What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (11 page)

These were not quite the headlines I would have chosen, but the approach was gaining rightful attention.

The Amber Hill students faced many difficulties in the examination, which they were not expecting, as they had worked so hard in lessons. In class the Amber Hill students had always been shown methods and then practiced them. In the examination they needed to choose methods to use and many of them found that difficult. As Alan explained to me: “It’s stupid really, ’cause when you’re in the lesson, when you’re doing work—even when it’s hard—you get the odd one or two wrong, but most of them you get right and you think, ‘Well, when I go into the exam, I’m gonna get most of them right,’ ’cause you get all your chapters right. But you don’t.” Even in the examination questions when it was obvious which methods to use, the Amber Hill students would frequently confuse the steps they had learned. For example, when the Amber Hill students answered a question on simultaneous equations, they attempted to use the standard procedure they had been taught, but only 26
percent of the students used the procedure correctly. The rest of the students used a confused and jumbled version of the procedure and received no credit for the question.

The Phoenix Park students had not met all of the methods they needed in the examination, but they had been taught to solve problems and they approached the examination questions in the same flexible way as they approached their projects—choosing, adapting, and applying the methods they had learned. I asked Angus whether he thought there were things in the exam that they hadn’t seen before. He thought for a while and said: “Well, sometimes I suppose they put it in a way which throws you. But if there’s stuff I actually haven’t done before, I’ll try and make as much sense of it as I can, try to understand it, and answer it as best as I can, and if it’s wrong, it’s wrong.”

The Phoenix Park students didn’t only do better on the examinations. As part of my research, I investigated the usefulness of the approach to students’ lives. One way I measured this was by giving a range of assessments to students over the three years that were designed to assess students’ use of mathematics in real-world situations. In the “architectural activity,” for example, students had to measure a model house, use a scale plan, estimate, and decide upon appropriate house dimensions. The Phoenix Park students outperformed the Amber Hill students on all of the different assessments. By the time I was completing the research, most of the students had jobs in the evenings and on weekends. When I interviewed the students at both schools about their use of mathematics outside school, there were stark differences. All forty of the Amber Hill students that I interviewed said that they would never, ever make use of their school-learned methods in any situation outside school. As Richard told me: “Well, when I’m out of school, the maths from here is nothing to do with it, to tell you the truth. . . . Most of the things we’ve learned in school we would never use anywhere.” The Amber Hill students thought school mathematics was a
strange sort of code that you would use in one place—the mathematics classroom—and they developed the idea that their school mathematics knowledge had boundaries or barri- ers surrounding it, which kept it firmly within the mathematics classroom.
2,
3

At Phoenix Park, the students were confident that they would utilize the methods they learned in school. They gave me examples of their use of school-learned mathematics in their jobs and lives. Indeed, many students’ descriptions suggested that they had learned mathematics in a way that transcended the boundaries that generally exist between the classroom and real situations.
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Mathematics for Life

Some years later I caught up with the ex-students from Amber Hill and Phoenix Park. By then they were about twenty-four years old. We talked about the usefulness of the math teaching they had experienced. I had frequently been asked about the future of the students after they had left their schools and so I decided to find out. I sent surveys to the ex-students’ addresses and followed up the surveys with interviews. As part of the survey I asked the young people what jobs they were doing. I then classified all the jobs and put them onto a scale of social class, which gives some indication of the professionalism of their jobs and their salaries. This showed something very interesting. When the students were in school, their social class levels (determined from parents’ jobs) had been equal. Eight years after my study, the Phoenix Park young adults were working in more highly skilled or professional jobs than the Amber Hill adults, even though the school achievement range of those who had replied to the surveys from the two schools had been equal. Comparing the jobs of the children to their parents, 65 percent of the Phoenix Park adults had moved into jobs that were more
professional than their parents’, compared with 23 percent of Amber Hill adults. Fifty-two percent of the Amber Hill adults were in less professional jobs than their parents, compared with only 15 percent of the Phoenix Park adults. At Phoenix Park, there was a distinct upward trend in careers and economic well-being. At Amber Hill, there was not, which is especially noteworthy considering that Phoenix Park was in a less prosperous area.

When I traveled back to England to conduct the follow-up interviews, I contacted a representative group from each school, choosing young adults with comparable examination grades. In interviews the Phoenix Park adults communicated a positive approach to work and life, describing the ways they used the problem-solving approach they had been taught in their mathematics classrooms to solve problems and make sense of mathematical situations in their lives. Adrian, who had studied economics at a university, told me that “you often get lots of stuff where there will be graphs of economic situations in countries and stuff like that. And I would always look at those very critically. And I think the maths that I’ve learned is very useful for being able to actually see exactly how it’s being presented, or whether it’s being biased.”

