to explain how they found a solution to a problem or why a rule works, it does not follow that a child who initially learns an operation by rote has no hope of learning the concept later. I learned fractions by rote and I think I can claim to understand fractions.
The idea that it is always harmful to teach rules before concepts is not supported by the actual practice of mathematics. John von Neumann, one of the great mathematicians of the 20th century, said that understanding mathematics is generally a matter of getting used to things. Mathematicians often approach difficult concepts by becoming familiar first with the rules and operations associated with the concepts. (In many branches of abstract mathematics, rules and concepts are hard to distinguish from each other.) To test a child’s understanding of mathematics only by testing their grasp of concepts (while overlooking their mastery of rules and operations) is to neglect an important part of a child’s education. Students who can add and subtract fractions without making errors already understand a good deal about fractions.
Mathematical discoveries are often based on ideas that are incomplete or partially false. Even in the mind of a mathematician, a concept will seldom arrive full-blown. A well-designed curriculum, therefore, will not penalize children academically or cause them to become discouraged if they fail to understand a concept immediately. As children mature, they acquire concepts naturally, through experience in the world. I have taught a number of adults to understand fractions in less than an hour. Even those who claimed to be hopeless in mathematics at school were shocked to see how quickly they could learn as adults.
In spite of the views I have expressed above, I believe that it is generally better to teach concepts before rules and that, whenever possible, children should be allowed to discover mathematical principles on their own. In schools where JUMP has raised the level of the students, mathematics is taught more conceptually each year. But in schools where children have been neglected, the benefits of starting with small steps far outweigh the disadvantages. All students are taken through the elementary curriculum at the same pace, with as much guidance as they need, until they have developed the confidence and skills they will require to explore more substantial mathematics.