Learning is a process of identity formation

 

The Swiss scholar Etienne Wenger created an important framework to help people think about learning differently. He states that when we learn something, it is more than just acquiring knowledge or accumulating facts and information, because learning changes us as people. When we learn new ideas, we see the world differently—we have a different way of thinking and a different way of interpreting every event in our lives. As Wenger says, learning is a process of identity formation. Psychologists used to see identity as a static concept, maintaining that we all have “an” identity that we develop as children and keep throughout our lives. But more recent work has given identity a more fluid meaning, suggesting that we can all have different identities in different parts of our lives. You might, for example, present yourself differently as a member of a sports team than you do in your job or in a family role. I wrote this book because I know that when we learn about brain growth, mindset, and multidimensional deep and collaborative thinking, it unlocks aspects of our true selves. These ideas do not turn us into different people, but they can set free what was in us already, what was always possible for us but in many cases not being realized.

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When we give up on something and decide we cannot do it, it is rarely because of actual limits; instead, it is because we have decided we cannot do it. We are all susceptible to this negative and fixed thinking, but we become particularly susceptible to it when we age and start to feel that we are not as physically or mentally strong as we once were.

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I am often asked by teachers what they can do with their “unmotivated” students. It is my firm belief that all students want to learn, and they only act unmotivated because someone, at some time in their lives, has given them the idea that they cannot be successful. Once students let go of these damaging ideas and someone opens a learning pathway for them, the lack of motivation goes away.

Collaboration in learning

 

Collaboration is vital for learning, for college success, for brain development, and for creating equitable outcomes. Beyond all of this, it is beneficial to establish interpersonal connections, especially in times of conflict and need.

Victor and Mildred Goertzel studied seven hundred people who had made huge contributions to society, choosing those who had been the subject of at least two biographies, people such as Marie Curie and Henry Ford. They found, incredibly, that less than 15 percent of the famous men and women had been raised in supportive families; 75 percent had grown up in families with severe problems such as “poverty, abuse, absent parents, alcoholism, serious illness,” and other major issues. Their study was conducted in the 1960s. Clinical psychologist Meg Jay, in her interesting Wall Street Journal article on resilience, reports that similar results would be found today and cites Oprah Winfrey, Howard Schultz, and LeBron James as examples of people who grew up in extreme hardship.

Jay has studied resilience over many years and points out that people who survive hardship often do better, but not through “bouncing back,” as some think, because the recovery process takes time and is more of a battle than a bounce. She also points out those who ultimately benefit from hardship, becoming stronger and resilient, do so when they maintain self-belief, when they “own the fighter within,” and when they connect with other people. The thing that people who overcome hardship and do not become defeated by it have in common is that in times of need they all reached out to someone—a friend, a family member, or a colleague—and those connections helped them survive and develop strength.

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Shane was beginning high school when he hit an all-time low. He had just started at a typical large US high school with high expectations for his experience there, but within weeks he said it was the “loneliest experience he had ever felt.” Shane, in a powerful video that has had tens of thousands of views, describes feeling like an outsider, someone who didn’t belong. It was this deep-seated feeling of emptiness that led Shane to an appointment with his guidance counselor. Shane agreed to this because he thought it might result in his transferring to another school.

Instead, he walked out of the meeting with the recommendation to join five different clubs at the school. Shane was skeptical at first, but he joined the clubs and started to notice some things shifting. He had people to say hello to in the corridors, and the more he got involved in school life, the more he felt he was part of the school community.

Shane discovered that the more he did, the better he felt about himself; the more involved he got, the more “connected, driven, and motivated” he felt. He now reflects that he felt like an outsider because he was one; the only thing that changed was that he put himself on the inside—and that shifted everything. This was such a powerful change that Shane became inspired to share his experience with others and to start what is now a global movement—to help young people to become more personally connected with others.

Shane initially had an idea to hold an assembly at his school to help other students know what can happen when they connect with others and then match them with clubs of interest to them. They expected about fifty students, but word spread and four hundred students from seven different schools attended the assembly. The following year that number grew to a thousand, and the numbers have continued to grow every year. Shane started the movement “Count Me In,” which has now impacted more than ten million people, with speaking programs that have reached students in over one hundred countries. When I interviewed Shane for this book, he highlighted the challenges today’s young people face in forming meaningful connections:

Teenagers today have it harder than any other generation by far, in my educated opinion. Not only are they dealing with all the same issues we’ve seen for generations, but also things like peer pressure, bullying, social isolation that can really be harmful on your upbringing and the trajectory of your life. These are now 24/7 issues for every kid because of technology and smartphones and how much they are plugged in online, yet unplugged in reality and in community. Those community connections, I think, are the key in forging something—so we can see the world just differently enough that we begin to feel a greater sense of self-acceptance and belonging.

