What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (3 page)

Studies of people using mathematics in their lives—when shopping and performing other routine tasks—lead to similar recommendations. Researchers have found that adults cope well with mathematical demands, but they draw from school knowledge infrequently. In real-world situations, such as in street markets and shops, individuals have rarely made use of any school-learned mathematical methods or procedures. Instead, they have created methods that work given the constraints of the situations they faced.
14,
15,
16,
17
Jean Lave, a professor at the University of California, Berkeley, found that shoppers used their own methods to work out which were better deals in shops, without using any formal methods learned in school, and that dieters used informal methods that they created when needing to work out measures of servings. For example, a dieter who was told he could eat 3/4 of a 2/3 cup of cottage cheese did not perform the standard algorithm for multiplying the fractions. Instead, he emptied 2/3 of a cup onto a measuring board, patted it into a circle, marked a cross on it, and took one quadrant away, leaving 3/4 of it.
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Lave gives many examples from her various studies of people using informal methods such as these, without any recourse to school-learned methods.

The ways in which people use mathematics in the world will probably sound familiar to most readers, as they are the ways many of us use mathematics. Adults rarely stop to remember formal algorithms; instead, successful math users size up situations and adapt and apply mathematical methods, using them flexibly.

The math of the world is so different from the math taught in most classrooms that young people often leave school ill-prepared for the demands of their work and lives. Children learn, even when they are still in school, about the irrelevance of the work they are given, an issue that becomes increasingly important to children as they move through adolescence. As part of a research study in England,
19
I interviewed students
who had learned traditionally, as well as those who had learned through a problem-solving approach, about their use of math in their part-time after-school jobs. The students from the traditional approach all said that they used and needed math out of school but that they would never, ever make use of the mathematics they were learning in their school classrooms. The students regarded the school math classroom as a separate world with clear boundaries that separated it from their lives. The students who had learned through a problem-solving approach did not regard the mathematics of school and the world as different and talked with ease about their use of the school-learned math in their jobs and lives.

When students from sixty-four countries were given tests of mathematical problem solving in 2012, the United States came in thirty-sixth place. This is despite the fact that problem-solving approaches not only teach young people how to solve problems using mathematics but also prepare them for examinations—as well as or better than traditional approaches.
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What’s math got to do with it? It has a lot to do with children having low self-esteem throughout their lives because they are made to feel bad in math classes; it also has a lot to do with children not enjoying school as they are made to sit through uninspiring lessons, and it has a lot to do with the future of the country, given that we urgently need many more mathematical people to help with jobs in science, medicine, technology, and other fields. Math has many things to answer for and this book is all about giving teachers, parents, and others the knowledge of good ways to work in schools and homes, so that we can start improving our children’s and our country’s futures.

We need to bring
mathematics
back into math classrooms and children’s lives, and we must treat this as a matter of urgency. The classroom characteristics that I am arguing for in this book are not at either of the poles of a “traditional” or “reform” debate, and they could take place in any math classroom or home because they are all about
being mathematical.
Children need to solve complex problems, to ask many forms of questions, and to use, adapt, and apply standard methods, as well as to make connections between methods and to reason mathematically—and they can engage in such methods at home and at school.

But let us return for a moment to Emily Moskam’s classroom, the one that I described at the start of this chapter. The class was in a public high school that offered students a choice between a “traditional” and a problem-solving approach. When I took a senior professor from Stanford to visit Emily’s class, he simply described it as “magical.” Perhaps his enthusiasm was not surprising. Emily Moskam had won awards for her teaching, including a Presidential Award, and her students regularly went on to pursue mathematical careers. What is surprising and
tragic
is that soon after I videotaped this lesson, Emily was told that she could no longer teach in this way. A small group of parents had worked hard to convince others at the school that all students needed to learn mathematics using only traditional methods—that students should sit in rows and not be asked to solve complex problems and only the teacher should talk. From that year on, math classes at Emily’s school
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looked very different. In some ways they were indistinguishable from classrooms in the 1950s. Teachers stood it the front, explaining procedures to students who sat and quietly practiced them individually. The problem-solving approach that Emily had shown to be so successful for her students is no longer an option in her school.

How Can This Book Help?

I conduct longitudinal studies of children’s mathematics learning. These studies are very rare. Typically researchers visit classrooms at a particular point in time to observe children learning, but I have followed thousands of American and British students through
years of middle and high school math classes to observe how their learning develops over time. I am a professor of mathematics education at Stanford University and the cofounder of YouCubed, a center I formed to get free and accessible research ideas to teachers and parents. I was formerly the Marie Curie Professor for Mathematics Education in Europe and prior to that a middle and high school math teacher.

In my studies I monitor how students are learning, finding out what’s helpful to them and what’s not. In recent years, I returned to a middle school in the Bay Area to teach school math again, with some of my graduate students. We taught classes of largely disaffected students who hated math and were getting D’s and F’s in it. They started the class saying they didn’t want to be there, but they ended up loving it, telling us that it had transformed their view of math. One boy told us that if math was like that during the school year, he would take it all day and every day. One of the girls told us that math class had always appeared so black and white to her, but in this class it was “all the colors in the rainbow.” Our teaching methods were not revolutionary: we talked with the children about math, and we worked on algebra and arithmetic through puzzles and problems such as the chessboard problem:

How many squares are there in a chessboard?

(The answer is not 64! See
appendix A
for details.)

Successful teachers use teaching methods that more people should know about. Good students also use strategies that make them successful—they are not just people who are born with some sort of math gene, as many people think. High-achieving students are people who learn, whether through great teachers, role models, family, or other sources, to use the particular strategies that I will share in this book.

Based on my studies of thousands of children,
What’s Math Got to Do with It?
will identify the problems that American students encounter, and it will share some solutions. I know that many parents and some teachers are afraid of math and don’t think that they know enough to help children or even to talk with them about math, especially when high school courses begin. But parents and teachers, whatever their level of math anxiety, can change everything for students by giving positive messages and sharing helpful math strategies that I provide in this book. I hope that this book will spark an interest in people who have been wounded by math experiences, motivate those who already know and enjoy math, and guide teachers and parents to inspire children through mathematical exploration and connection making.

And Why Do We
All
Need It?

I
n my different research studies I have asked hundreds of children, taught traditionally, to tell me what math is. They will typically say such things as “numbers” or “lots of rules.” Ask mathematicians what math is and they will more typically tell you that it is “the study of patterns” or “a set of connected ideas.” Students of other subjects, such as English and science, give descriptions of their subjects that are similar to those of professors in the same fields. Why is math so different? And why is it that students of math develop such a distorted view of the subject?

Reuben Hersh, a philosopher and mathematician, has written a book called
What Is Mathematics, Really?
in which he explores the true nature of mathematics and makes an important point: people don’t like mathematics because of the way it is
misrepresented
in school. The math that millions of Americans experience in school is an impoverished version of the subject and it bears little resemblance to the mathematics of
life or work or even the mathematics in which mathematicians engage.

What Is Mathematics, Really?

Mathematics is a human activity, a social phenomenon, a set of methods used to help illuminate the world, and it is part of our culture. In Dan Brown’s bestselling novel
The Da Vinci Code,
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the author introduces readers to the “divine proportion,” a ratio that is also known as the Greek letter phi. This ratio was first discovered in 1202 when Leonardo Pisano, better known as Fibonacci, asked a question about the mating behavior of rabbits. He posed this problem:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The resulting sequence of pairs of rabbits, now known as the Fibonacci sequence, is

1, 1, 2, 3, 5, 8, 13, . . .

What is fascinating about this number pattern is that after the sequence starts, any number can be found from the sum of the previous two numbers. Even more interesting, as we move along the sequence of numbers, dividing each number by the one before it, produces a ratio that gets closer and closer to 1.618, also known as phi or
the golden ratio.
What is amazing about this ratio is that it exists throughout nature. When flower seeds grow in spirals, they grow in the ratio 1.618:1. The ratio of spirals in seashells, pinecones, and pineapples is exactly the same. For example, if you look very carefully at the photograph of a daisy here, you will see that the seeds in the center of the flower form spirals, some of which curve to the left and some to the right.

If you map the spirals carefully, you will see that close to the center there are twenty-one running counterclockwise. Just a little further out, there are thirty-four spirals running clockwise. These numbers appear next to each other in the Fibonacci sequence.

Daisy showing twenty-one counterclockwise spirals

Daisy showing thirty-four clockwise spirals

Remarkably, the measurements of various parts of the human body have the exact same relationship. Examples include a person’s height divided by the distance from belly button to the floor; and the distance from shoulders to fingertips, divided by the distance from elbows to fingertips. The ratio turns out to be so pleasing to the eye that it is also ubiquitous in art and architecture, even being featured in the United Nations building, the Parthenon in Athens, and the pyramids of Egypt.

Ask most mathematics students in middle or high school about these relationships, and they will not even know they exist. This is not their fault, of course. They have never been taught about them. Mathematics is all about illuminating relationships such as those found in shapes and in nature. It is also a powerful way of expressing relationships and ideas in numerical, graphical, symbolic, verbal, and pictorial forms. This is the wonder of mathematics that is denied to most children.

Those children who do learn about the true nature of mathe- matics are very fortunate, and it often shapes their lives. Margaret Wertheim, a science reporter for the
New York Times
, reflects upon an Australian mathematics classroom from her childhood and the way that it changed her view of the world:

When I was ten years old I had what I can only describe as a mystical experience. It came during a math class. We were learning about circles, and to his eternal credit our teacher, Mr. Marshall, let us discover for ourselves the secret image of this unique shape: the number known as pi. Almost everything you want to say about circles can be said in terms of pi, and it seemed to me in my childhood innocence that a great treasure of the universe had just been revealed. Everywhere I looked I saw circles, and at the heart of every one of them was this mysterious number. It was in the shape of the sun and the moon and the earth; in mushrooms, sunflowers, oranges, and pearls;
in wheels, clock faces, crockery, and telephone dials. All of these things were united by pi, yet it transcended them all. I was enchanted. It was as if someone had lifted a veil and shown me a glimpse of a marvelous realm beyond the one I experienced with my senses. From that day on I knew I wanted to know more about the mathematical secrets hidden in the world around me.
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How many students who have sat through American math classes would describe mathematics in this way? Why are they not enchanted, as Wertheim was, by the wonder of mathematics, the insights it provides into the world, the way it elucidates the patterns and relationships all around us? It is because they are misled by the image of math presented in school mathematics classrooms and they are not given an opportunity to experience real mathematics. Ask most school students what math is and they will tell you it is a list of rules and procedures that need to be remembered.
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Their descriptions are frequently focused on calculations. Yet as Keith Devlin, mathematician and writer of several books about math, points out, mathematicians are often not even very good at calculations as they do not feature centrally in their work. As I had mentioned before, ask mathematicians what math is and they are more likely to describe it as
the study of patterns.
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Early in his book
The Math Gene
,
Devlin tells us that he hated math in elementary school. He then recalls his reading of W. W. Sawyer’s book
Prelude to Mathematics
during high school, which captivated his thinking and even made him consider becoming a mathematician himself. One of Devlin’s passages begins with a quote from Sawyer’s book:

“Mathematics is the classification and study of all possible patterns.”
Pattern
is here used in a way that everybody may agree with. It is to be understood in a very wide sense, to cover almost
any kind of regularity that can be
recognized by the mind
[emphasis added]. Life, and certainly intellectual life, is only possible because there are certain regularities in the world. A bird recognizes the black and yellow bands of a wasp; man recognizes that the growth of a plant follows the sowing of a seed. In each case, a mind is aware of pattern.
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Reading Sawyer’s book was a fortunate event for Devlin, but insights into the true nature of mathematics should not be gained
in spite
of school experiences, nor should they be left to the few who stumble upon the writings of mathematicians. I will argue, as others have done before me, that school classrooms should give children a sense of the nature of mathematics, and that such an endeavor is critical in halting the low achievement and participation that extends across America. Schoolchildren know what English literature and science are because they engage in authentic versions of the subjects in school. Why should mathematics be so different?
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What Do Mathematicians Do, Really?

Fermat’s last theorem, as it came to be known, was a theory proposed by the great French mathematician Pierre de Fermat in the 1630s. Proving (or disproving) the theory that Fermat set out became the challenge for centuries of mathematicians and caused the theory to become known as “the world’s greatest mathematical problem.”
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Fermat (1601–1665) was famous in his time for posing intriguing puzzles and discovering interesting relationships between numbers. Fermat claimed that the equation
a
n
+
b
n
=
c
n
has no whole number solutions for
n
when
n
is greater than 2. So, for example, no numbers could make the statement
a
3
+
b
3
=
c
3
true. Fermat developed his theory through consideration of Pythagoras’s famous case of
a
2
+
b
2
=
c
2
. Schoolchildren are typically introduced to the Pythagorean formula
when learning about triangles, as any right-angled triangle has the property that the sum of squares built on the two sides (
a
2
+
b
2
) is equal to the square of the hypotenuse,
c
2
.

So, for example, when the sides of a triangle are 3 and 4, then the hypotenuse must be 5 because 3
2
+ 4
2
= 5
2
. Sets of three numbers that satisfy Pythagoras’s case are those where two square numbers (e.g., 4, 9, 16, 25) can combine to produce a third.

Fermat was intrigued by the Pythagorean triples and explored the case of cube numbers, reasonably expecting that some pairs of cubed numbers could be combined to produce a third cube. But Fermat found that this was not the case and the resulting cube always has too few or too many blocks. For example,