**Authors: **Jo Boaler

The traditional book gives two pages of explanations like these before presenting twenty-six “oral exercise” questions and forty-nine “written exercise” questions,

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such as:

Simplify 9 + (18 – 2) and 2 • (b + 2)

In contrast, the IMP book introduced students to variables through a particular situation—that of the nineteenth-century settlers who traveled from Missouri to California to set up life on the West Coast. The students are told, “You will encounter some very important mathematical ideas—such as graphs, different use of variables, lines of best fit, and rate problems—as you travel across the continent.”

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They are then given various exercises that require them to represent the settlers’ situations using mathematical tools such as graphs and variables. For example, the students are told about families and some general conventions, such as “anyone more than fourteen years old is considered an adult.”

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They are then introduced to variables through this question:

The Hickson household contains 3 people of different generations. The total age of the 3 family members is 90.

a) Find reasonable ages for the 3 Hicksons.

b) Find another reasonable set of ages for them.

One student in solving this problem wrote:

C + (C + 20) + (C + 40) = 90

What do you think C means here?

How do you think the student got 20 and 40?

What set of ages do you think the student came up with? Try this question, it can be solved in many ways.

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In the IMP curriculum, students are

gradually

introduced to the concept of a variable—one of the most important concepts in the mathematics curriculum—before being asked to interpret and use variables in representing a situation mathematically. The students are also encouraged to

discuss

variables, exploring their meaning and when they are used, and generally to raise any questions or problems they have. The traditional curriculum, by contrast, tells students what variables are on page

1

of the book and then leads students through seventy-five practice questions. In requiring that students consider

situations

and that they discuss the

meaning

of concepts such as variables, the IMP curriculum focuses more on understanding and less on practicing methods. In the traditional curriculum, students practice a lot more. And this is where the crux of the disagreement lies, with one group of people believing that students need to spend a lot of time practicing, and the other group believing that it is better to understand an idea than it is to practice it by rote.

My aim in this book is not to promote a curriculum. I am aware that traditional and nontraditional books can be taught

well or badly and that any book requires a knowledgeable and caring teacher. But we would not be facing our current crisis if the people who were concerned about the newer books had worked with the mathematicians to improve them, rather than declaring war.

The wars in California were instigated by organizations such as Mathematically Correct, which hosted a Web site explaining that they are involved in a “great educational war” to save traditional math teaching. The site, which included no contact names or people, was full of articles attacking any new mathematics approaches and gave instructions on ways to get rid of any reforms in schools. Behind the Web site was a group of activists who scoured the country, looking for schools that were using new books so that they could land in the area, with a wealth of resources, and organize parents to crush the reforms. One of the most active members of Mathematically Correct wrote several threatening messages to me, because my published research has shown that students need opportunities to learn actively. He wrote that I pose a great threat, as I am a professor from a top university and have data. He recommended that readers of his Web pages visit university education departments and “nuke ’em all, dammit.”

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Others have told me that I had “better not” talk about my research publicly in America. Such threats and attempts to suppress research evidence may seem incredible, but they are characteristic of the events that make up the math wars.

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For more detailed accounts of this phenomenon that continues to suppress progress in the education of our children, I recommend reading University of California, Berkeley, professor Alan Schoenfeld’s article “The Math Wars” and Michigan State professor Suzanne Wilson’s book

California Dreaming,

both very readable accounts of a truly unfortunate set of events.

How It All Started

In the 1980s there was widespread awareness that students were failing mathematics in shockingly high numbers, and a range of reforms were introduced into schools, prompted by the National Council of Teachers of Mathematics (NCTM), which issued a new set of curriculum standards in 1989. Math books were quickly rewritten by publishers and filled with bright colors and real-world contexts. Teachers were instructed to be facilitators rather then lecturers and to have children work in groups. The reforms were introduced relatively quickly and often without consultation with parents. Some teachers were not trained to work in the new ways and so found it difficult. Critics claimed that mathematics was being threatened, that students were no longer learning standard methods, and that they were wasting time in groups chatting with friends instead of working. But rather than opening a dialogue between the different people who cared about mathematics teaching, battle lines were drawn and certain organizations such as Mathematically Correct declared war.

The same groups are now fighting the introduction of Common Core Mathematics.

Learning without Thought

Those involved in the math wars think of different versions of mathematics teaching as either traditional or reform, and debates revolve around these two imaginary poles. In my research I have found that such categories do not actually mean much and that both camps include many types of teachers, teaching, and methods, some of which are highly effective and some not. Certain teachers might be described as traditional because they lecture and they have students work individually, but they also ask students great questions, engage them in interesting mathematical inquiries, and give students opportunities to solve

problems, not just rehearse standard methods. Such teachers are wonderful and I wish there were many more of them. The type of traditional teaching that concerns me greatly and that I have identified from decades of research as highly ineffective is a version that encourages

passive learning.