He makes a crucial point, and his movement to create greater connections among young people serves a much-needed purpose, as he stated in our interview:

The more you get involved, the more you immerse yourself in the community, the more connected you feel, and the more different you see things, the stronger that lens becomes, and the softer things become. The real defining moment for me that I can pinpoint is when I started living from this place of: My life is bigger than this moment, and it doesn’t matter what’s going on, how dark or desperate I feel inside. I know for a fact, with absolute conviction and certainty, that my life is bigger than this moment, than any one moment.

Shane’s movement has been particularly helpful for young people who feel isolated, who are going through difficult times at home, or who are facing any of the myriad issues that impact young people. He reflected that the main response that differentiates those who change positively from those who don’t is their perspective, or their mindset. Shane’s movement is also a helpful reminder that even—or perhaps particularly—in a world of online connectivity, genuine human connections are something that everybody needs and that changes people’s lives. Shane found that they helped young people know that their lives are bigger than the moments they are in now and that no matter how hard a situation may be, connections with people bring you out of it.

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Everybody has a different way of approaching things and you can always learn and grow.” She told me that these ideas had caused students to be less egocentric. In their interactions now, instead of insisting on their way of thinking or working and closing down because others have a different idea, students think: “Oh, you know what? This is how I’m thinking about it, but I know others think about it another way.” This acceptance of different ways of thinking has led to greater tolerance and appreciation of each other. As Holly reflected:

They know that other people have good ideas too and they also know that they should open their mind to hearing other people’s solutions, because that might be a new idea for them that they hadn’t yet thought of. And so that mindset of, “Hey, maybe your idea is something that I could add to my idea” is a huge one for kids.

Many reformers in education who work to change student experiences in classrooms work on content, finding new ways to approach topics, often with cool technology. But imagine what students’ learning and lives outside school would be like if they learned to collaborate with others more productively, going into conversations with an openness to hear and understand what others have to say. This would change classroom dynamics as well as many other aspects of students’ lives.

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Prior to learning about the value of struggle and brain growth, Jenny “felt like an island.” She described to me a mindset that I am sure many share—of feeling that she had to be an expert when interacting with others, of being fearful of revealing a lack of knowledge. And as a teacher in a classroom, she felt she had to be the one who knew everything. But Jenny’s perspective has changed, and she now embraces uncertainty and opens up more to her community of colleagues. Part of this change has involved letting go of the idea that she is being judged. Jenny described her new perspective:

Being willing to feel uncomfortable with not knowing something and still know that I don’t have to give up on something just because I don’t understand it right away. And I have other resources that I can utilize to increase my learning as an educator, as a person. So for me, it’s just . . . I always felt like I was an island and I had to show up knowing. . . . I think for me, it’s changed the way I navigate life in terms of I listen better, I think. I feel like I grow and learn by collaborating, so I think I’ve opened up a different way of connecting to my community of colleagues so that I can learn better, and sharing is really learning. That whole idea of letting go of judgment and knowing your worth changed me as a person.

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This new approach—of embracing uncertainty instead of pretending to know everything, of looking for resources to learn more—seems to enhance people’s connections with each other as well as people’s way of being in the world.

Approaching content with uncertainty and vulnerability is a trait I also recommend to teachers I work with. When students see their teacher present correct content all the time, always knowing the answer to any student question, always being right, never making mistakes, and never struggling, it creates a false image of what it means to be a good learner, in any subject. Teachers should embrace uncertainty and be open about not knowing something or making a mistake.

If you are a teacher, share these times with students so that they know such times are an important part of having expertise. When I teach my undergraduates at Stanford, I give them open mathematics problems to explore. They take them in all sorts of directions, some of which are new to me. I embrace these moments and admit that I do not know, saying, “How interesting. I have not seen that before. Let’s explore it together.”

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Mathematics is often depicted as the most solitary of subjects, but it is a discipline, like all others, that has been built through connections between ideas. New ideas and directions come from people reasoning with each other, setting out ideas, and considering the ways they are connected to each other. Parents, particularly of high-achieving students, often say to me, “My child can work out the answers correctly. Why should she have to explain them?” But such parents are missing an important point—mathematics is all about communication and reasoning.

Conrad Wolfram, well known for his work with Wolfram Alpha, the online computational knowledge site, and director of Wolfram Research, told me that people who are unable to communicate their mathematical thinking and ideas are of no use to him as employees, because they cannot take part in team problem solving. In team problem solving, when people communicate their thinking, others can connect with their ideas. Critical evaluation by many minds also guards against incorrect or irrelevant ideas. When people cannot communicate an idea or come up with the reasoning that led them to it, they are not particularly useful in a team of problem solvers. I am sure this principle is true of all areas—people who can explain and communicate their ideas to others, whether in math, science, art, history, or any other area, are more effective problem solvers and are able to make a larger contribution to the work in companies and other groups.

