What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (4 page)

The sum of the volumes of cubes of dimensions 9 and 10 almost equals the volume of a cube of dimension 12, but not quite. (It is one short!)

Indeed, Fermat went on to claim that even if every number in the world was tried, no one would ever find a solution to
a
3
+
b
3
=
c
3
nor to
a
4
+
b
4
=
c
4
,
or to any higher power. This was a bold claim involving the universe of numbers. In mathematics it is not enough to make such claims, even if the claims are backed up by hundreds of cases, as mathematics is all about the construction of time-resistant proofs. Mathematical proofs
involve making a series of logical statements from which only one conclusion can follow and, once constructed, they are always true. Fermat made an important claim in the 1630s, but he did not provide a proof—and it was the proof of his claim that would elude and frustrate mathematicians for more than 350 years. Not only did Fermat not provide a proof, but he scribbled a note in the margin of his work saying that he had a “marvelous” proof of his claim but that there was not enough room to write it. This note tormented mathematicians for centuries as they tried to solve what some have claimed to be the world’s greatest mathematical problem.
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Fermat’s last theorem stayed unsolved for more than 350 years, despite the attentions of some of the greatest minds in history. In recent years it was dramatically solved by a shy English mathematician, and the story of his work, told by a number of biographers, captures the drama, the intrigue, and the allure of mathematics that is unknown by many. Any child—or adult—wanting to be inspired by the values of determination and persistence, enthralled by the intrigue of puzzles and questions, and introduced to the sheer beauty of living mathematics should read Simon Singh’s book
Fermat’s Enigma.
Singh describes “one of the greatest stories in human thinking,”
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providing important insights into the ways mathematicians work.

Many people had decided that there was no proof to be found of Fermat’s theorem and that this great mathematical problem was unsolvable. Prizes were offered from different corners of the globe, and men and women devoted their lives to finding a proof, to no avail. Andrew Wiles, the mathematician who would write his name into history books, first encountered Fermat’s theory as a ten-year-old boy while reading in his local library in his hometown of Cambridge. Wiles described how he felt when he read the problem: “It looked so simple, and yet all the great mathematicians in history could not solve it. Here was a problem that I, as a ten-year-old, could understand and I
knew from that moment that I would never let it go, I had to solve it.”
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Years later, Wiles earned a PhD in mathematics from Cambridge and then took a position in Princeton’s mathematics department. But it was only some years later that Wiles realized that he could devote his life to the problem that had intrigued him since childhood. As Wiles set about trying to prove Fermat’s last theorem, he retired to his study and started reading journals and gathering new techniques. He began exploring and looking for patterns, working on small areas of mathematics, and then standing back to see if they could be illuminated by broader concepts. Wiles worked on a number of techniques over the next few years, exploring different methods for attacking the problem. Some seven years after starting the problem, Wiles emerged from his study one afternoon and announced to his wife that he had solved Fermat’s last theorem.

The venue that Wiles chose to present his proof of the 350-year-old problem was a conference at the Isaac Newton Institute in Cambridge, England, in 1993. Some people had become intrigued about Wiles’s work, and rumors had started to filter through that he was actually going to present a proof of Fermat’s last theorem. By the time Wiles arrived, there were more than two hundred mathematicians crammed into the room, and some had sneaked in cameras to record the historic event. Others—who could not get in—peered through windows. Wiles needed three lectures to present his work and at the conclusion of the last the room erupted into great applause. Singh described the atmosphere of the rest of the conference as “euphoric” with the world’s media flocking to the institute. Was it possible that this great and historical problem had finally been solved? Barry Mazur, a number theorist and algebraic geometer, reflected on the event, saying that “I’ve never seen such a glorious lecture, full of such wonderful ideas, with such dramatic tension, and what a buildup. There was only one possible punch line.” Everyone who had witnessed the event thought that Fermat’s last theorem
was finally proved. Unfortunately, there was an error in Wiles’s proof, which meant that he had to plunge himself back into the problem. In September 1994, after more months of work, Wiles knew that his proof was complete and correct. Using many different theories, making connections that had not previously been made, Wiles had constructed beautiful new mathematical methods and relationships. Ken Ribet, a Berkeley mathematician whose work had contributed to the proof, concluded that the landscape of mathematics had changed and mathematicians in related fields could work in ways that had never been possible before.

Wiles’s fascinating story is told in detail by Simon Singh and others. But what do such accounts tell us that could be useful in improving children’s education? One clear difference between the work of mathematicians and schoolchildren is that mathematicians work on long and complicated problems that involve combining many areas of mathematics. This stands in stark contrast to the short questions that fill the hours of math classes and that involve the repetition of isolated procedures. Long and complicated problems are important to work on for many reasons—one being that they encourage persistence, a critical trait for young people to develop that will stand them in good stead in life and work. In interviews, mathematicians often speak of their enjoyment in working on difficult problems. Diane Maclagan, a professor at the University of Warwick, was asked, “What is the most difficult aspect of your life as a mathematician?” She replied, “Trying to prove theorems.’’ The interviewer then asked what the most fun is. “Trying to prove theorems,’’ she replied.
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Working on long and complicated problems may not sound like fun, but mathematicians find such work enjoyable because they are often successful. It is hard for schoolchildren to enjoy a subject if they experience repeated failure, which of course is the reality for many young people in mathematics classrooms. But the reason that mathematicians are successful
is that they have learned something very important—and very learnable. They have learned to problem solve.