In many mathematics classrooms across America the same ritual unfolds: teachers stand at the front of class demonstrating methods for twenty to thirty minutes of class time each day while students copy the methods down in their books, then students work through sets of near-identical questions, practicing the methods. Students in such classrooms quickly learn that

thought

is not required in math class and that the way to be successful is to watch the teachers carefully and copy what they do. In interviews with hundreds of students from such classes, I have asked them what it takes to be successful in math class, and they almost always give the exact same answer:

pay careful attention.

As one of the girls I interviewed told me, “In math you have to remember; in other subjects you can think about it.”

Students taught through passive approaches follow and memorize methods instead of learning to inquire, ask questions, and solve problems. I have interviewed hundreds of students taught in such ways and they usually reflect on their experiences saying such things as “I’m just not interested in, just, you give me a formula, I’m supposed to memorize the answer, apply it, and that’s it,” and “You have to be willing to accept that sometimes things don’t look like—they don’t seem that you should do them. Like they have a point. But you have to accept them.” Students who are taught using passive approaches do not engage in sense making, reasoning, or thought (acts that are critical to an effective use of mathematics), and they do not view themselves as active problem solvers. This passive approach, which characterizes math teaching in America, is widespread and ineffective.

When students try to memorize hundreds of methods, as students do in classes that use a passive approach, they find it extremely hard to use the methods in any new situations, often resulting in failure on exams as well as in life. The secret that good mathematics users know is that only a few methods need to be memorized and that most mathematics problems can be tackled through the understanding of mathematical concepts and active problem solving. I am now working with the PISA (Programme for International Student Assessment) team, based in the OECD (Organisation for Economic Co-operation and Development) in Paris. The PISA team not only collects data on students’ math achievements, but it also collects data on the strategies students use in mathematics and relate these to performance.

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One of the strategies the PISA team has identified is “memorization.” Some students think their role in math classrooms is to memorize all the steps and methods. Other students think their role is to connect ideas. These different strategies link, unsurprisingly, to achievement, and the students who memorize are the lowest achieving in the world. The highest-achieving students are the ones who think about the big ideas in mathematics. Unfortunately, in most US classrooms mathematics is presented as a set of procedures, leading students to think that their role in learning mathematics is to memorize. This PISA result gives us an interesting insight into the low performance of students in the United States, relative to the other countries tested.

I have spent my research career conducting unusual studies of learning. They are unusual because, instead of dropping in on students to see what they are doing in math classes, I have followed students through years of middle and high school, performing

longitudinal

studies. I have spent thousands of hours, with teams of graduate students, collecting data on students learning mathematics in different ways. This has included watching hundreds of hours of math classes, interviewing students about their experiences, giving them questionnaires on

mathematical beliefs, and performing assessments to probe students’ understanding. These studies have revealed that many math classrooms leave students cold, disinterested, or traumatized. In hundreds of interviews with students who have experienced passive approaches, they have told me that thought is not required, or even

allowed,

in math class. Children emerge from passive approaches believing that they only have to be obedient and memorize what the teacher tells them to do. They learn that they must simply memorize methods even when methods do not make sense. It is ironic that math—a subject that should be all about inquiring, thinking, and reasoning—is one that students have come to believe requires

no thought.

In 1982, before teaching reforms were introduced in classrooms, students were asked in a national assessment to estimate the answer to

and they were given a choice of answers:

1, 2, 19, or 21.

Both numbers, 12/13 and 7/8, are close to 1, so an estimate tells us that their sum is close to 2. In the national assessment only 24 percent of thirteen-year-olds and a stunning 37 percent of seventeen-year-olds answered the question correctly. Most students chose the nonsensical answers of 19 or 21.

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The seventeen-year-olds did not appear to make sense of the question and estimate, probably because they were trying to follow a rule and they made mistakes in the enactment of the rule.

The fact that students are drilled in methods and rules that do not make sense to them is not just a problem for their understanding of mathematics. Such an approach leaves students frustrated, because most of them want to understand what they are learning. Students want to know how different mathematical methods fit together and why they work. This is especially true for girls and women, as I shall explain in

chapter 6

. The following response from Kate, a girl taking calculus in a traditional class, resembles those that I have received from many young people I have interviewed:

We knew how to do it. But we didn’t know why we were doing it and we didn’t know how we got around to doing it. Especially with limits, we knew what the answer was, but we didn’t know why or how we went around doing it. We just plugged into it. And I think that’s what I really struggled with—I can get the answer, I just don’t understand why.

Young people are naturally curious and their inclination—at least before they experience traditional teaching—is to make sense of things and to understand them. Most American math classes rid students of this worthy inclination. Kate was at least fortunate to still be asking “Why?” even though she, like others, was not given opportunities to understand why the methods worked. Children begin school as natural problem solvers and many studies have shown that students are better at solving problems

before

they attend math classes.

17,

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They think and reason their way through problems, using methods in creative ways, but after a few hundred hours of passive math learning students have their problem-solving abilities drained out of them. They think that they need to remember the hundreds of rules they have practiced and they abandon their common sense in order to follow the rules.

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