Creativity and Flexibility

 

Neuroscientist Sian Beilock has studied the brain when people are working under pressure. A particular area of the brain called the “working memory” is needed when we do calculations. The working memory is sometimes referred to as the “search engine of the mind” and, like all areas of our brains, is developed through practice. What Beilock has shown is that when we are stressed or under pressure, our working memory is impeded. The students who are the most compromised are those with the most working memory. This means that when students are given timed math tests and they become anxious, as many do, their working memory is compromised, and they cannot calculate the answers. Anxiety sets in, and a pattern of harmful beliefs soon follows.

The feeling of stress impeding your brain may be something you have known yourself. Have you ever had to work on a math calculation under pressure and felt as though your mind “went blank”? That is the feeling of stress blocking your working memory. When we give timed tests to young children, many of them experience stress, their working memory is compromised, and they cannot recall math facts. When they realize they cannot achieve, anxiety sets in.

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The irony of the unfortunate speed-based math activities in schools, where children are turned away from a lifetime of mathematical and scientific thinking because they don’t produce math facts quickly and under pressure, is that mathematics is not a subject that requires speed. Some of the strongest mathematical thinkers are very slow with numbers and other aspects of mathematics. They do not think quickly; they think slowly and deeply.

In recent years, some of the world’s greatest mathematicians, including those who have won the Fields Medal, such as Laurent Schwartz and Maryam Mirzakhani, have talked openly about how slow they are with math. After Schwartz won the Fields Medal, he wrote an autobiography about his school days in which he talked about feeling stupid in school because he was one of the slowest thinkers. He says:

I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Toward the end of the eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time.

 

I’m still just as slow. . . . At the end of the eleventh grade, I took the measure of the situation and came to the conclusion that rapidity doesn’t have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant.

**

At the time I was speeding through my math questions, I was myself working under the myth that speed is what is important. In our archaic school system, it is not surprising that millions of students believe speedy performance is what is valued. Now many years on, I have learned to approach content differently. I no longer look at math problems as something to answer quickly, but as something to think about deeply and creatively. That change has helped me greatly. I now get more from not only mathematical thinking, but any scientific or technical reading or work. The change in my approach has helped me so much and has fueled my passion to help others disarm this pervasive myth in the pursuit of understanding, creativity, and connections.

Medical doctor Norman Doidge says that when people learn something quickly, they are probably strengthening existing neural connections. These he describes as “easy come, easy go” neural connections, which can be rapidly reversed. This is what is happening when we study for a test, and we go over something we have already learned. We cram information in and reproduce it in a day or so, but it does not last and is quickly forgotten. More permanent brain changes come from the formation of new structures in the brain—the sprouting of neural connections and synapses. This is always a slow process. 

Mathematical Mindsets [Jo Boaler] (4): What is mathematics?

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?

**

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

**

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.

**

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.

**

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.

**

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.

Mathematical Mindsets [Jo Boaler] (5): Real Mathematicians are Slow Thinkers!

Mathematical Mindsets [Jo Boaler] (3):The Power of Mistakes and Struggle

Mathematical Mindsets [Jo Boaler] (2):The Brain and Mathematics Learning

Mathematical Mindsets [Jo Boaler] (1):Traumatized by Math

After Mirzakhani

Complicating the global image of a nation

Much of the reactions to the death of Mirzakhani was both normal and predictable. The Iranian President Hassan Rouhani and the Stanford University President Marc Tessier-Lavigne both issued solemn statements of condolences. The Iranian media competed in showering her with accolades. The Iranian oppositional venues in the United States and Europe began to use and abuse the occasion to denounce the Islamic Republic for its various policies, especially for the conditions conducive to "brain drain".

Maryam Mirzakhani left Iran to pursue her advanced mathematical studies in the US. She would have gone to Baghdad to do so if she were born one thousand years ago - and she would have probably gone to Beijing if she were born just a couple of decades from now. The question of "brain drain" is, of course, a serious malady in Iran and many other similar countries. But Maryam Mirzakhani was no "brain drain". Hers was a superior intelligence and she travelled where she could nourish it best - and that travelling did not suddenly turn her into this strange thing called "Iran-born", instead of just plain "Iranian".

But something else, something quite simple and significant, was also happening, just like Maryam Mirzakhani herself, gently and quietly.

Mirzakhani is comparable to Omar Khayyam not just because they were both Iranian mathematicians. But because like Khayyam, Mirzakhani too complicates the vision of their common homelands in the European and now American imagination in unpredictable ways.

Markedly brilliant minds and beautiful souls like Maryam Mirzakhani both upon their global recognitions and perforce upon their early and tragic passing become a symbol, a sign, a citation far beyond who they are and what they have achieved in their professional calling. From the time that she achieved her coveted Fields Medal, Maryam Mirzakhani had begun to complicate the global image of her homeland against the backdrop of the pervasive demonisation of Iran by one brand of warmongering or another.