Problem solving is at the core of mathematicians’ work, as well as the work of engineers and others, and it starts with the making of a guess. Imre Lakatos, mathematician and philosopher, describes mathematical work as “a process of ‘conscious guessing’ about relationships among quantities and shapes.”
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Those who have sat in traditional math classrooms are probably surprised to read that mathematicians highlight the role of guessing, as I doubt whether they have ever experienced any encouragement to guess in their math classes. When an official report in the UK was commissioned to examine the mathematics needed in the workplace, the investigator found that estimation was the most useful mathematical activity.
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Yet when children who have experienced traditional math classes are asked to estimate, they are often completely flummoxed and try to work out exact answers, then round them off to look like an estimate. This is because they have not developed a good
feel
for numbers, which would allow them to estimate instead of calculate, and also because they have learned, wrongly, that mathematics is all about precision, not about making estimates or guesses. Yet both are at the heart of mathematical problem solving.

After making a guess, mathematicians engage in a zigzagging process of conjecturing, refining with counterexamples, and then proving. Such work is exploratory and creative, and many writers draw parallels between mathematical work and art or music. Robin Wilson, a British mathematician, proposes that mathematics and music “are both creative acts. When you are sitting with a bit of paper creating mathematics, it is very like sitting with a sheet of music paper creating music.”
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Devlin agrees, saying that “Mathematics is not about numbers, but about life. It is about the world in which we live. It is about ideas. And far from being dull and sterile, as it is so often portrayed, it is full of creativity.”
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The exhilarating, creative pathways that mathematicians follow as they solve problems are often hidden in the end point of mathematical work, which just shows results. These pathways cannot be the exact same ones as schoolchildren experience, as children need to be taught the methods they need, as well as to use them in the solving of problems, but neither should school mathematics be so different as to be unrecognizable. As George Pólya, the eminent Hungarian mathematician, reflected,

A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.
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Another interesting feature of the work of mathematicians is its collaboratory nature. Many people think of mathematicians as people who work in isolation, but this is far from the truth. Leone Burton, a British professor of mathematics education, interviewed seventy research mathematicians and found that they generally challenged the solitary stereotype of mathematical work, reporting that they preferred to collaborate in the production of ideas. More than half of the papers they submitted to Burton as representative of their work were written with colleagues. The mathematicians interviewed gave many reasons for collaboration, including the advantage of learning from one another’s work, increasing the quality of ideas, and sharing the “euphoria” of problem solving. As Burton reflected, “They offered all the same reasons for collaborating on research that are to be found in the educational literature advocating
collaborative work in classrooms.”
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Yet silent math classrooms continue to prevail across America.

Something else that we learn from various accounts of mathematicians’ work is that an important part of real, living mathematics is the posing of problems. Viewers of
A Beautiful Mind
may remember John Nash (played by Russell Crowe) undergoing an emotional search to form a question that would be sufficiently interesting to be the focus of his work. People commonly think of mathematicians as solving problems, but as Peter Hilton, an algebraic topologist, has said: “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.”
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Such work requires creativity, original thinking, and ingenuity. All the mathematical methods and relationships that are now known and taught to schoolchildren started as questions, yet students do not see the questions. Instead, they are taught content that often appears as a long list of answers to questions that nobody has ever asked. Reuben Hersh puts it well:

The mystery of how mathematics grows is in part caused by looking at mathematics as answers without questions. That mistake is made only by people who have had no contact with mathematical life. It’s the questions that drive mathematics. Solving problems and making up new ones is the essence of mathematical life. If mathematics is conceived apart from mathematical life, of course it seems—dead.
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Bringing mathematics back to life for schoolchildren involves giving them a sense of living mathematics. When students are given opportunities to ask their own questions and to extend problems into new directions, they know mathematics is still alive, not something that has already been decided and just needs to be memorized. If teachers pose and extend problems
of interest to students, they enjoy mathematics more, they feel more ownership of their work, and they ultimately learn more. English schoolchildren in math classes used to work on long problems that they could extend into directions that were of interest to them. For example, in one problem students were asked to design any type of building. This gave them the opportunity to consider interesting questions involving high-level mathematics, such as the best design for a fire station with a firefighter’s pole. Teachers would submit the students’ work to examination boards, and it was assessed as part of the students’ final examination grades. When I asked English schoolchildren about their work on these problems, they reported that they were not only enjoyable and they learned a lot from them but also that their work made them “feel proud” and that they could not feel proud of their more regular textbook work.

Another important part of the work of mathematicians that enables successful problem solving is the use of a range of representations such as symbols, words, pictures, tables, and diagrams, all used with precision. The precision required in mathematics has become something of a hallmark for the subject and it is an aspect of mathematics that both attracts and repels. For some schoolchildren it is comforting to be working in an area where there are clear rules for ways of writing and communicating. But for others it is just too hard to separate the precision of mathematical language with the uninspiring drill-and-kill methods that they experience in their math classrooms. There is no reason that precision and drilled teaching methods need to go together, and the need for precision with terms and notation does not mean that mathematical work precludes open and creative exploration. On the contrary, it is the fact that mathematicians can rely on the precise use of language, symbols, and diagrams that allows them to freely explore the
ideas
that such communicative tools produce. Mathematicians do not play with the notations, diagrams, and words as a poet or artist might. Instead, they explore the relations and
insights that are revealed by different arrangements of the notations. As Keith Devlin reflects: