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

Another serious problem we face in math education is that people believe that mathematics is all about calculating and that the best mathematics thinkers are those who calculate the fastest. Some people believe something even worse—that you have to be fast at math to be good at math. There are strong beliefs in society that if you can do a calculation quickly then you are a true math person and that you are “smart.” Yet mathematicians, whom we could think of as the most capable math people, are often slow with math. I work with many mathematicians, and they are simply not fast math thinkers. I don't say this to be disrespectful to mathematicians; they are slow because they think carefully and deeply about mathematics.

Laurent Schwartz won the Fields Medal in mathematics and was one of the greatest mathematicians of his time. But when he was in school, he was one of the slowest math thinkers in his class. In his autobiography, A Mathematician Grappling with His Century (2001), Schwartz reflects on his school days and how he felt “stupid” because his school valued fast thinking, but he thought slowly and deeply:

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. Towards 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. (Schwartz, 2001)

Schwartz writes, as have many other mathematicians, about the misrepresentation of mathematics in classrooms, and about mathematics being about connections and deep thinking, not fast calculation. There are many students in math classrooms who think slowly and deeply, like Laurent Schwartz, who are made to believe that they cannot be math people. Indeed, the idea that math is about fast calculations puts off large numbers of math students, especially girls. Yet mathematics continues to be presented as a speed race, more than any other subject—timed math tests, flash cards, math apps against the clock. It is no wonder that students who think slowly and deeply are put off mathematics.

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When people are asked about how mathematics is used in the world, they usually think of numbers and calculations—of working out mortgages or sale prices—but mathematical thinking is so much more. Mathematics is at the center of thinking about how to spend the day, how many events and jobs can fit into the day, what size of space can be used to fit equipment or turn a car around, how likely events are to happen, knowing how tweets are amplified and how many people they reach. The world respects people who can calculate quickly, but the fact is, some people can be very fast with numbers and not be able to do great things with them, and others, who are very slow and make many mistakes, go on to do something amazing with mathematics. The powerful thinkers in today's world are not those who can calculate fast, as used to be true; fast calculations are now fully automated, routine, and uninspiring. The powerful thinkers are those who make connections, think logically, and use space, data, and numbers creatively.

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

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

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

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?

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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.

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

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

Every time a student makes a mistake in math, they grow a synapse. (Carol Dweck)

Psychologist Jason Moser studied the neural mechanisms that operate in people's brains when they make mistakes (Moser et al., 2011).

Moser's study shows us that we don't even have to be aware we have made a mistake for brain sparks to occur. When teachers ask me how this can be possible, I tell them that the best thinking we have on this now is that the brain sparks and grows when we make a mistake, even if we are not aware of it, because it is a time of struggle; the brain is challenged, and this is the time when the brain grows the most.

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In Moser and his colleagues' study, the scientists looked at people's mindsets and compared mindsets with their ERN and Pe responses when they made mistakes on questions. Moser's study produced two important results. First, the researchers found that the students' brains reacted with greater ERN and Pe responses—electrical activity—when they made mistakes than when their answers were correct. Second, they found that the brain activity was greater following mistakes for individuals with a growth mindset than for individuals with a fixed mindset. Figure 2.1 represents brain activity in individuals with a fixed or growth mindset, with the growth mindset brains lighting up to a much greater extent when mistakes were made.

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The fact that our brains react with increased activity when we make a mistake is hugely important. I will return to this finding in a moment.

The study also found that individuals with a growth mindset had a greater awareness of errors than individuals with a fixed mindset, so they were more likely to go back and correct errors. This study supported other studies (Mangels, Butterfield, Lamb, Good, & Dweck, 2006) showing that students with a growth mindset show enhanced brain reaction and attention to mistakes. All students responded with a brain spark—a synapse—when they made mistakes, but having a growth mindset meant that the brain was more likely to spark again, showing awareness that a mistake had been made. Whether it is mathematics, teaching, parenting, or other areas of your life, it is really important to believe in yourself, to believe that you can do anything. Those beliefs can change everything.

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Moser's study, showing that individuals with a growth mindset have more brain activity when they make a mistake than those with a fixed mindset, tells us something else very important. It tells us that the ideas we hold about ourselves—in particular, whether we believe in ourselves or not—change the workings of our brains. If we believe that we can learn, and that mistakes are valuable, our brains grow to a greater extent when we make a mistake. This result is highly significant, telling us again how important it is that all students believe in themselves—and how important it is for all of us to believe in ourselves, particularly when we approach something challenging.

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“Peter Sims, a writer for the New York Times, has written widely about the importance of mistakes for creative, entrepreneurial thinking (Sims, 2011). He points out: 

“Imperfection is a part of any creative process and of life, yet for some reason we live in a culture that has a paralyzing fear of failure, which prevents action and hardens a rigid perfectionism. It's the single most disempowering state of mind you can have if you'd like to be more creative, inventive, or entrepreneurial.”

He also summarizes the habits of successful people in general, saying that successful people:

Feel comfortable being wrong

Try seemingly wild ideas

Are open to different experiences

Play with ideas without judging them

Are willing to go against traditional ideas

Keep going through difficulties”

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One of the most powerful moves a teacher or parent can make is in changing the messages they give about mistakes and wrong answers in mathematics.

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Another strategy for celebrating mistakes in class is to ask students to submit work of any form—even test papers (although the less we test students the better); teachers then highlight their “favorite mistakes.” Teachers should share with students that they are looking for their favorite mistakes, which should be conceptual mistakes, not numerical errors. Teachers can then share the mistakes with the class and launch a class discussion about where the mistake comes from and why it is a mistake. This is also a good time to reinforce important messages—that when the student made this mistake, it was good, because they were in a stage of cognitive struggle and their brain was sparking and growing. It is also good to share and discuss mistakes, because if one student makes a mistake we know others are making them also, so it is really helpful for everyone to be able to think about them.

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

In the last decade we have seen the emergence of technologies that have given researchers new access into the workings of the mind and brain. Now scientists can study children and adults working on math and watch their brain activity; they can look at brain growth and brain degeneration, and they can see the impact of different emotional conditions upon brain activity. One area that has emerged in recent years and stunned scientists concerns “brain plasticity.” It used to be believed that the brains people were born with couldn't really be changed, but this idea has now been resoundingly disproved. Study after study has shown the incredible capacity of brains to grow and change within a really short period (Abiola & Dhindsa, 2011; Maguire, Woollett, & Spiers, 2006; Woollett & Maguire, 2011).

When we learn a new idea, an electric current fires in our brains, crossing synapses and connecting different areas of the brain.

 
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If you learn something deeply, the synaptic activity will create lasting connections in your brain, forming structural pathways, but if you visit an idea only once or in a superficial way, the synaptic connections can “wash away” like pathways made in the sand. Synapses fire when learning happens, but learning does not happen only in classrooms or when reading books; synapses fire when we have conversations, play games, or build with toys, and in the course of many, many other experiences.

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The new findings that brains can grow, adapt, and change shocked the scientific world and spawned new studies of the brain and learning, making use of ever-developing new technologies and brain scanning equipment. In one study that I believe is highly significant for those of us in education, researchers at the National Institute for Mental Health gave people a 10-minute exercise to work on each day for three weeks. The researchers compared the brains of those receiving the training with those who did not. The results showed that the people who worked on an exercise for a few minutes each day experienced structural brain changes. The participants' brains “rewired” and grew in response to a 10-minute mental task performed daily over 15 weekdays (Karni et al., 1998). Such results should prompt educators to abandon the traditional fixed ideas of the brain and learning that currently fill schools—ideas that children are smart or dumb, quick or slow. If brains can change in three weeks, imagine what can happen in a year of math class if students are given the right math materials and they receive positive messages about their potential and ability.

 
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The new evidence from brain research tells us that everyone, with the right teaching and messages, can be successful in math, and everyone can achieve at the highest levels in school. There are a few children who have very particular special educational needs that make math learning difficult, but for the vast majority of children—about 95%—any levels of school math are within their reach. And the potential of the brain to grow and change is just as strong in children with special needs. Parents and teachers need to know this important information.

 
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I am often asked whether I am saying that everyone is born with the same brain. I am not. What I am saying is that any brain differences children are born with are nowhere near as important as the brain growth experiences they have throughout life. People hold very strong views that the way we are born determines our potential; they point to well-known people who were considered geniuses—such as Albert Einstein or Ludwig van Beethoven. But scientists now know that any brain differences present at birth are eclipsed by the learning experiences we have from birth onward (Wexler in Thompson, 2014). Every second of the day our brain synapses are firing, and students raised in stimulating environments with growth mindset messages are capable of anything. Brain differences can give some people a head start, but infinitesimally small numbers of people have the sort of head start that gives them advantages over time. And those people who are heralded as natural geniuses are the same people who often stress the hard work they have put in and the number of mistakes they made. Einstein, probably the most well known of those thought to be a genius, did not learn to read until he was nine and spoke often about his achievements coming from the number of mistakes he had made and the persistence he had shown. He tried hard, and when he made mistakes he tried harder. He approached work and life with the attitude of someone with a growth mindset. A lot of scientific evidence suggests that the difference between those who succeed and those who don't is not the brains they were born with, but their approach to life, the messages they receive about their potential, and the opportunities they have to learn. The very best opportunities to learn come about when students believe in themselves. For far too many students in school, their learning is hampered by the messages they have received about their own potential, making them believe they are not as good as others, that they don't have the potential of others. 

When students are given fixed praise—for example, being told they are smart when they do something well—they may feel good at first, but when they fail later (and everyone does) they think that means they are not so smart after all.

 
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The impact of the praise students receive can be so strong that it affects their behavior immediately. In one of Carol's studies, researchers asked 400 fifth graders to take an easy short test, on which almost all performed well. Half the children were then praised for “being really smart.” The other half were complimented on “having worked really hard.” The children were then asked to take a second test and choose between one that was pretty simple, that they would do well on, or one that was more challenging, that they might make mistakes on. Ninety percent of those who were praised for their effort chose the harder test. Of those praised for being smart, the majority chose the easy test (Mueller & Dweck, 1998).

Praise feels good, but when people are praised for who they are as a person (“You are so smart”) rather than what they did (“That is an amazing piece of work”), they get the idea that they have a fixed amount of ability. Telling students they are smart sets them up for problems later. As students go through school and life, failing at many tasks—which, again, is perfectly natural—they evaluate themselves, deciding how smart or not smart this means they really are. Instead of praising students for being smart, or any other personal attribute, it's better to say things like: “It is great that you have learned that,” and “You have thought really deeply about this.”

 

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There is no preordained pace at which students need to learn mathematics, meaning it is not true that if they have not attained a certain age or emotional maturity they cannot learn some mathematics. Students may be unready for some mathematics because they still need to learn some foundational, prerequisite mathematics they have not yet learned, but not because their brain cannot develop the connections because of their age or maturity. When students need new connections, they can learn them.

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

When I taught my online class, and I read all the responses from the people who took it, I realized more strongly than ever before that many people have been traumatized by math. Not only did I find out how widespread the trauma is, but the evidence I collected showed that the trauma is fuelled by incorrect beliefs about mathematics and intelligence. Math trauma and math anxiety is kept alive within people because these incorrect beliefs are so widespread that they permeate society in the United States, the United Kingdom, and many other countries in the world.

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We all know that math trauma exists and is debilitating for people; numerous books have been devoted to the subject of math anxiety and ways to help people overcome it (Tobias, 1978). It would be hard to overstate the number of people who walk on our planet who have been harmed by bad math teaching, but the negative ideas that prevail about math do not come only from harmful teaching practices. They come from one idea, which is very strong, permeates many societies, and is at the root of math failure and underachievement: that only some people can be good at math. That single belief—that math is a “gift” that some people have and others don't—is responsible for much of the widespread math failure in the world.

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The idea that people who can do math are the smartest or cleverest people. This makes math failure particularly crushing for students, as they interpret it as meaning that they are not smart. We need to dispel this myth. The combined weight of all the different wrong ideas about math that prevail in society is devastating for many children—they believe that mathematics ability is a sign of intelligence and that math is a gift, and if they don't have that gift then they are not only bad at math but they are unintelligent and unlikely to ever do well in life.