**Authors: **Jo Boaler

Mathematical notation no more

is

mathematics than musical notation is music. A page of sheet music represents a piece of music, but the notation and the music are not the same; the music itself happens when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive; it exists not on the page but in our minds. The same is true for mathematics.

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Mathematics is a performance, a living act, a way of interpreting the world. Imagine music lessons in which students worked through hundreds of hours of sheet music, adjusting the notes on the page, receiving checks and crosses from the teachers, but never playing the music. Students would not continue with the subject because they would never experience what music

is.

Yet this is the situation that continues, seemingly unabated, in mathematics classes.

Those who use mathematics engage in mathematical

performances.

They use language in all its forms, in the subtle and precise ways that have been described, in order to do something with mathematics. Students should not just be memorizing past methods; they need to engage, do, act, perform, and problem solve, for if they don’t

use

mathematics as they learn it, they will find it very difficult to do so in other situations, including examinations.

Maryam Mirzakhani is a mathematician at Stanford who was featured in newspapers across the world by winning the Fields Medal, the most prestigious prize in mathematics. When news of the award spread,*t*

22

and I told the newspaper that just weeks before I had chaired a PhD defense for one of Maryam’s students in Stanford’s

math department. A PhD defense is the occasion when a student defends her PhD dissertation to a committee of scholars. That day I sat with the other members of the committee, all mathematicians, and watched as Maryam’s student, a young woman, paced the room, showing visual representations and sharing conjectures, using mathematics creatively to connect different ideas. Many times in the defense she was asked a question to which she replied, “I don’t know.” This answer was perfectly reasonable and accepted by the mathematicians on the committee because she was exploring new territory to which no one had answers. This struck me as highly significant because schoolchildren everywhere would be shocked to see that high-level mathematics involves such creativity and uncertainty.

The erroneous thinking behind many school approaches is that students should spend years being drilled in a set of methods that they can use later. The mathematicians who oppose change are most concerned about the students who will enter graduate programs in mathematics. At that point students will encounter real mathematics and use the tools they have learned in school to work in new, interesting, and authentic ways. But by this time most students have given up on math. We cannot keep pursuing an educational model that leaves the best and the only real taste of the subject to the end, for the rare few who make it through the grueling years that precede it. If students were able to work for at least some of the time in the ways mathematicians do—posing problems, making conjectures using intuition, exploring with and refining ideas, and discussing ideas with others—then they would not only be given a sense of true mathematical work, which is an important goal in its own right,

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they would also be given the opportunities to enjoy mathematics and learn it in the most productive way.

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25,

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wavebreakmedia

Identifying the Problems

The Night of Nonsense

S

oon after I first became a Stanford professor, I was introduced to what I believe to be a damaging and very strange phenomenon. The phenomenon—which began in California, but has now taken hold across the United States—is known as the math wars,

1

a series of unproductive and heated exchanges between advocates of different mathematics approaches. Participants in the math wars have been known to engage in hunger strikes, secret meetings, and extended campaigns of abuse, all in the pursuit of their preferred method of teaching. One outcome of the math wars is that good teachers have been driven out of the teaching profession after years of bullying by extremists. Another result is that any questions or discussions about change in mathematics teaching have been suppressed and paths to improvement have been blocked.

Ironically, the issue that has gotten so many people hot under the collar is not even the most important one. The subject of the math wars is the

curricula

that teachers use in their classrooms—the published sets of textbooks that schools and districts choose to adopt. Of course, the books used in classrooms are important and we want our children to be working with high-quality materials that teach mathematics well, but the most important factor in school effectiveness, proved by study after study, is not the curriculum but the teacher.

2,

3,

4,

5

Good teachers can make mathematics exciting even with a dreary textbook. Conversely, bad teachers do not become good just because a book is written well, but the math wars have prevented people from focusing upon and supporting good teaching; instead, they have, quite deliberately, turned attention away from teachers and tried to

force

particular curricula upon them. Emily Moskam, whose classroom I described in the introduction, has now left teaching. A teacher who could inspire children to love and use mathematics, at the highest levels, like no other I have seen could not stand being forced to use textbooks that she knew were ineffective and that were damaging her students’ learning. She could see no way forward and ultimately chose to leave the profession.

I experienced my first personal contact with the math wars on a cool November evening in California. I was thinking about including Emily Moskam’s high school in my research study when I learned of a meeting taking place at her school to discuss its mathematics curriculum. Strangely, although the meeting was for parents to consider the school’s math approach, no teachers or administrators had been invited. As parents walked into the hall they saw three women standing at the front, passing papers back and forth. All three were parents of freshmen in the school and one later confided to me that she had spent a year gathering data for that meeting.

