What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (17 page)

It was once suggested that girls are made of “sugar and spice and all things nice”—a harmless nursery rhyme perhaps, but the idea that girls are sugary sweet and lack the intellectual rigor for mathematics and science is still around. It is time that such ideas are buried and that girls are encouraged to go into math and science, for their sake as well as the sake of the disciplines themselves. Mathematics is and has always been about deep inquiry, connection making, and rigorous thought. Girls are ideally suited to the study of high-level mathematics—and the only reason that they drop math in high numbers is because the subject is misrepresented and taught badly in too many classrooms in America.

I was invited to the White House recently to present to the Commission for Women and Girls on the reasons too few women choose STEM subjects. That day I told the gathered group that mathematics is one of the reasons girls do not go forward in STEM because girls look for a depth of understanding that is often unavailable in math classrooms. I also urged the group to pay more attention to teaching (instead of only focusing on stereotypes, messages, and role models) because I believe this is neglected in policy discussions. I look at initiatives to involve girls, such as in summer camps and after-school clubs, and I am aware that they have a positive impact, but they are working by trying to
change the girls
rather than by changing the teaching environments that push girls away from STEM. We urgently need to reorient mathematics and other subjects so
that they focus on understanding and deep inquiry. When such changes are made girls choose STEM subjects in equal numbers to boys, and this is a goal that we should prioritize in the United States, not only for the futures of girls and women but for the future of the STEM disciplines.

In my work with the PISA team in Paris we have found that students take very different approaches to math and that the different approaches they take influence greatly whether they are successful or not. The incredible data set from PISA, including thirteen million students worldwide, shows that students who use memorization as their primary strategy are the lowest-achieving students in the world. This is consistent with my analyses of mathematics teaching and learning over the decades. I think of students as setting out on two contrasting pathways with mathematics: one of them leads to success and enjoyment and the other leads to frustration and failure. Students start on these pathways from a very early age. In this chapter I will review a study that illustrates the different pathways beautifully, as well as their relationship with achievement, before describing the time that my graduate students and I set out to change students’ pathways through a summer school teaching program. What we achieved in a five-week period, in a challenging
setting, is eminently achievable by teachers over the course of a year, and it is also achievable by parents armed with the right knowledge from research and the ideas I will share. I also include the stories of some children who changed as a result of the summer school, as their different circumstances and reasons for underachieving in mathematics may remind readers of their own children and students and the barriers they often face.

A Critical Way of Working

In an influential research study, two British researchers, Eddie Gray and David Tall, identified the reasons why many children struggle with math.
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The results were so important that they should be shouted from the rooftops and posted in every math classroom across America.

Gray and Tall conducted a study of seventy-two students between the ages of seven and thirteen. They asked teachers in England to identify children from their classes whom they regarded as above average, average, or below average, and they interviewed the seventy-two children. The researchers gave children various addition and subtraction problems to do. One type of problem required adding a single-digit number, such as 4, to a teen-digit number such as 13. They then recorded the different strategies children used. These strategies turned out to be critical in predicting children’s achievement.

For example, let’s take 4 + 13.

One strategy for solving this addition problem is called “counting all.” With this strategy students look at the 4 dots and
count them (1-2-3-4); they then look at 13 dots and count them (1-2-3-4-5-6-7-8-9-10-11-12-13); they then look at all the dots and count them, from 1 to 17. This is often the first strategy that children use as they are learning to count.

A more sophisticated strategy that develops from “counting all” is called “counting on.” A student using this strategy would count from 1 to 4 and then continue from 5 to 17.

A third strategy is called “known facts”—some people just know, without adding or thinking, that 4 and 13 is 17 because they remember these number facts.

The fourth strategy is called “derived facts.” This is where students decompose and recompose the numbers to make them more familiar numbers for adding and subtracting. So they may say, “Well, I know 10 and 4 is 14,” and then they add on the 3.

This sort of strategy of decomposing and recomposing numbers is helpful when you are given calculations to do, especially when doing them in your head. For example, you may need to know the answer to 96 + 17. To most people that is a nasty addition sum that looks daunting, but if 4 is taken from the 17 first and added to the 96, the problem becomes 100 + 13, which is much more reasonable. People who are good at mathematics decompose and recompose numbers all the time. This is the strategy that the researchers called “derived facts” as the students changed the numbers into ones that they knew the answers to, by decomposing and then recomposing the numbers. A better, well-known term for this flexible use of numbers is “number sense.” Students with number sense are able to use numbers flexibly, grouping numbers, decomposing and recomposing.

The researchers found that the above-average children in the 8+ age group counted on in 9 percent of the cases, they used known facts 30 percent of the time, and they used number sense 61 percent of the time. In the same age group the
students who were below average counted all 22 percent of the time, counted on 72 percent of the time, used known facts 6 percent of the time, but they never used number sense. It was this absence of number sense that was critical to their low achievement.

