What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (15 page)

You’re putting this psychological prison around them. . . . People don’t know what they can do, or where the boundaries are, unless they’re told at that kind of age.

It kind of just breaks all their ambition. . . . It’s quite sad that there’s kids there that could potentially be very, very smart and benefit us in so many ways, but it’s just kind of broken down from a young age. So that’s why I
dislike the set system so much—because I think it almost formally labels kids as stupid.

The impact that ability grouping has upon students’ lives—in and beyond school—is profound. Researchers in England revealed that 88 percent of children placed into ability groups at age four remain in the same groupings until they leave school.
15
This is one of the most chilling statistics I have ever read. The fact that children’s futures are decided for them by the time they are placed into groups, at an early age, derides the work of schools and contravenes basic knowledge about child development and learning. Children develop at different rates, and they reveal different interests, strengths, and dispositions at various stages of their development. In the United States such decisions are usually made during middle school, but they still prejudge children’s potential before they have had a chance to develop. One of the most important goals of schools is to provide stimulating environments for all children—environments in which children’s interest can be piqued and nurtured, with teachers who are ready to recognize, cultivate, and develop the potential that students show at different times and in different areas. This can only be done through a flexible system of grouping that does not prejudge a child’s achievement and that uses multileveled mathematics materials that individual students take to their own highest level. Only such an approach will enable America to become a fairer society in which all children are given the chance to be successful.

Volt Collection

How Girls and Women Are Kept Out of Math and Science

W
hen I started my research at Amber Hill, Caroline was fourteen years old, eager to learn, and very accomplished. When the students had entered the school some three years earlier, they had all been given a math test. Caroline had earned the highest score in her year, but three years after coming to the school she was the lowest-achieving student in her class. How could it all have gone so badly wrong? When I met Caroline, she had just been placed into the top set—a group that was taught by Tim, the friendly and well- qualified math department chair. But Tim was a traditional teacher and he, like most math teachers, demonstrated methods on the board and then expected students to work through
exercises practicing the techniques. Caroline sat at a table of girls, six of them in total. All of the girls were high achievers and they all wanted to do well in math. From the beginning, Caroline looked uncomfortable in class. She was an inquisitive and thoughtful girl, and whenever Tim explained methods to the class, she, like many girls I have taught and observed over the years, had questions: Why does that method work? Where does it come from? And how does that fit with the methods we learned yesterday? Caroline asked Tim these questions from time to time, but he would generally just re-explain the method, not really appreciating why she was asking. Caroline became less and less happy with math and after a while her achievement started to decline.

In one of their lessons the students were learning about the multiplication of binomial distributions. Tim had taught students to multiply binomial expressions such as

(
x
+ 3)(
x
+ 7)

by telling students that they:

1. Multiply the first terms (
x
times
x
)

2. Multiply the outer terms (
x
times 7)

3. Multiply the inner terms (3 times
x
) and then

4. Multiply the last terms (3 times 7) Then add all the terms together (x
2
+ 7x + 3x + 21 = x
2
+ 10x + 21)

Students are often taught to remember this sequence with the pneumonic FOIL (first, outer, inner, last). These are the sorts of procedures in math class that seem meaningless to students—they are hard to remember and easy to muddle up. I approached Caroline’s group one day as she was sitting with her head in her hands. I asked her if she was all right, and she looked at me
with an agonized expression. “Ugh. I hate this stuff,” she said. “Can you tell me why it works like this? Why does it have to be in that order, with all of that adding?” She had tried asking Tim, who had told her that that is how the formula works and you just need to remember it.

I knelt beside Caroline’s desk and asked her whether I could draw a diagram for her. I explained that we could think about the multiplication visually by thinking of the two expressions as sides of a rectangle. She sat up and watched as I drew a sketch:

Soon all the girls at the table were watching. Before I had finished the drawing, Caroline said, “Oooh, I see it now,” and the others made similar appreciative noises. I felt a bit bad about offering this drawing, as my role in the classroom was not to help students and certainly not to undermine Tim’s teaching, but it was an opportunity that I took on that one occasion. The drawing was simple but it offered a lot—it allowed the girls to see
why
the method worked and that was important for them.

