What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (14 page)

In a mixed-ability group, the teacher has to
open
the work, making it suitable for students working at different levels and different speeds. Instead of prejudging the achievement of students and delivering work at a
particular
level, the teacher has to provide work that is multileveled and that enables students to work at the highest levels they can reach. This means that work can be at the right level and pace for all students.

Borderline Casualties

When teachers assign students to different groups, they make decisions that affect their long-term achievement and their life chances. Despite the importance of such decisions, they are often made on the basis of insufficient evidence. In many cases, students are assigned to groups on the basis of a single test score with some students missing the high groups because of one point. That one missed point, which students may have scored on another day, ends up limiting their achievement for the rest of their lives.

Researchers in Israel and the UK found that students on the borders of different groups had essentially the same understandings, but the ones entering higher groups ended up scoring at significantly higher levels at the end of school because of the group they were placed into. Indeed, the group that students were put into was more important for their eventual achievement than the school they attended.
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Borderline casualties are those students who just miss the high groups and become
casualties of the grouping system from that day on. There are many of these students in schools and for them the almost arbitrary assignment to a lower group effectively ends their chances of success.

JJ pixs

Student Resources

In a tracked class the main sources of help are the teacher and the textbook. Students are presumed to have the same needs and to work in the same way, so the teacher feels comfortable lecturing to the class for longer periods of time and requiring the class to work in silence or in very quiet conditions, denying them the many advantages of talking through problems, as set out in in
chapter 2
.

In mixed-ability classes the students are organized to work with each other and help each other. Instead of one person serving as the resource to thirty or more students, there are many. The students who do not understand as readily have access to many helpers. The students who do understand serve as helpers to classmates. This may seem like it is wasting the time of high achievers, but the reason these students end up achieving at higher levels in such classrooms is because the act of explaining work to others deepens understanding. As students explain to others, they uncover their own areas of weakness
and are able to remedy them and they strengthen what they know. In the two longitudinal studies I conducted, the high-achieving students told me that they learned more and more deeply from having to explain work to others. Some of the high achievers from Railside High School talked about their experiences in mixed-ability groups in the United States. Zane had said, “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.”

Some of the higher achievers entered Railside thinking it unfair that they had to explain work to others, but they changed their minds within the first year as they realized that the act of explaining was helping them. Imelda reflected: “So maybe in ninth grade it’s like ‘Oh, my God. I don’t feel like helping them. I just wanna get my work done. Why do we have to take a group test?’ But once you get to AP Calc, you’re like ‘Ooh, I need a group test before I take a test.’ So, like, the more math you take and the more you learn, you grow to appreciate, like, ‘Oh, thank God I’m in a group!’”

The high achievers also learned that different students could add more than they thought to discussions. As Ana said, “It’s good working in groups because everybody else in the group can learn with you, so if someone doesn’t understand—like if I don’t understand but the other person does understand—they can explain it to me, or vice versa, and I think it’s cool.”

Students who work together, supporting each other’s learning, provide a tremendous resource for each other, maximizing learning opportunities at the same time as learning important principles of communication and support.

Student Respect

When we consider the role of ability grouping and the difference it makes in students’ lives, there is another dimension besides
achievement that it is critical to consider. For ability grouping not only limits opportunities, it influences the sorts of people our children will become. As students spend thousands of hours in their mathematics classrooms, they do not only learn about mathematics, they learn about ways of acting and ways of being.

Mathematics classrooms influence, to a high and regrettable degree, the confidence students have in their own intelligence. This is unfortunate both because math classrooms often treat children harshly but also because we know that there are many forms of intelligence, and math classrooms tend to value only one. In addition to the power that math classrooms have to build or crush children’s confidence, they also influence to a large extent the ideas students develop about other people.

Through my own research I have found that students in tracked classes in American high schools not only developed ideas about their own potential, but they began to categorize others in unfortunate ways—as smart or dumb, quick or slow. Comments such as this came from students in tracked classes: “I don’t want to feel like a retard. Like if someone asks me the most basic question and I can’t do it, I don’t want to feel dumb. And I can’t stand stupid people either. Because that’s one of the things that annoy me. Like stupid people. And I don’t want to be a stupid person.”

The students who had worked in mixed-ability classes at Railside High School did not talk in these ways and they developed impressive levels of respect for each other. Any observer to the classrooms could not fail to notice the respectful ways students interacted with one another, seeming not to notice the usual dividing lines of social class, ethnicity, gender, or “ability.” The ethnic cliques that often form in multicultural schools did not form at Railside, and students talked about the ways their math classes had taught them to be respectful of different people and ideas. In learning to consider different approaches to math problems, students also learned to respect different ways of thinking
more generally and the people making such contributions. Many of the students talked about the ways they had opened their minds and their ideas—for example:

Tanita:
You got everyone’s perspective, ’cause like when you’re debating a rule or a method, you get someone else’s perspective of what they think instead of just going off your own thoughts. That’s why it was good with, like, a lot of people.