When I asked Paul, who was a senior regional hotel manager, whether he found the mathematics he had learned in school useful, he said that he did: “I suppose there was a lot of things I can relate back to maths in school. You know, it’s about having a sort of concept, isn’t it, of space and numbers and how you can relate that back. And then, okay, if you’ve got an idea about something and how you would then use maths to work that out. . . . I suppose maths is about problem solving for me. It’s about numbers; it’s about problem solving; it’s about being logical.”

Whereas the Phoenix Park young adults talked of maths as
a problem-solving tool, and they were generally very positive about their school’s approach, the Amber Hill students could not understand why their school’s mathematics approach had prepared them so badly for the demands of the workplace. Bridget spoke sadly: “It was never related to real life, I don’t feel. I don’t feel it was. And I think it would have been a lot better if I could have seen what I could use this stuff for . . . because then it helps you to know
why.
You learn
why
that is that, and
why
it ends up at that. And I think definitely relating it to real life is important.”

Marcos was also puzzled as to why the school’s maths approach had seemed so removed from the students’ lives and work: “It was something where you had to just remember in which order you did things, and that’s it. It had no significance to me past that point at all—which is a shame. Because when you have parents like mine who keep on about maths and how important it is, and having that experience where it just seems to be not important to anything at all really. It was very abstract. And it was kind of almost purely theoretical. As with most things that are purely theoretical, without having some kind of association with anything tangible, you kind of forget it all.”

My book of the study of the approaches at Phoenix Park and Amber Hill,
Experiencing School Mathematics,
won a national book award in England and has been read by thousands of British, American, and other readers. Many teachers have contacted me saying that they would like to teach through a problem-solving approach similar to Phoenix Park’s but they cannot because of lack of support from departments, administration, and parents. But now is the time to make changes, to grasp the optimism and enthusiasm existing around math change, and to make decisions that are informed by research about the best math education we can provide to children.

The two teaching approaches I have reviewed were the subject of comprehensive research studies and, although they were
conducted in different countries, the findings pointed to the same conclusion: students need to be actively involved in their learning and they need to be engaged in a broad form of mathematics—using and applying methods, and representing and communicating ideas. As a professor at Stanford, I am frequently contacted by teachers, district officials, and parents wanting to know which curricula are good to use in math classrooms. This is a question I find difficult to answer, as I strongly believe that teachers are the most important part of an approach and it is hard to recommend a book or a curriculum approach without knowing how a teacher is using it. But it is the case that some books have been written to involve students more actively and some books include very good mathematics problems. In
appendix B
, I list some of these books, including those that I personally regard to be of a high quality.

New Forms of Testing That Encourage Learning

D
oes it matter that American children are tested more than they ever used to be? Does it matter that they are tested more than students in the rest of the world? And does it matter that the tests used in America are rejected by most other countries? The answer to all of these questions is, of course, yes. America is out of sync with the rest of the world, not because the United States has a better system of testing, as one might expect from such a well-resourced country, but because the testing system in the United States is disastrous. Students are overtested to a ridiculous degree, and the tests that are used are damaging—to schools and teachers and, most important, to the health, hearts, and minds of students.

Alfie Kohn, author of numerous books on education, writes that “Standardized testing has swelled and mutated, like a creature in one of those old horror movies, to the point that it now
threatens to swallow our schools whole.”
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The testing movement has indeed swallowed many of our schools whole, and it is time to do something about it. Fortunately, there is a new form of assessment that is so powerful and so effective, not only at finding out what students know but also at diagnosing and improving learning, that it has energized a whole new movement. It is called “assessment for learning” and it has been shown to have an effect on learning that, if implemented, would raise America’s ranking in international comparisons from the middle of the pack to a place in the top five. In England, a booklet called
English
Inside the Black Box
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(which detailed the new assessment methods) sold tens of thousands of copies in a very short space of time. The schools that used the methods and were part of a careful research study
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significantly improved their students’ achievement and attitudes toward work.

Assessment for learning aims to create self-regulating learners—learners who have the knowledge and power to monitor their own learning. The testing monster has the biggest impact on math classrooms and math learning, and we know that math has the power to crush the spirit of young people in ways that no other subject can. How wonderful, then, that there is a new way of assessing math (and other subjects) that doesn’t terrorize children, doesn’t distort the curriculum, and propels students to higher levels of learning. If it sounds too good to be true, please read on.

What Is Wrong with What We Have Now?