When the researchers looked at ten-year-olds, they found that the below-average group used the same number of known facts as the above-average eight-year-olds, so you could think of them as having learned more facts over the years but, noticeably, they were still not using number sense. Instead, they were counting. What we learn from this, and from other research, is that the high-achieving students don’t just know more, but they work in very different ways—and, critically, they engage in flexible thinking when they work with numbers, decomposing and recomposing numbers.

The researchers drew two important conclusions from their findings. One was that low achievers are often thought of as
slow
learners, when in fact they are not learning the same things slowly. Rather, they are learning a
different
mathematics. The second is that the mathematics that low achievers are learning is a more difficult subject.

As an example of the very difficult mathematics that the below-average children were using, consider the strategy of counting back, which they frequently used with subtraction problems. For example, when they were given problems such as 16 – 13, they would start at the number 16 and count down 13 numbers (16-15-14-13-12-11-10-9-8-7-6-5-4-3). The cognitive complexity of this task is enormous and the room for mistakes is huge. The above-average children did not do this. They said, “16 take away 10 is 6, and 6 take away 3 is 3,” which is much easier. The research showed that the students who were achieving at high levels were those who had worked out that numbers can be flexibly broken apart and put together again. The problem for
the low-achieving children was simply that they had not learned to do this. The researchers also found that when low achievers failed at their methods, they did not change their method; instead, they fell back into counting more and more. Indeed, many of the low achievers became very efficient with small numbers, which lulled them into a sense of security. The low-achieving students came to believe that in order to be successful, they needed to count very precisely. Unfortunately, problems become more and more difficult in mathematics, so as time went on, the low achievers were trying to count in more and more complex situations. Meanwhile, the high achievers had forgotten counting strategies and were working with numbers flexibly. This is an easier task, but it is also a more important way of working in mathematics. As the low achievers continued to count, the high achievers worked flexibly and pulled further and further ahead.

Not surprisingly, the researchers found that the lower-achieving students who were not using numbers flexibly were also missing out on other important mathematical activities. For example, one of the important things that people do as they learn mathematics is compress ideas. What this means is, when we are learning a new area of math, such as multiplication, we may initially struggle with the methods and the ideas and have to practice and use it in different ways, but at some point things become clearer, at which time we compress what we know, and move on to harder ideas. At a later stage when we need to use multiplication, we can use it fairly automatically, without thinking about the process in depth.

William Thurston was a professor of mathematics and computer science at Cornell University who won the highest honor awarded in mathematics—the Fields Medal. He described the process of learning mathematics well:

Mathematics is amazingly compressible: you may struggle a long time, step-by-step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics.
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One way of thinking about the learning of mathematics, visually, is to think of a triangle such as that shown in the following diagram. The larger space at the top of the triangle is new mathematics you learn, that you need to think about and connect to other areas, and it takes up a big space in your brain. The smaller area at the bottom of the triangle represents mathematics that you know well and has been compressed.

shutterstock/molars

It is this compression that makes it easy for people to use concepts they learned many years ago, such as addition or multiplication, without having to think about how they work every time they use them. Gray and Tall found that the low-achieving students were not compressing ideas. Instead, they were so focused on remembering different methods, stacking one new method on top of the
next. Our brains can only compress concepts, not rules or methods, and the low-achieving students were not thinking conceptually, probably because they had been led to believe that mathematics is all about rules. For the low-achieving students the learning of mathematics would not be represented as a triangle. It would be more like a never-ending ladder stretching up to the sky, with every rung of the ladder being another rule or procedure to learn.

Bruce Rolff

Students who are not taught to flexibly use numbers often cling to methods and procedures they are taught, believing that each method is equally important and must simply be remembered and reproduced carefully. For these students the mathematics they are learning is much more difficult.

Students who struggle to master more and more procedures, without using numbers flexibly or compressing concepts, are working with the wrong model of mathematics. These students need to work with someone who will change their pathway to mathematics and show them how to use numbers flexibly and how to think about mathematical concepts. But instead of working with people who will change the students’ approach, what typically happens is that they get labeled as low-achieving students and people decide they need more drill, putting them into classes where they repeat methods over
and over again. This is the last thing these students need and it simply feeds into their faulty worldview of math. Instead, they need opportunities to play with numbers, as I will suggest in the next chapter, and to develop “number sense.” Fortunately, students can learn to use numbers flexibly and to consider concepts at any age, and it was with this knowledge that my graduate students and I set out to work with students who had previously been low achievers. We sought to give them the experience of using mathematics flexibly. This is the story of what happened.