Carol:
I liked it too. Most people opened up their ideas.

Undeniably, one of the goals of schools is to teach students subject knowledge and understanding, but schools also have a responsibility to teach students to be good citizens—to be people who are open-minded, thoughtful, and respectful of others who are different from themselves. The Carnegie Corporation of New York, based on a report from the Council for Adolescent Development, recommended that detracking occur in schools in order to create “environments that are caring, healthy, and democratic as well as academic.”
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The Coalition for Essential Schools also recommended that schools abandon tracking for both academic and moral reasons. Its former director Ted Sizer argued that “if we want citizens who take an active and thoughtful part in our democracy they must get trained for this in school—working together on equally challenging problems, and using every possible talent toward their solutions.”
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Although it seems to make sense to place students into groups where they have similar needs, the negative consequences of tracking decisions—in terms of students’ achievement and their moral development—are too strong to ignore. Teachers of mathematics are often inexperienced with different systems of grouping, but this is not a good reason for perpetuating a flawed approach. Teachers need to be supported in gearing their teaching toward high
achievement for all. I have visited math classes that have recently become detracked and seen teachers using the same approach they have always used, to disastrous effect. These teachers gave out worksheets targeting small areas of content appropriate for only a few students in the class, leaving the rest floundering or bored.

For mixed-ability classes to work well, two critical conditions need to be met. First, the students must be given open work that can be accessed at different levels and taken to different levels. Teachers have to provide problems that people will find challenging in different ways, not small problems targeting a small, specific piece of content. These are also the most interesting problems in mathematics so they carry the additional advantage of being more engaging. See www.youcubed .org for many examples of these engaging tasks. I have seen such problems used to great effect in a number of classrooms. These sorts of multilevel problems are used in Japanese classrooms in order to promote high achievement for all, as Steve Olson, author of the bestselling book
Count Down,
reflects:

In Japanese classrooms . . . teachers
want
their students to struggle with problems because they believe that’s how students come to really understand mathematical concepts. Schools do not group students into different ability levels, because the differences between students are seen as a resource that can broaden the discussion of how to solve a problem. Not all students will learn the same thing from a lesson; those who are interested in and talented at math will achieve a different level of proficiency from their classmates. But each student will learn more by having to struggle with the problem than by being force-fed a simple, predigested procedure.”
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Japanese students are not all expected to learn the same from each lesson, which is the unrealistic expectation for many
American classes; instead, they are given challenging problems and each student gets the most from them that he or she can.

In addition to open, multilevel problems, the second critical condition for mixed-ability classes to work is that students are taught to work respectfully with each other. I have observed many math classrooms where students are working in groups, but the students do not listen to each other. The teachers of such classrooms have given students good problems to work on together, and they have asked students to discuss the problems, but the students have not learned to work well together. This can result in chaotic classrooms with groups where only some of the students do the work, or, even worse, groups in which some students are ignored or ridiculed by other students because they are deemed to be low status. Teaching students to work respectfully requires careful and consistent building of good group behavior. Some teachers do this by highlighting the need for respect and hard work in groups, some teachers employ additional strategies such as “complex instruction” (an approach designed for use with mixed-ability groups), which is aimed at reducing status differences between students. Whatever the approach, when students learn to work well and respectfully together and their different strengths are seen as a resource rather than a point to ridicule, then children are helped by being able to achieve at high levels and society is helped through the development of respectful, caring young people.

Psychological Prisons

In my study of Amber Hill and Phoenix Park in England, I was able to follow students through a school that used ability grouping (Amber Hill) and one that did not (Phoenix Park). At Amber Hill the teachers also taught traditionally, whereas at Phoenix Park the teachers used complex, open problems. As I described in
chapter 3
, I was fortunate in being able to catch up with the students eight years after my initial study and talk to them about the
impact of their school experiences on their jobs and lives. During that study I found that one of the most important differences between the students, perhaps not surprisingly, was the grouping approaches they experienced. At Amber Hill, where they used ability grouping, the adults talked about how the grouping had shaped their whole school experience and many of those from set 2 downward talked not only about the ways their achievement had been constrained by the grouping but also the ways they had been set up for low achievement in life. In a statistical comparison of the jobs that the ex-students were working in, I found that those who had experienced mixed-ability grouping, despite growing up in one of the poorest areas in the country, were now in more professional jobs than those who had experienced tracking.
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Interviews with some of the young adults gave meaning to these interesting differences. The students from Phoenix Park talked about the ways the school had excelled at finding and promoting the potential of different students; they said that teachers had regarded everyone as a high achiever. The young adults communicated a positive approach to work and life, describing the ways they used the problem-solving approaches they had been taught in school to get on in life. The young adults who had attended Amber Hill, which had put them into sets, told me that their ambitions were broken at school and their expectations lowered. One young man, Nikos, spoke passionately about the ability grouping experience: