What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (13 page)

In another interesting study, Ruth Butler, a professor of education at the Hebrew University of Jerusalem,
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compared three ways of giving feedback to different groups of students. One group received grades; one group received comments on their work, saying whether carefully explained criteria were matched; and one group received both grades and comments. What the researchers found was that the group receiving comments increased their performance significantly, whereas those who received grades did not. More surprising, perhaps, those who
received both grades and comments did as poorly as those who only received grades. It turned out that those who received both grades and comments only focused upon the grades they received, which served to override any comments. Diagnostic, comment-based feedback is now known to promote learning, and it should be the standard way in which students’ progress is reported. Grades may be useful for communicating where students are in relation to one another, and it is fine to give them at the end of a semester or term, but if they are given more frequently than that, they will reduce the achievement of many. More typical reporting should be made up of feedback on the mathematics that students have learned, clear insights into the mathematics students are working toward, and advice on how to improve.

One assessment expert puts it simply: “Feedback to learners should focus on what they need to do to improve, rather than on how well they have done, and should avoid comparison with others.”
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This makes perfect sense. Coaches tell athletes how to be better; they don’t just tell them a grade. So why should teachers just say, in so many words, “You are a low achiever”? As Royce Sadler, a professor of higher education, says, “The indispensable conditions for improvement are that the student comes to hold a concept of quality roughly similar to that held by the teacher, is continuously able to monitor the quality of what is being produced during the act of production itself, and has a repertoire of alternative moves or strategies from which to draw at any given point.”
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The beauty of the assessment-for-learning approach is that it not only gives critical information to learners but that it also gives the same important information to teachers, helping them gear their teaching to their students’ needs.

The main aim of assessment for learning is the improvement of
classroom
assessment, but the methods have also been used
to improve wide-scale assessments such as those used at national levels. This is important because we know that teachers tend to mimic state and national assessments in their own classrooms. In Queensland, Australia, for example, the government recognized the need for good assessments that could inform and improve learning rather than traditional assessments that simply ranked students, and they designed a system to ensure objectivity. The system was instigated in 1971 and has continued to evolve and be improved upon since then. It involves students taking two assessments: a school-based “course of work” (in which teachers assess their students’ work in class, which is moderated by a committee) and a test of core skills. The test is used for the purpose of comparing performance across subjects. If there is a discrepancy between performance on the school work and the skills test, then the former takes precedence. Importantly, the school-based work offers opportunity for good assessment tasks, criteria that students can consider as they work, and diagnostic teacher feedback, combined with good, reliable information for parents and others. Other countries are now developing large-scale assessments that encourage learning but that can also be used to give unbiased, objective measurements of students’ learning through well-developed systems to ensure that graders are following the same standards.

Good assessments are critical at national and state levels as well as in classrooms. They should help students know what they are learning, provide students with the opportunity to show their understanding thoroughly, show students (and others) where they are in their learning, and give feedback on how to improve. Importantly, students should be given information that refers to the content area being assessed, not to other students. Assessment for learning transforms students from passive receivers of knowledge to active learners who regulate their own progress and knowledge and propel themselves to higher levels
of understanding. It also broadens teachers’ testing strategies, encouraging them to focus less on simple tests and more on the broad ways in which they can monitor student learning, including class work and student discussions and presentations. National and state assessments should lead the way in educating teachers about good assessments, and these powerful new methods should be in place in all classrooms in America.

How American Grouping Systems Perpetuate Low Achievement

W
hether or not to group students by prior attainment (often called ability) is one of the most controversial topics in education. Many parents support ability grouping because they want their high-achieving, motivated children to be working with similar children. This is completely understandable and makes perfect sense. But we know from several international studies that countries that reject ability grouping—nations as varied as Japan and Finland—are among the most successful in the world, whereas countries that employ ability grouping, such as America, are among the least successful. Is ability grouping part of the problem of America’s low achievement? And if it is, how can it be the case, when the idea seems to make so much sense?

In the Third International Mathematics and Science Study (TIMSS) researchers collected a wide range of data on eighth-grade students in thirty-eight countries.
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The achievement results,
with the United States coming in nineteenth place, received a huge amount of attention and prompted widespread concern. But analysts of the TIMSS database also uncovered some interesting facts about ability grouping. For example, in a study of achievement variability, the United States was found to have the greatest amount of variability between classes—that is, the United States had the most tracking.
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The highest-achieving country in the group, Korea, was the country with the least tracking and the most equal grouping. The United States also had the strongest links between achievement and socioeconomic status (SES), a fact that has been attributed to the widespread use of tracking in American schools. The high achievement of countries that do not use ability grouping and the low achievement of those that do was also a finding of the Second International Mathematics and Science Study (SIMSS). This caused leading analysts to conclude that countries that leave grouping to the
latest
possible moment or that use the
least
amount of grouping by ability are those with the highest achievement.

Some have argued that the high degree of tracking in America is a reflection of our desire to sort students at an early stage in order to find and focus upon the high-achieving students. But such an approach has some serious flaws, including the difficulty of identifying students correctly when children develop at different rates, and the creation of highly unequal schooling, which is one of the results identified by the international studies. In Japan, by contrast, the main priority is to promote high achievement for all, and teachers refrain from prejudging achievement, instead providing all students with complex problems that they can take to high levels. Japanese educators are bemused by the Western goal of sorting students into high and low “abilities,” as Gerald Bracey, a former professor at George Mason University noted:

In Japan there is strong consensus that children should not be subjected to measuring of capabilities or aptitudes and subsequent remediation or acceleration during the nine years of compulsory education. In addition to seeing the practice as inherently unequal, Japanese parents and teachers worried that ability grouping would have a strong negative impact on children’s self-image, socialization patterns, and academic competition.
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Lisa Yiu, a student of mine at Stanford, was interested in the very different ways Japan and the United States treat the grouping of students. She visited Japan and interviewed some of the mathematics teachers she observed, who explained to her why they do not use ability grouping. “In Japan what is important is balance. Everyone can do everything; we think that is a good thing. . . . So we can’t divide by ability.”

Japanese education emphasizes group education, not individual education. Because we want everyone to improve, promote, and achieve goals together, rather than individually. That’s why we want students to help each other, to learn from each other . . . , to get along and grow together—mentally, physically and intellectually.
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The Japanese approach of teaching students “to help each other, to learn from each other, . . . to get along and grow together—mentally, physically and intellectually” is part of the reason for their high achievement. Research tells us that approaches that keep students as equal as possible and that do not group by “ability” help not only those who would otherwise be placed in low tracks, which seems obvious, but also those who would be placed in high tracks too.

In neither of my two longitudinal studies of students working
in different mathematics approaches did I intend to research the impact of ability grouping, but it emerged as a powerful factor in both of them. In both studies the two successful schools were those that chose against ability grouping and used a different system of grouping, a result that mirrors the larger-scale international findings. England uses an overt system of ability grouping, one much harsher in its impact on low achievers than any system in the United States. In England, students are placed into “sets” that are ranked by achievement. I studied students in sets across the range from set 1 (the highest) to set 8 (the lowest). It did not surprise many people that students who were put into low sets achieved at low levels, partly because of the low-level work they were given and partly because students gave up when they knew they had been put into a low set and labeled a low achiever. This was true of students from set 2 downward. What was more surprising to people was that many of the students in set 1, the highest group, were also disadvantaged by the grouping. Students in set 1 reported that they felt too much pressure from being in the top set. They felt the classes were too fast, they were unable to admit to not following or understanding, and many of them started to dread and hate math lessons. The students in the highest group should have felt good about their understanding of math, but instead they felt pressured and inadequate. After three years of ability- grouped classes, the students achieved at significantly lower levels than students who had been grouped in mixed-ability classes.

In the United States the ability grouping that is used is nowhere near as overt as that in England, but it still has a significant impact. At around seventh and eighth grade, students in the United States typically get placed into different levels of classes, which determines their future for many years to come. Despite the importance of the different placements, the names of the classes often sound innocuously similar. Some seventh-
grade students may be in something called “math 7” while their peers are in pre-algebra, a higher class. Or eighth graders may be placed in “math 8” or in pre-algebra, while their peers are in algebra. The critical information that schools rarely provide is that in most American high schools, students cannot take calculus unless they have already passed algebra in middle school. If, like in most high schools, classes are a year long, then students who take algebra in ninth grade will take four years of courses without ever reaching calculus (algebra, geometry, advanced algebra, and precalculus). Thus the tracking decisions made by middle school teachers impact the classes reached in high school and, from there, students’ chances of being admitted to colleges of their choice. Middle school students should hear a strange sound when they are placed into lower-level math classes. It is the sound of doors closing.

As educators become more aware of the disadvantages of tracking, more schools are trying different approaches. Carol Burris, from South Side High School in New York, with Columbia University professors Jay Heubert and Hank Levin, conducted a study of a detracking innovation in mathematics.
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They compared the performance of students who were in tracked classes with those who were in mixed-ability classes. The researchers compared six annual cohorts of students in a middle school in a district of New York. The students attending the school in the first three years were taught in tracked classes with only high-track students being taught the advanced curriculum. But in the next three years all students in grades six through eight were taught the advanced curriculum in mixed-ability classes and all of the eighth graders were taught an accelerated algebra course. The researchers looked at the impact of these different middle school experiences upon the students’ completion of high school courses and their achievement, using four achievement measures, including scores on the Advanced Placement calculus examinations. They found that the students from
detracked classes took more advanced classes, pass rates were significantly higher, and students passed exams a year earlier than the average in New York State. Their scores were also significantly higher on various achievement tests. The increased success from detracking applied to students across the achievement range—from the lowest to the highest achievers.

In my own work in the United States and England, I have studied high schools that have shown a commitment to more equal schooling by placing all incoming students into mixed-ability classes. Railside High School in California started all students in a class called algebra regardless of prior achievement. This was a challenging class for all students, even those who had taken algebra in middle school, because it involved working on complex, multidimensional, multilevel problems. The school also taught classes that were ninety minutes long but that only took half a year. This meant that all students could reach calculus as they had eight opportunities to take a math class during their four years of high school. The results were outstanding, including 47 percent of the seniors taking calculus and precalculus advanced classes, compared to 28 percent of students in the more typical tracked high schools. Interestingly, the students who were most advantaged by the mixed-ability grouping were the highest-achieving students, who achieved at higher levels than students who were placed into high tracks in the other schools, and who improved their achievement more than any other students in their school.

Many parents fear mixed-ability grouping and cannot see the logic of grouping students with very different needs and limited teacher resources into one class. So why is it that mixed-ability grouping is repeatedly found to be associated with higher achievement? The most important reasons are explained next.

Opportunity to Learn

Researchers consistently find that the most important factor in school success is what they call “opportunity to learn.”
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If students are not given opportunities to learn challenging and high-level work, then they do not achieve at high levels. We know that when students are in lower groups, they receive low-level work and this, in itself, is damaging. In addition, teachers inevitably have lower expectations for students. In the 1960s Robert Rosenthal and Lenore Jacobson, two sociologists, conducted an experiment to look at the impact of teacher expectations. Students in a San Francisco elementary school were assigned to two groups. Both groups were taught the same work, but teachers were told that one group included all the students identified as especially talented. In reality, the students had been randomly assigned to the two groups. After the experimental period, the students in the group identified as talented showed better results and scored at higher levels on IQ tests. The authors concluded that this effect was entirely due to the different expectations the teachers held for the students.
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In a study of six schools in England, a team of researchers and I were saddened to find that teachers routinely underestimated children in low groups. One student reported to us: “Sir treats us like we’re babies, puts us down, makes us copy stuff off the board, puts up all the answers like we don’t know anything. And we’re not going to learn from that, ’cause we’ve got to think for ourselves.”

The students talked openly to us about the low expectations teachers held for them and the way their achievement was being held back, and observations of the classes confirmed this to be true. The students simply wanted to be given opportunities to learn: “Obviously we’re not the cleverest, we’re group five, but still—it’s still maths, we’re still in year nine, we’ve still got to learn.”
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When teachers have lower expectations for students and they teach them low-level work, the children’s achievement is suppressed. This is the reason that ability grouping is illegal in many countries in the world, including Finland, a country that tops the world in the international achievement tests.
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Tracking Leads to Damaging Fixed Mind-sets

When students are put into ability groups, they get hit with a damaging message—that their “ability” is fixed. This leads many students with healthy growth mind-sets—that is, they believe that the harder they work the smarter they get—to change to a fixed mind-set, believing that they are either smart or that they are not. Fixed mind-sets, as explained in the preface, result in significantly lower achievement for students and disadvantages that continue throughout life. In one of Carol Dweck’s then doctoral students’ studies, Carissa Romero found that when students moved into ability groups in grade seven, positive growth-mind-set thinking reduced, and the students who were most negatively affected were those going into the top track. Many more of the students started to believe that they were smart, and this sets students on a very vulnerable pathway, where they become fearful of making mistakes and start to avoid more challenging work. The consequences, especially for high-achieving girls, are devastating.

Student Differences

When students are placed into a tracked group, high or low, assumptions are made about their potential achievement. Teachers tend to pitch their teaching to students in the middle of the group, and they teach a particular level of content, assuming that all students are more or less the same. In such a system, the work inevitably is at the wrong pace or the wrong level for many students within a group. The lower students struggle to catch up,
while others are held back. While a teacher of any class, including a mixed-ability class, can wrongly assume that the students all have the same needs, tracking is
based on this erroneous assumption, and when teachers have a tracked group, they often feel license to treat all students the same. This is true even when students clearly have different needs and would benefit from working at different paces.