What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (8 page)

The bottom line on talking is that it is critical to math learning and to giving students the depth of understanding they need. This does not mean that students should be talking all the time or that just any form of talking is helpful. Math teachers need to organize productive mathematical discussions, and they should give children some time to discuss math and some time to work alone. But the distorted version of mathematics that is conveyed in silent math classrooms is one that makes mathematics inaccessible and extremely boring to most children.

Learning without Reality

A problem with both old and newer mathematics approaches that has emerged from my research is the ridiculous problems that are used in mathematics classrooms. Just like stepping through the wardrobe door and entering Narnia, in math classrooms trains travel toward each other on the same tracks and people paint houses at identical speeds all day long. Water fills tubs at the same rate each minute, and people run around tracks at the same distance from the edge. To do well in math class, children know that they have to suspend reality and accept the ridiculous problems they are given. They know that if they think about the problems and use what they understand from life, then they will fail. Over time, schoolchildren realize that when you enter Mathland you leave your common sense at the door.

Contexts started to become more common in math problems in the 1970s and ’80s. Up until that point most mathematics had been taught through abstract questions with no reference to the world. The abstractness of mathematics is synonymous for many people with a cold, detached, remote body of knowledge. Some believed that this image may be broken down through the use of contexts, and so mathematics questions were placed into contexts with the best of intentions. But instead of giving students realistic situations that they could analyze, textbook authors began to fill books with make-believe contexts—contexts that students were meant to believe but for which they should not use any of their real-world knowledge. Students are frequently asked to work on questions involving, for example, the price of food and clothes, the distribution of pizza, the numbers of people who can fit into an elevator, and
the speeds of trains as they rush toward each other, but they are not meant to use any of their actual knowledge of clothing prices, people, or trains. Indeed, if they do engage in the questions and use their real-world knowledge, they will fail. Students come to know this about math class. They know that they are entering a realm in which common sense and real-world knowledge are not needed.

Here are the sorts of examples that fill math books:

Joe can do a job in 6 hours and Charlie can do the same job in 5 hours. What part of the job can they finish by working together for 2 hours?

A restaurant charges $2.50 for
1
⁄
8
of a quiche. How much does a whole quiche cost?

A pizza is divided into fifths for 5 friends at a party. Three of the friends eat their slices, but then 4 more friends arrive. What fractions should the remaining 2 slices be divided into?

Everybody knows that people work together at a different pace than when they work alone, that food sold in bulk such as a whole quiche is usually sold at a different rate than individual slices, and that if extra people turn up at a party more pizza is ordered or people go without slices—but none of this matters in Mathland. One long-term effect of working on make-believe contexts is that such problems contribute to the mystery and otherworldliness of Mathland, which curtails people’s interest in the subject. The other effect is that students learn to ignore contexts and work only with the numbers, a strategy that would not apply to any real-world or professional situation. An illustration of this phenomenon is given by this famous question asked in a national assessment of students:

An army bus holds 36 soldiers. If 1,128 soldier are being bussed to their training site, how many buses are needed?

The most frequent response from students was 31 remainder 12, a nonsensical answer when dealing with the number of buses needed.
23
Of course, the test writers wanted the answer of 32, and the “31, remainder 12” response is often used as evidence that American students do not know how to interpret situations. But it may equally be given as evidence that they have been trained in Mathland, where such responses are sensible.

My argument against pseudocontexts does not mean that contexts should not be used in mathematics examples; they can be extremely powerful. But they should only be used when they are realistic and when the contexts offer something to the students, such as increasing their interest or modeling a mathe matical concept. A realistic use of context is one where stud-ents are given real situations that need mathematical analysis, for which they do need to consider (rather than ignore) the variables. For example, students could be asked to use mathematics to predict population growth. This would involve interpreting newspaper data on the US population, investigating the amount of growth over recent years, determining rates of change, building linear models (
y
=
mx
+
b
), and using these to predict population growth into the future. Such questions are excellent ways to interest students, motivate them, and give them practice in using mathematics to solve problems. Contexts may also be used to give a visual representation, helping to convey meaning. It does not hurt to suggest that a circle is a pizza that needs dividing into fractions, but it does hurt when students are invited into the world of parties and friends while at the same time being required to ignore everything they know about parties and friends.

There are also many wonderful mathematics problems with
no context or barely any context at all that can engage students. The famous four-color problem, which intrigued mathematicians for centuries, is a good example of a gripping, abstract math problem. The problem came about in 1852 when Francis Guthrie was trying to color a map of the counties of England. He did not want to color any adjacent counties with the same color and noticed that he needed only four colors to cover the map. Mathematicians then set out to prove that only four colors would be needed in any map or any set of touching shapes. It took more than a century to prove this, and some still question the proof.

This is a great problem that can be given to students to investigate. They can work with a map of touching countries, such as Europe, or draw their own shapes. For example,

Can you color this map using only 4 colors, with no 2 adjacent “countries” colored the same?

Other examples of problems I have used in this book, like the chessboard problem in the introduction, use contexts sensibly and responsibly. The contexts give meaning to the problems and provide realistic constraints—students do not have to partly believe them and partly ignore them.

The artificiality of mathematics contexts may seem to be a small concern, but the long-term impact of such approaches can be devastating for students of mathematics. Hilary Rose, a sociologist and newspaper columnist, illustrates this point well. She recalls that as a young child she had been something of “a mathematical wunderkind”
24
and loved to explore patterns, numbers, and shapes. She describes how her sense of mathematical magic ended when real-world problems were used in school. At first she enthusiastically engaged with the problems, drawing upon her knowledge of the situations described, but then found that such engagement was not allowed:

Thinking about it, it was those so called practical problems that irritated me the most. It was obvious to me that many of the questions simply indicated that the questioner did not know enough about the craft skills involved in real-world solutions. Lawn rollers being pulled up slopes, wallpapering rooms by calculating square feet and inches: these were tedious and as far as that highly practical child could see, stupid . . . I know that the price I paid was to lose my sense of confidence that school maths and everyday maths were part of one world.
25

[Some of the studies I refer to in this book took place in the UK and Australia, where
mathematics
is shortened to
maths
as opposed to the American abbreviation
math
.]

If the ridiculous contexts were taken out of math books across America—be they the traditional or newer books—they
would probably fall in size by more than half. The elimination of ridiculous contexts would be good for many reasons. Most important, students would realize that they are learning an important subject that helps make sense of the world, rather than a subject that is all about mystification and non-sense.

 • • • 

As the world changes and technology becomes more and more pervasive in our jobs and lives, it is impossible to know exactly which mathematical methods will be most helpful in the future. This is why it is so important that schools develop flexible thinkers who can draw from a variety of mathematical principles in solving problems. The only way to create flexible mathe-matical thinkers is to give children experience in working in these ways, both in school and at home. In the next chapter I will consider two very different school approaches that were extremely effective in achieving this goal.

ra2studio

3 / A Vision for a Better Future

Effective Classroom Approaches

I
magine a time when children were eager to go to math lessons at school, excited to learn new mathematical ideas, and able to use math to solve problems outside of the classroom. Adults would feel comfortable with math, be happy to be given mathematical problems at work, and refrain from saying at parties, “I am terrible at math.” America would have the number and range of people good at math to fill the various jobs needing mathematical and scientific understanding that our technological age requires. All of this might sound far-fetched given the number of mathematically damaged and phobic people in the population and the scores of schoolchildren who dread math lessons. But a very different mathematical reality is achievable, and teachers and parents can bring about this reality. In the pages that follow I will describe two highly successful approaches that offered students an experience of real mathematical work. They are middle and high school level, but the principles of the two approaches apply to all levels. In the
following pages we will learn about students from a wide range of backgrounds who came to love math, to achieve at high levels, and to view mathematics as an important part of their future.

The Communicative Approach

Photo courtesy of Jo Boaler

Railside High School is an urban school in California and lessons are frequently interrupted by the sound of speeding trains. As with many urban schools, the buildings look as though they are in need of repair, but Railside is not like other urban schools. Calculus classes are often badly attended or nonexistent in many schools, but at Railside they are packed with eager and successful students. When I have taken visitors to the school and we have stepped inside the math classrooms, they have been amazed to see all the students hard at work, engaged, and excited about math. I first visited Railside because I had learned that the teachers collaborated and planned teaching ideas together, and I was interested to see their lessons. I saw enough in that visit to invite the school to be part of a new Stanford research project investigating the effectiveness of different mathematics approaches. Some four years later, after we had followed seven hundred students through three high schools, observing,
interviewing, and assessing them, we knew that Railside’s approach was both highly successful and highly unusual.

