**Authors: **Jo Boaler

Photo courtesy of Jo Boaler

After a while the teacher came over and looked at the boys’ work, talking with them about their diagrams, graphs, and algebraic expressions, and probing their thinking to make sure they

understood the algebraic relationships. He asked Pedro where the 7 (from 3

x

+ 7) was represented on his graph. He showed the teacher and then decided to show the +7 in the same color on his tile patterns, on his graph, and in his algebraic expression. The communication of key features of functions using color coding was something all students were taught in the Railside approach, to give meaning to the different representations. This helped the students learn something important—that the algebraic expression shows something tangible and that the relationships within the expression can also be seen in the tables, graphs, and diagrams.

Juan, sitting at the same table, had been given a more complicated, nonlinear pattern that he had color coded in the following way:

See if you can work out how the pattern is growing and the algebraic expression that represents it!

As well as producing posters that showed linear and non-linear patterns, the students were asked to find and connect patterns, within their own pile patterns, and across all four teammates’ patterns, and to show the patterns using technical writing tools. Because some of the pile patterns were nonlinear, this was a complicated task for ninth graders, and it provoked much discussion, consternation, and learning! One aim of the lesson was to teach students to look for patterns within and among representations and to begin to understand generalization.

The tasks at Railside were designed to fit within the separate subject areas of algebra and geometry, in line with the tradition of content separation in American high schools. Still, the problems the students worked on were open enough to be thought

about in different ways, and they often required that students represent their thinking using different mathematical representations, emphasizing the connections between algebra and geometry.

The Railside classrooms were all organized in groups, and students helped each other as they worked. The Railside teachers paid a lot of attention to the ways the groups worked together, and they taught students to respect the contributions of other students, regardless of their prior achievement or their status with other students. One unfortunate but common side effect of some classroom approaches is that students develop beliefs about the inferiority or superiority of different students. In the other classes we studied, classes that were taught traditionally, students talked about other students as smart and dumb, quick and slow. At Railside the students did not talk in these ways. This did not mean that they thought all students were the same, but they came to appreciate the diversity of the class and the various attributes that different students offered. As Zane described to me: “Everybody in there is at a different level. But what makes the class good is that everybody’s at different levels, so everybody’s constantly teaching each other and helping each other out.” The teachers at Railside followed an approach called

complex instruction,

1

a method designed to make group work more effective and to promote equity in classrooms. They emphasized that all children were smart and had strengths in different areas and that everyone had something important to offer when working on math.

As part of our research project we compared the learning of the Railside students to that of similar-size groups of students in two other high schools who were learning mathematics through a more typical, traditional approach. In the traditional classes the students sat in rows at individual desks, they did not discuss mathematics, they did not represent algebraic relationships

in different ways, and they generally did not work on problems that were applied or visual. Instead, the students watched the teacher demonstrate procedures at the start of lessons and then worked through textbooks filled with short, procedural questions. The two schools using the traditional approach were more suburban, and students started the schools with higher mathematics achievement levels than the students at Railside. But by the end of the first year of our research study, the Railside students were achieving at the same levels as the suburban students on tests of algebra; by the end of the second year, the Railside students were outperforming the other students on algebra and geometry tests.

In addition to the high achievement at Railside, the students learned to enjoy math. In surveys administered at various times during the four years of the study, the students at Railside were always significantly more positive and more interested in mathe-matics than the students from the other classes. By their senior year, a staggering 41 percent of the Railside students were in advanced classes of precalculus and calculus, compared with only 23 percent of students from the traditional classes. Further, at the end of the study, when we interviewed 105 students (mainly seniors) about their future plans, almost all of the students from the traditional classes said that they had decided not to pursue mathematics as a subject—even when they had been successful. Only 5 percent of students from the traditional classes planned a future in mathematics compared with 39 percent of the Railside students.

There were many reasons for the success of the Railside students. Importantly, they were given opportunities to work on interesting problems that required them to think (not just to reproduce methods), and they were required to discuss mathematics with one another, increasing their interest and enjoyment. But there was another important aspect of the school’s approach that is much more rare—the teachers enacted an expanded

conception of mathematics and “smartness.” The teachers at Railside knew that being good at mathematics involves many different ways of working, as mathematicians’ accounts tell us. It involves asking questions, drawing pictures and graphs, rephrasing problems, justifying methods, and representing ideas, in addition to calculating with procedures. Instead of just rewarding the correct use of procedures, the teachers encouraged and rewarded all of these different ways of being mathematical. In interviews with students from both the traditional and the Railside classes, we asked students what it took to be successful in math class. Students from the traditional classes were unanimous: they all said that it involved paying careful attention—watching what the teacher did and then doing the same. When we asked students from the Railside classes, they talked of many different activities such as asking good questions, rephrasing problems, explaining ideas, being logical, justifying methods, representing ideas, and bringing a different perspective to a problem. Put simply, because there were many more ways to be successful at Railside, many more students

were

successful.

Janet, one of the freshmen, described to me the way that Railside was different from her middle school experience: “Back in middle school the only thing you worked on was your math skills. But here you work socially and you also try to learn to help people and get help. Like you improve on your social skills, math skills, and logic skills.” Another student,

Jasmine, also talked about the variety in Railside’s approach, saying that “with math you have to interact with everybody and talk to them and answer their questions. You can’t be just like ‘Oh, here’s the book. Look at the numbers and figure it out.’”

We asked Jasmine why math was like that. She answered:

“It’s not just one way to do it. . . . It’s more interpretive. It’s not just one answer. There’s more than one way to get it. And then it’s like ‘Why does it work?’”

The students highlighted the different ways that mathematics problems could be

solved and the important role played by mathematical justification and reasoning. The students at Railside recognized that helping, interpreting, and justifying were critical and valued in their math classes.

In addition, the teachers at Railside were very careful about identifying and talking to students about all the ways they were smart. The teachers knew that students—and adults—are often severely hampered in their mathematical work by thinking they are not smart enough. They also knew that all students could contribute a great deal to mathematics, so they took it upon themselves to identify and encourage students’ strengths. This paid off. Any visitor to the school would have been impressed by the motivated and eager students who believed in themselves and who knew they could be successful in mathematics.

The Project-Based Approach

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Phoenix Park School

The day that I walked into Phoenix Park School, in a low-income area of England, I didn’t know what to expect. I had invited the maths department at the school to be a part of my research

project. I knew that the department used a project-based approach, but I did not know much more than that. I made my way across the playground and into the school buildings on that first morning with some trepidation. A group of students congregated outside their maths classroom at break time and I asked them what I should expect from the lesson I was about to see. “Chaos,” said one of the students. “Freedom,” said another. Their descriptions were curious, and they made me more excited to see the lesson. Some three years later, after moving with the students through school, observing hundreds of lessons, and researching the students’ learning, I knew exactly what they meant.

This was to be my first longitudinal research project on different ways of learning mathematics and, as with the Railside study, I watched hundreds of hours of lessons, interviewed and gave surveys to students, and performed various assessments. I chose to follow an entire cohort of students in each of two schools, from when they were thirteen to when they were sixteen years of age. One of the schools, Phoenix Park, used a project-based approach, the other, Amber Hill, used the more typical traditional approach. The two schools were chosen because of their different approaches, but also because their student intakes were demographically very similar, the teachers were well qualified, and the students had followed exactly the same mathematics approaches up to the age of thirteen, when my research began. At that time, the students at the two schools scored at the same levels on national mathematics tests. Then their mathematical pathways diverged.

The classrooms at Phoenix Park did look chaotic. The project-based approach meant a lot less order and control than in traditional approaches. Instead of teaching procedures that students would practice, the teachers gave the students projects to work on that

needed

mathematical methods. From the beginning of year 8 (when students started at the school) to three-quarters of

the way through year 10, the students worked on open-ended projects in every lesson. The students didn’t learn separate areas of mathematics, such as algebra or geometry, as English schools do not separate mathematics in that way. Instead, they learned “maths,” the whole subject, every year. The students were taught in mixed-ability groups, and projects usually lasted for about three weeks.

At the start of the different projects, the teachers would introduce students to a problem or a theme that the students explored, using their own ideas and the mathematical methods that they were learning. The problems were usually very open so that students could take the work in directions that interested them. For example, in “Volume 216,” the students were simply told that the volume of an object was 216 and they were asked to go away and think about what the object could be, what dimensions it could have, and what it could look like. Sometimes, before the students started a new project, teachers taught them mathematical content that could be useful to them. More typically though, the teachers would introduce methods to individuals or small groups when they encountered a need for them within the particular project on which they were working. Simon and Philip of year 10 described the school’s maths approach to me in this way:

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