Nations need to be simplified to be targeted for military strikes. Afghanistan was reduced to Mullah Omar. Iraq was reduced to Saddam Hussein. The more the image of a nation is complicated the more difficult it is for warmongers in Washington, Tel Aviv, or Riyadh targeting it for destruction.

Precisely in the quiet dignity of her work, her avoiding publicity like a plague, the tiny, cancer-ravaged body of Maryam Mirzakhani shined like a beautiful star on the dark planet of her earthly life. An Iranian, a Muslim, a woman of modest middle-class background, rising gloriously to put a big brilliant question mark in front of everything mobilised against her people!

Those exonerating Iranian people when targeting "Iran" for demonisation should take a look at what they have done to Afghanistan, Iraq, Syria, or Libya before they might believe their own delusions. Ordinary people, not governments, are the primary targets and the final victims of warmongering anywhere and everywhere.

To be sure, much to the chagrin of warmongers, the image of Iran has been complicated by other prominent Iranians, in particular, those who have put Iranian cinema on the global map. I recall vividly when another accomplished Iranian young woman, Samira Makhmalbaf first appeared in Cannes Film Festival at the age of seventeen to premier her film "Apple" in 1998. She too succeeded that year, seriously altering the image of Iran from that of a bearded angry man to a gifted young woman.
 In the realm of art, no one, of course, did more to complicate the image of Iran than the late master Abbas Kiarostami who was the principal engine bringing the rest of Iranian art to global spotlight.

But much of that complexity has been in the realm of arts, not sciences. In the realm of science, the only thing publicly related to Iran is, of course, the nuclear scientists who are the targets of assassinations presumably by a settler colony on planet Mars that is very concerned about the Iranian nuclear programme. These Martians are suspected to have occasionally sent their assassins to kill these Iranian nuclear scientists.

Mirzakhani was not a nuclear physicist. A breed apart, she was a world-renowned mathematician. Her accomplishments, as a result, assume entirely historical proportions comparable to other Iranian and Muslim scientists at the historical level of Muhammad ibn Musa al-Khwarizmi, Muhammad ibn Zakariyya al-Razi, and above all to Omar Khayyam, the towering astronomer and mathematician.




Mirzakhani's uplifting of her homeland to its historical memories alerts the world to a whole different register of consciousness about Iran as a Muslim a country in circumstances that, because of pervasive Islamophobia in the US, even Rumi is read as if he were a New Age guru from California.

Mirzakhani is comparable to Omar Khayyam not just because they were both Iranian mathematicians. But because like Khayyam, Mirzakhani too complicates the vision of their common homeland in the European and now American imagination in unpredictable ways.
The reputation of Omar Khayyam as a poet, however, widely outshines his fame as a mathematician. But the difference is only on the surface. The beauty of Khayyam's mathematical mind, it now seems, had to be translated, as it were, into poetic scepticism to be registered for mortal beings, whereas Maryam Mirzakhani's poetry was and remained in pure mathematics.

**
The mathematician, poet, painter

On another occasion, she said: "Of course, the most rewarding part is the 'Aha' moment, the excitement of discovery and enjoyment of understanding something new - the feeling of being on top of a hill and having a clear view. But most of the time, doing mathematics for me is like being on a long hike with no trail and no end in sight". That is the mind of a mathematician in the soul of a poet. That is Khayyam incarnate.

Mirzakhani's mathematical equations were her poetry - a poetry only a happy few can decipher to their delight. The world at large is baffled at the beauty of that poetry. Thus, the other more recent kindred soul of Mirzakhani was, of course, the Indian mathematician Srinivasa Ramanujan (1887-1920) who died even younger than her.

The famous story narrated by Ramanujan's English colleague, the prominent mathematician GH Hardy, is now known as "Hardy-Ramanujan number 1729". According to Hardy, he was once going to see Ramanujan when he was bedridden. Hardy had just ridden in a cab number 1729 and upon arrival, he remarked to his friend that the number seemed to be quite dull and that he hoped it was not an unfavourable omen. "No", Ramanujan replied: "It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways". The two different ways are:

1729 = 13+123 = 93+103.

Now, that is pure poetry - for what is poetry other than a truth so obviously beautiful that mortals cannot see it. The joyous eyes of Maryam Mirzakhani's little daughter could see what her mother's beautiful mind knew with mathematical precision: "At the family's home, near Stanford University," Roberts tells us in her essay on the Iranian mathematician, "Mirzakhani would spend hours on the floor with supersized canvases of paper, sketching out ideas, drawing diagrams and formulae, often leading Anahita, now six, to exclaim, 'Oh, Mommy is painting again!"



Hamid Dabashi
aljazeera