I observed Tim’s class, and other classes at the school, many times over the three years. As I interviewed more and more of the boys and girls, I started to notice that the desire to know
why
was something that separated the girls from the boys. The girls were able to accept the methods that were shown to them and practice them, but they wanted to know
why
they worked,
where
they came from, and how they
connected
with other methods. Some of the boys were also curious about the ways methods were connected and how they worked, but they seemed willing to adapt to a teaching approach that did not offer them such insights. In interviews the girls would frequently say such things as “He’ll write it on the board and you end up thinking, ‘Well, how come this and this? How did you get that answer? Why did you do that?’”

Many of the boys, on the other hand, would tell me that they were happy as long as they were getting answers correct. The boys seemed to enjoy completing work at a fast pace and competing with other students, and they did not seem to need the same depth of understanding. John (year ten) spoke for many of the boys when he said, “I dunno, the only maths lessons you like are when you’ve really done a lot of work and you’re proud of yourself because you’ve done so much work, you’re so much ahead of everyone else.”

In a questionnaire I gave to the whole of the year cohort, I asked the students to rank five ways of working in mathematics. Ninety-one percent of girls chose “understanding” as the most important aspect of learning mathematics, compared with only 65 percent of the boys, which was a statistically significant difference.
1
The rest of the boys said that the memorization of rules was the most important. The girls and boys also acted differently in lessons. During the hundred or so lessons that I observed, I would often see boys racing through their textbook questions, trying to work as quickly as possible and complete as many questions as they could. I would, just as frequently, observe girls looking lost and confused, struggling to understand their work, or giving up all together. In lessons I would
often ask students to explain to me what they were doing. The vast majority of the time, the students would tell me the chapter title and, if I asked them questions like “Yes, but what are you actually
doing
?” they would tell me the number in the exercise; neither girls nor boys would be able to tell me why they were using methods or what they meant.

On the whole the boys were unconcerned by this as long as they were getting their questions right, as Neil told me in an interview: “Some of the stuff you do, it’s just hard, and some of it’s really easy and you can remember it every time. I mean, sometimes you try and race past the hard bits and get it mostly wrong, to go onto the easy bits that you like.”

The girls would get questions right, but they wanted more. As Gill explained: “It’s like you have to work it out and you get the right answers but you don’t know what you did. You don’t know how you got them, you know?”

At the end of the three years, the students took a national examination. In the top set (similar to honors class in the United States), which I had observed so closely, the girls achieved at significantly lower levels than the boys, which was a pattern that was repeated in different groups across the year cohort. Caroline, once the glittering star of the group, achieved the lowest grade of all. By the time she was finishing the course, she had decided that she was no good at math, despite the fact that she had been the highest achiever in the school. Caroline, and many of the other girls, had underachieved because they had not been given the opportunity to ask why methods worked, where they came from, and how they were connected. Their requests were not at all unreasonable—they wanted to locate the methods they were being shown within a broader sphere of understanding. Neither the boys nor the girls particularly liked the traditional mathematics lessons at Amber Hill—math was not a popular subject at the school—but
the boys worked within the procedural approach they were given whereas many of the girls resisted it. When they could not get access to the depth of understanding they wanted, the girls started to turn away from the subject. National statistics tell us that girls now do very well in mathematics, achieving at equal or higher levels than boys. This high achievement, given the inequitable approach that most girls experience, is testament to their capability and impressive motivation to do well. But the high achievement of girls often masks a worrying reality—the approaches they experience make many girls uncomfortable and their lack of opportunity to inquire deeply is the reason that so few women take mathematics to high levels.

At Phoenix Park School—where students were taught through longer, more open problems and they were encouraged to ask why, when, and how—the girls and boys achieved equally, and both groups achieved at higher levels than the students at Amber Hill on a range of assessments, including national exams.

Some years later, I was sitting in a math class myself, having decided to take an applied statistics course. We had a wonder- ful teacher, a woman who explained why and how methods worked—almost all of the time. I remember one day sitting in class when our teacher showed the formula for standard deviation and then said, “By the way, would you like to know why it works?” Then a funny thing happened. The women in the class chorused yes and most of the men chorused no. The women joked with the men, asking, “What is wrong with you?” One of the men responded quickly: “Why do we need to know why? It is better just to learn it and move on.” It was then that I realized we were playing out the same gender roles as the students I had been observing at Amber Hill.

In one of my first research studies at Stanford, I decided to
learn more about the experiences of high-achieving students in American high school classes. I chose six schools and interviewed forty-eight boys and girls about their experiences in AP calculus classes. In four of the schools the teachers used traditional approaches, giving the students formulas to memorize without discussing why or how they worked. In the other two schools the teachers used the same textbooks, but they would always encourage discussions about the methods that students were using. I was not investigating or looking for gender differences, but I was struck again by the reflections of the girls in the traditional classes and their need to inquire deeply, as Kate at the Lewis school described:

We knew
how
to do it. But we didn’t know
why
we were doing it and we didn’t know how we got around to doing it. Especially with limits, we knew what the answer was, but we didn’t know
why
or
how
we went around doing it. We just plugged into it. And I think that’s what I really struggled with—I can get the answer, I just don’t understand
why.

Again, many of the girls told me that they needed to know
why
and
how
methods worked, and they talked about their dislike of classes in which they were just asked to memorize formulas, as Kristina and Betsy at the Angering school described:

K:
I’m just not interested in, just, you give me a formula, I’m supposed to memorize the answer and apply it, and that’s it.

JB:
Does math have to be like that?

B:
I’ve just kind of learned it that way. I don’t know if there’s any other way.

K:
At the point I am right now, that’s all I know.

Kristina went on to tell me that her need to explore and to understand phenomena was due to being a young woman:

Math is more, like, concrete, it’s so “It’s that and that’s it.” Women are more, they want to explore stuff and that’s life kind of, like, and I think that’s why I like English and science. I’m more interested in, like, phenomena and nature and animals and I’m just not interested in, just, you give me a formula, I’m supposed to memorize the answer and apply it, and that’s it.

It was unfortunate for Kristina that mathematics was not one of her school subjects that allowed her to “explore” or to consider phenomena, when it should have been.

David Sela, from the ministry of education in Israel, and Anat Zohar, from the Hebrew University in Jerusalem, conducted an extensive investigation of gender differences in the learning of physics. They took my notion of the quest for understanding that I had found to be prevalent among girls in math classes and considered whether it was also prevalent among girls in physics classes. They found, resoundingly, that it was. The researchers drew from a database of approximately four hundred high schools in Israel that offered advanced-placement physics classes. They sampled fifty students from the schools and interviewed twenty-five girls and twenty-five boys. They found that the girls in physics classes were exhibiting the same preferences that I had found in mathematics classes, resisting the requirement to memorize without understanding, saying that it was “driving them nuts.” The girls talked about wanting to know why methods worked and how they were linked. The authors concluded that “although both girls and boys in the advanced-placement physics classes share a quest for understanding, girls strive for it much more urgently than boys, and seem to suffer academically more than boys do in a classroom culture that does not value it.”
2

Neither the female math students I interviewed nor the female physics students interviewed in Israel wanted an easier science or math. They did not need or want softer versions of the subjects. In fact, the versions of mathematics and science they wanted required considerable depth of thought. In both cases the girls wanted opportunities to inquire deeply, and they were averse to versions of the subjects that emphasized rote learning. This was true of boys and girls, but when girls were denied access to a deep, connected understanding, they turned away from the subject.