The mathematics teachers at Railside used to teach using traditional methods, but they were unhappy with their students’ high failure rates and lack of interest in math, so they worked together to design a new approach. Teachers met over several summers to devise a new algebra curriculum and later to improve all the courses they offered. They also detracked classes and made algebra the first course that
all
students would take on entering high school—not just higher achieving students. In most algebra classes students work through questions designed to give practice in mathematical techniques such as factoring polynomials or solving inequalities. At Railside the students learned the same methods, but the curriculum was organized around bigger mathematical ideas, with unifying themes such as “What is a linear function?” The focus of the Railside approach was “multiple representations,” which is why I have described it as communicative—the students learned about the different ways that mathematics could be communicated through words, diagrams, tables, symbols, objects, and graphs. As they worked, the students would frequently be asked to explain work to each other, moving between different representations and communicative forms. When we interviewed students and asked them what they thought math was, they did not tell us that it was a set of rules, as most students do. Instead, they told us that math was a form of communication, or a language. As one young man explained: “Math is kind of like a language, because it has got a whole bunch of different meanings to it, and I think it is communicating. When you know the solution to a problem, I mean that is kind of like communicating with your friends.”

In one of the lessons I observed, students were learning about functions. The students had been given what the teachers referred to as “pile patterns.” Different students had been given different patterns to work with. Pedro was given the following pattern, which includes the first three cases:

The aim of the activity was for students to work out how the pattern was growing (you could try this too) and to represent this as an algebraic rule, a t-table, a graph, and a generic pattern. They also needed to show the hundredth case in the sequence, having been given the first three cases.

Pedro started by working out the numbers that went with the first 3 cases, and he put these in his t-table:

Case Number	Number of Tiles
1	10
2	13
3	16

He noted at this point that the pattern was “growing” by 3 each time. Next he tried to
see
how the pattern was growing in his shapes, and after a few minutes he saw it! He could see that each of the 3 sections grew by 1 each time. He represented the first two cases in this way:

He could see that there were seven tiles that always stayed the same and were present in the same positions (this was the way he visualized the pattern growth, but there are other ways of visualizing it). In addition to the “constant” seven, there were lines of tiles that grew with every case number. So, for example, if we just look at the vertical line of tiles:

we see that in case 1, there is 1 at the bottom + 3. In case 2, there are 2 + 3. In case 3 there would be 3 + 3; and in case 4, there would be 4 + 3, and so on. The 3 is a constant, but there is one more added to the lower section of tiles each time. We can also see that the growing section is the same size as the case number each time. When the case is 1, the total number of tiles is 1 + 3; when the case is 2, the total is 2 + 3. We can assume from this that when it is the hundredth case, there will be 100 + 3 tiles. This sort of work—considering, visualizing, and describing patterns—is at the heart of mathematics and its applications.

Pedro represented his whole pattern in the following way:

where
x
stood for the case number. By adding together the three sections, he could now represent the whole function as 3
x
+ 7.

At this point I should explain something about algebra for those who find this example totally bewildering. When a friend of mine read about this pattern, she was utterly lost and I realized that her confusion came from the way she had learned algebra in her traditional math classes. She looked at the pattern with me, saw that the student had represented it as 3
x
+ 7, and asked me, “So what is
x?
” I said that
x
was the case number, so in the first case
x
is 1, in the second
x
is 2, and so on. This completely confused her, which I realized was because to her
x
was always meant to be a
single
number. She had spent so many years of math classes “solving for
x
”—rearranging equations to find out what number
x
was—that she, like millions of schoolchildren, had missed the most important point about algebra—that
x
is used to represent a
variable.
The reason that algebra is used so pervasively by mathematicians, scientists, medics, computer programmers, and many other professionals is because patterns—that grow and change—are central to their work and to the world, and algebra is a key method in describing and representing them. My friend could see that the pattern had a different number of tiles each time but was just not used to using algebra to represent a
changing
quantity. But the task in this problem—to find a way of visualizing and representing the pattern, using algebra to describe the changing parts of the pattern—is extremely important algebraic work. The way that most people learn algebra hides the meaning of algebra, it stops them from using it appropriately, and it hinders their ability to see the usefulness of algebra as a problem-solving tool in mathematics and science.

Pedro was pleased with his work and he decided to check his algebraic expression with his t-table. Satisfied that 3
x
+ 7 worked, he set about plotting his values on a graph. I left the
group as he was eagerly reaching for graph paper and colored pencils. The next day in class I checked in with him again. He was sitting with three other boys and they were designing a poster to show their four functions. Their four desks were pushed together and covered by a large poster that was divided into four sections. From a distance the poster looked like a piece of mathematical artwork with color-coded diagrams, arrows connecting different representations to each other, and large algebraic symbols.

Photo courtesy of Jo Boaler