The meeting had been arranged because the math department at Emily’s school had abandoned the traditional books and methods of teaching that they had used for many years, with very little success, and started using an award-winning curriculum that engaged students in real—and impressively hard—problems to solve (IMP). Students responded well to the new curriculum. They reported enjoying math more and many more of them were taking high-level classes. The teachers told me they were the happiest they had ever been—they were attending professional development workshops on weekends as well as spending time discussing good teaching methods with one another, and their students were doing well. It was around this time that a group of extreme traditionalists heard about the school’s new approach and started to plot its downfall. They needed parents at the school to be the public face of their attack, and the women who were running the meeting that night had stepped forward.

In that first momentous meeting the women bombarded the gathered parents with data that had been fed to them, telling them that if their children continued with the new math program, then they would not be eligible to go to college and that test scores would fall. They showed graphs that had been constructed to give the impression of falling student test scores. It is easy to manipulate educational data to give the illusion of a bad program, as it is always possible to find a set of students somewhere whose scores have declined—even if for a particular reason or for a very brief period of time—and then to produce graphs with inappropriate axes that make tiny differences appear huge or that generalize from ten students. To support their claim that students would be ineligible for college, the women had phoned a range of prestigious colleges and asked them a question to the effect of “Would you accept a student who had not taken any math in high school but had just talked about math?” Many of the colleges said no and the women started composing a list. These tactics may sound incredible,

but the people involved felt justified in trying to promote their position, using any method that they could, as they believed that they were involved in a “great educational war”

6

and that any tactics, no matter how underhanded, are admissible when at war.

That night they also suggested something to the parents that greatly upset the teachers when they heard about it—they implied that the teachers were being paid by the program and they were supporting it for personal gain. The parents left the hall in a somber mood that night, some of them skeptical, many of them scared. I was amazed that such a meeting could take place, but that was just the beginning.

In the weeks that followed, many of the parents, sensibly, asked for proper data. They knew of other students who had taken this math approach and been very successful. They contacted some of the colleges that would, apparently, not accept students from the math program and they found out that the admissions officers had not been asked about the program, only a ridiculous question about talking instead of learning math. Stanford University was one of the universities being used as an example of a university that would not accept students who had studied with the new curriculum approach—despite having admitted many students who had studied with the curriculum the teachers were using. When Stanford heard that it was being used as an example of a university that would not admit such students, its admissions officers quickly wrote a letter saying that this was not true and that the admissions office did not discriminate against any high school math program. Unfortunately, the damage had already been done.

It was around this time that I was sitting in a local coffee shop with a group of my graduate students discussing the research we were planning to conduct at the school. A girl and her mother walked over to our table. The girl asked us: “Are you discussing that new math program at the school? It has ruined my life!” When we asked her why, she and her mother explained that she would no longer be eligible to go to college. We explained to them both that this was not true. They thanked us for the information, but they continued to look upset.

Sensing that they had not won over all of the parents, the three women tried another line of attack. They moved on to the students, trailing after them at break times, telling them they would be ineligible for college, and asking them to sign a petition to end the math program. The students became more and more confused and scared, so many of them signed the petitions. By the time the teachers were told anything about the secret campaign, it was all over. The women, supported and organized by the extreme traditionalists, had scared enough parents and convinced the school board that the teachers must go back to teaching traditionally. Now desks are in rows, teachers lecture, students silently copy methods and then practice with lots of examples, and the problem solving that students used to love is no longer in evidence. The teachers at the school were demoralized and defeated.

There were two types of books that were at the center of this. Both introduced students to the same mathematical methods and procedures, but the IMP books attempt to do so in a way that made the mathematics more meaningful for students. Consider, for example, the way that the IMP book and the traditional book (brought in by the parent activists) introduced algebraic variables to the students. The traditional book begins with a brief example of rental charges at an ocean shop and then has the following text:

The letter

h

stands for the hours shown in the table: 1, 2, 3, or 4. Also,

h

can stand for other hours not in the table. We call

h

a variable.

A**variable**is a symbol used to represent one or more numbers. The numbers are called

h,

is called a

Another way to indicate multiplication is to use a raised dot, for example, 4.50 • 4. In algebra, products that contain a variable are usually written without the multiplication sign because it looks too much like the other

x,

which is often used as a variable.

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