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| Jenya Sapir |
When we ask students what math is, they will typically give descriptions that are very different from those given by experts in the field. Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what math is, they will say it is the study of patterns; that it is an aesthetic, creative, and beautiful subject (Devlin, 1997). Why are these descriptions so different?
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Recently I was chairing the PhD viva of one of Maryam's students. A viva is the culminating exam for PhD students when they “defend” the dissertations they have produced over a number of years in front of their committee of professors. I walked into the math department at Stanford that day, curious about the defense I was to chair. The room in which the defense was held was small, with windows overlooking Stanford's impressive Palm Drive, the entrance to the university, and it was filled with mathematicians, students, and professors who had come to watch or judge the defense. Maryam's student was a young woman names Jenya Sapir, who strode up and down that day, sharing drawings on different walls of the room, pointing to them as she made conjectures about the relationships between lines and curves on her drawings. The mathematics she described was a subject of visual images, creativity, and connections, and it was filled with uncertainty
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Three or four times in the defense, professors asked questions, to which the confident young woman simply answered, “I don't know.” Often the professor added that she or he did not know either. It would be very unusual in a defense of an education PhD for a student to give the answer: “I don't know,” and it would be frowned upon by some professors. But mathematics, real mathematics, is a subject full of uncertainty; it is about explorations, conjectures, and interpretations, not definitive answers. The professors thought it was perfectly reasonable that she did not know the answers to some of the questions, as her work was entering uncharted territories. She passed the PhD exam with flying colors.
This does not mean that there are no answers in mathematics. Many things are known and are important for students to learn. But somehow school mathematics has become so far removed from real mathematics that if I had taken most school students into the mathematics department defense that day, they would not have recognized the subject before them. This wide gulf between real mathematics and school mathematics is at the heart of the math problems we face in education. I strongly believe that if school math classrooms presented the true nature of the discipline, we would not have this nationwide dislike of math and widespread math underachievement.
Mathematics is a cultural phenomenon; a set of ideas, connections, and relationships that we can use to make sense of the world. At its core, mathematics is about patterns. We can lay a mathematical lens upon the world, and when we do, we see patterns everywhere; and it is through our understanding of the patterns, developed through mathematical study, that new and powerful knowledge is created.
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Knowledge of mathematical patterns has helped people navigate oceans, chart missions to space, develop technology that powers cell phones and social networks, and create new scientific and medical knowledge, yet many school students believe that math is a dead subject, irrelevant to their futures.
To understand the real nature of mathematics it is helpful to consider the mathematics in the world—the mathematics of nature. The patterns that thread through oceans and wildlife, structures and rainfall, animal behavior, and social networks have fascinated mathematicians for centuries.and social networks have fascinated mathematicians for centuries.
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Numerous research studies (Silver, 1994) have shown that when students are given opportunities to pose mathematics problems, to consider a situation and think of a mathematics question to ask of it—which is the essence of real mathematics—they become more deeply engaged and perform at higher levels. But this rarely happens in mathematics classrooms. In A Beautiful Mind, the box office movie hit, viewers watch John Nash (played by Russell Crowe) strive to find an interesting question to ask—the critical and first stage of mathematical work. In classrooms students do not experience this important mathematical step; instead, they spend their time answering questions that seem dead to them, questions they have not asked.
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Over the years, school mathematics has become more and more disconnected from the mathematics that mathematicians use and the mathematics of life. Students spend thousands of hours in classrooms learning sets of procedures and rules that they will never use, in their lives or in their work. Conrad Wolfram is a director of Wolfram-Alpha, one of the most important mathematical companies in the world. He is also an outspoken critic of traditional mathematics teaching, and he argues strongly that mathematics does not equal calculating. In a TED talk watched by over a million people, Wolfram (2010) proposes that working on mathematics has four stages:
1. Posing a question
2. Going from the real world to a mathematical model
3. Performing a calculation
4. Going from the model back to the real world, to see if the original question was answered
The first stage involves asking a good question of some data or a situation—the first mathematical act that is needed in the workplace. The fastest-growing job in the United States is that of data analyst—someone who looks at the “big data” that all companies now have and asks important questions of the data. The second stage Wolfram describes is setting up a model to answer the question; the third is performing a calculation, and the fourth is turning the model back to the world to see whether the question is answered. Wolfram points out that 80% of school mathematics is spent on stage 3—performing a calculation by hand—when that is the one stage that employers do not need workers to be able to do, as it is performed by a calculator or computer. Instead, Wolfram proposes that we have students working on stages 1, 2, and 4 for much more of their time in mathematics classes.
What employers need, he argues, is people who can ask good questions, set up models, analyze results, and interpret mathematical answers. It used to be that employers needed people to calculate; they no longer need this. What they need is people to think and reason.
The Fortune 500 comprises the top 500 companies in the United States. Forty-five years ago, when companies were asked what they most valued in new employees, the list looked like this:
Computation has dropped to the second-from-the-last position, and the top places have been taken by teamwork and problem solving.
Parents often do not see the need for something that is at the heart of mathematics: the discipline. Many parents have asked me: What is the point of my child explaining their work if they can get the answer right? My answer is always the same: Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics. Scientists prove or disprove theories by producing more cases that do or do not work, but mathematicians prove theories through mathematical reasoning. They need to produce arguments that convince other mathematicians by carefully reasoning their way from one idea to another, using logical connections. Mathematics is a very social subject, as proof comes about when mathematicians can convince other mathematicians of logical connections.
A lot of mathematics is produced through collaborations between mathematicians; Leone Burton studied the work of mathematicians and found that over half of their publications were produced collaboratively (Burton, 1999). Yet many mathematics classrooms are places where students complete worksheets in silence. Group and whole class discussions are really important. Not only are they the greatest aid to understanding—as students rarely understand ideas without talking through them—and not only do they enliven the subject and engage students, but they teach students to reason and to critique each other's reasoning, both of which are central in today's high-tech workplaces. Almost all new jobs in today's technological world involve working with massive data sets, asking questions of the data and reasoning about pathways. Conrad Wolfram told me that anyone who cannot reason about mathematics is ineffective in today's workplace. When employees reason and talk about mathematical pathways, other people can develop new ideas based on the pathways as well as see if a mistake has been made. The teamwork that employers value so highly is based upon mathematical reasoning. People who just give answers to calculations are not useful in the workplace; they must be able to reason through them.
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