Students in American schools are subject to narrow standardized tests in mathematics—as well as other subjects—from a very young age. Although most countries in the world test students by posing questions that students respond to in writing and that are graded by trained experts, America uses multiple-choice tests that are graded by machines. It is hard to find a single multiple-choice
question used in Europe—in any national assessment, in any subject, at any level, in any country—yet almost all of America’s test questions are of a multiple-choice format. I will speculate that the reasons other countries do not use multiple-choice questions in their examinations are fourfold. First, they want to assess understanding, which includes the thinking that children do and express in words, numbers, and symbols. What children choose to put on paper is the best indicator of what they understand, not their choice of one out of four options, none of which may mean anything to them. Second, multiple-choice testing is known to be biased—particularly for ethnic minority students. There has been mixed evidence on the bias of multiple-choice tests against girls, but we do know that girls do less well on tests such as the SATs, designed to predict college performance, but then outperform boys to a significant degree in college.
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Third, undergoing timed multiple-choice tests causes anxiety and contributes to the stressed-out nation of schoolchildren that America now has.
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Fourth, the best thing that multiple-choice tests show is a student’s ability to complete multiple-choice tests. Some students are good at that type of test taking and do well on those tests, while other students, including those who are highly intelligent and knowledgeable, do badly. Martin Luther King, for example, one of the country’s most influential writers, scored in the lowest 10 percent of students on both the math and verbal sections of the Graduate Record Examination, despite being such a brilliant student that he entered college at the age of sixteen.
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Knowing that a student has done well, or badly, on a multiple-choice test does not tell you much, if anything, about how well they will handle more advanced material or solve complex problems in the workplace.

In addition to a reliance on multiple-choice formats, the mathematics tests used in most states across America are extremely narrow. They do not assess thinking, reasoning, or problem
solving, all of which are at the core of mathematics; instead, they assess the simple use of procedures, completed under timed conditions. Procedures are important, of course, but only if they can be used to solve problems. What is the point of knowing procedures if students don’t know when they should use them, or how to apply them to complex problems? One of the most important principles of good testing is that it assesses what is important. The tests that predominate in America do not. The worst of this is not that the tests provide little information but that they have a huge and damaging impact on what is taught in schools. Almost every mathematics teacher in America will tell you that the pressure to prepare students for standardized tests harms their teaching and their students’ learning. In mathematics the teachers have to focus upon knowledge that can be tested rather than knowledge that is important for work or for life. Students also suffer from an extreme narrowing of the curriculum. In North Carolina, for example, the pressure to perform on high-stakes tests resulted in widespread decline in the teaching of science, social studies, physical education, and the arts.
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Many people in America think that achievement is low because teachers are poorly qualified or lack knowledge. At Stanford I teach future math teachers who are highly qualified and committed. They are people with math degrees from some of the most prestigious universities in the world, people who have turned down much more lucrative careers so they can teach children. They know how important it is that students learn to think and reason, and that they are willing and able to solve complex problems. But at the same time the teachers often feel unable to spend any time teaching students to do these things while they are so constrained by California’s standardized tests. Alfie Kohn quotes from a teacher who had to stop one of her most successful teaching activities because of the high-stakes tests. She used to ask her middle school students to become experts in a subject, developing their researching and
writing skills. This was an experience that her students remembered for years, that they looked back on as a highlight of school, but that she was forced to end. As Kohn writes: “Within each classroom ‘the most engaging questions kids bring up spontaneously—“teachable moments”—become annoyances.’ Excitement about learning pulls in one direction; covering the material that will be on the test pulls in the other.”
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The elimination of powerful learning experiences because they cannot be reduced to testable knowledge is damaging education in America.

The tests used in America are particularly harmful for children of low income and for English language learners (ELL students). America is currently one of the
most unequal countries in the world, and the gap between the educational achievement of the wealthy and the poor is vast. The No Child Left Behind Act, brought in by President Bush’s administration, made standardized testing compulsory across the country, with the supposed aim of eradicating inequalities. The irony and the tragedy of this is that the testing movement has made inequalities worse.

There is also evidence that the standardized questions test language as much as they do mathematics. In California in 2004, there was a staggering correlation of 0.932 between students’ scores on the mathematics and language arts sections of the tests used. Correlations this high between two tests tell us that the tests are assessing virtually the same thing. Even the same test taken twice at different times might not give us such a high correlation. We should expect some correlation between people’s scores on mathematics and language arts tests, as good students sometimes do well in both, but a correlation of 0.932 is ridiculously high. And since the language arts questions have no mathematics in them but the mathematics questions use unnecessarily complex language, there can be only one reason that they are so highly correlated: the mathematics tests are really language tests. These are the tests that are being used to
judge students’ mathematical understanding and the tests that are driving the curricula in schools.

We are entering a new era of Common Core math and associated assessments. We will not be free from standardized testing, but the new assessments do promise to fundamentally change the way children are tested, replacing the dreaded multiple-choice items with free-response items in which students explain their thinking. The new assessments come from two different assessment companies, the Parthership for Assessment of Readiness for College and Careers (PARCC) and Smarter Balanced Assessment Consortium, which are promising to assess problem solving rather than procedure repetition. It is too early to say whether the assessment companies fulfill the promises made, but we have to hope that they do, because assessment drives instruction.

When I was researching Railside High School,
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with its outstanding mathematics department, I met Simon from Nicaragua, who had arrived in this country as a young boy. Simon told me that elementary school was a time of constant failure, as he couldn’t understand what the teachers were saying. Since then, though, he had attended wonderful schools and was excelling in all of his subjects. Always happy and smiling, Simon was one of those students whom teachers love to teach. He told us that the teachers at Railside convinced him that he was smart, and he started to believe in himself and achieve. When I met Simon, he told me that he loved math. In different assessments, including ours from Stanford, he performed extremely well. Despite all of this, he had not done well on the standardized test administered by the state of California—at that time the SAT-9 tests. The reason for this had little to do with mathematics understanding but instead with the unfamiliar language and contexts used in the test and the strange format of the questions.
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Many of the ELL students at Railside underperformed on the tests and the Railside teachers were put under pressure to spend less time on important mathematics and more time on
training students to complete a multiple-choice test. It was very sad to see the teachers and students spending their time in these ways, especially as they were being pulled away from important mathematical activities and learning opportunities.

In addition to the harm done by inadequate tests that do not assess mathematical understanding, additional punishment is dealt to many of the students by the harsh and comparative reporting of scores. When students are sent a label telling them where they stand compared to other students, rather than where they stand in their learning of mathematics, it offers no helpful information and is harmful to many students. A test should communicate to students
what
they have learned and how much they have learned
over a period of time.
At the very least, tests should set out for students what they know and do not know so that they give students information to work with.
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,
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Knowing that you are doing worse than others is simply demoralizing, as Simon discovered. Simon had learned a huge amount at Railside and worked extremely hard, but the odds were stacked against him when taking a test using convoluted language and strange contexts. He did not do well on the test, but instead of finding out which questions he did badly on, so that he could work on those, or how much he had improved over the past year (a great deal), he received a label telling him where he was ranked compared to others in the country. This practice was especially unfair as some states allow calculators in the tests but his state (California) did not. Simon was simply and devastatingly told that he was “below average.” The label that Simon’s parents received in the mail caused him to question his ability despite all of his achievements at school. As he told us: “My parents, they saw in the SAT-9 graph thing I was
below average
in the majority of the things, and especially math. I was like
below average.
Right there. The thing is like
below average
,
you want it to be a little bit above average.”

I asked him whether that affected how he thought of his
abilities as a mathematics learner. He said that it did: “You tried so hard and then suddenly they give you a paper where it says you’re
below average
and you’re like, ‘What? I did so much work.’”

Simon had reasonably assumed that the result he was given should tell him something about how hard he had worked or what he had learned in mathematics, but it did neither of these things. Testing and reporting measures such as those experienced by Simon can
create
low-achieving students, crushing students’ confidence and giving them an identity as a low achiever.

Claude Steele, an eminent psychologist, has demonstrated the damage caused by “stereotype threat.”
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He found that when students took a math test in which they recorded their gender at the start of the test, women underperformed. In his earlier studies, he found that women underperformed if they were reminded of gender differences in math at the start of the test, but in subsequent experiments he found that people did not need to be reminded of gender differences because stereotypes are always “in the air,” and a simple recording of gender on a test produced gender differences for women in math.
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Researchers have shown the same phenomenon for any group that feels that they could be lower performers. This includes white men given a golfing test alongside African American men, who the white men believe will be superior golfers.
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Educational research—a field that often produces conflicting results—shows remarkable consistency on this issue. If you tell students they are low achievers, they achieve at a lower level than if you do not. At the time that Simon received his label, almost half of all students in the state were told that they were below average. At the time, students were told that they were “basic,” “below basic,” or even “far below basic.” California is now proposing to use levels instead of words. What impact, I wonder, do those who design these systems think that they will have upon students’ confidence and
their future mathematics achievement? Research tells us that confidence in one’s ability to succeed in mathematics is an intrinsic part of success and motivation. The labels the students in America receive tell many of them that there is no point in trying.