The six keys that have been presented play an important role in changing people’s communication and, consequently, create a multitude of life opportunities. Many people are too locked up to be good communicators. They are fearful of saying something wrong, and they worry that what they say is an indication of their worth, that they are being judged by others. When people learn about mindset, brain growth, multidimensionality, and struggle, it often unlocks them, gives them a limitless perspective, and enables them to let go of the fear of being judged. Instead, they embrace openness and uncertainty and become more willing to share ideas, which, in collaboration with others, grow into solution pathways. Such collaborations enhance people’s lives, and the very best collaborations, it seems, start with a limitless approach to people and to ideas.

The Thinking of "Trailblazers"

 

Adam Grant has written a book called Originals: How Non-Conformists Move the World in which he argues that we have long valued the rule-following, memorizing students. He notes that the students in the US often regarded as “prodigies”—the ones who “learn to read at age two, play Bach at four, breeze through calculus at six”—rarely go on to change the world. When scholars study the most influential people in history, they are rarely those regarded as “gifted” or “geniuses” in their childhood. Instead, the people who excel in school often “apply their extraordinary abilities in ordinary ways, mastering their jobs without questioning defaults, and without making waves.” Grant concludes: “Although we rely on them to keep the world running smoothly, they keep us running on a treadmill.” Those who do go on to change the world are creative and flexible thinkers, people who think outside rather than inside the box.

Many people know that creative and flexible thinking is valuable, but they do not associate it with mathematics. Instead, they see math time as an area in which to follow rules and be compliant. But when we combine mathematics with creativity, openness, and out-of-the-box thinking, it is wonderfully liberating. This is something everybody deserves to know about and experience, and when they do, they do not look back.

The advantages of deep and flexible thinking apply to all subjects and avenues of life. We do not know what problems people will need to solve in the future, but they are likely to be problems we have never even dreamed of. Filling our minds with content that we can reproduce at speed is unlikely to help us solve the problems of the future; instead, training our minds to think deeply, creatively, and flexibly seems far more useful. The thinking of “trailblazers” whose brains were studied was found to be more flexible than that of regular people. They had learned to approach problems in different ways and not just rely on a memory. Speed and fixed approaches will only take us so far. In the education world and beyond, we must all challenge the assumptions about the benefits of speed and memorization and instead focus on flexible and creative learning. This will help us unlock our own and others’ potential as learners.

"Mathematics is amazingly compressible"

 

When we learn new knowledge, it takes up a large space in the brain—it literally occupies more room—as the brain works out what it means and where it connects with other ideas already learned. But as time goes on, the concepts we have learned are compressed into a smaller space. The ideas are still there so that when we need them, we can quickly and easily “pull” them from our brain and use them; they just take up less space. If I were to teach arithmetic to kindergarten students, the concepts would take up a large space in their brains. But if I asked adults to add 3 and 2, they would quickly do so, pulling the answer from their compressed knowledge of addition. William Thurston, a mathematician who won the Fields Medal, described compression in this way:

Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics.

You may be thinking that few students describe math as a “real joy,” and part of the reason is that we can only compress concepts. So when students are engaging in mathematics conceptually—looking at ideas from different perspectives and using numbers flexibly—they are developing a conceptual understanding, creating concepts that can be compressed in the brain. When students believe that mathematics is about memorization, they are not developing a conceptual understanding or forming concepts that can then be compressed. Instead of compressed concepts in the brain, their math knowledge is more like a ladder of memorized methods that stack one on top of another, stretching, as it may seem to these learners, to the sky. 

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.

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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. 

Multidimensional Approach in Math Education

 

 

Carol Dweck herself has written that the information on the value of changing mindsets needs to be accompanied by a different approach to teaching, one that enables students to learn differently. One of the things that she says keeps her up at night is when students are told to put in effort and that success is all about hard work, without their being given the tools by teachers to learn more effectively. As she says, “Effort is key for students’ achievement, but it is not the only thing. Students need to try new strategies and seek input from others when they are stuck.”

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Fluid and flexible brains, neuroscientists conclude, come from the synchrony that occurs when multiple brain areas are working together. Communication between brain areas comes about when we approach knowledge through multiple avenues, encountering ideas in different forms and representations.

A multidimensional approach can be used in the teaching of all subjects to bring about higher engagement and achievement. Many subject areas, particularly in the humanities, already value treating the subject in multiple ways by asking students to give their own interpretations of texts they read and employing such forms as group discussions, debates, and plays. In most cases they could still become more multidimensional, but they are rarely as narrowly taught as some other subjects. In my experience the subjects that seem most in need of change are mathematics, science, and language teaching. Coming at the subject matter from multiple angles is an ideal learning approach for all of these disciplines.

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Working in these multiple ways encourages brain communication while also bringing the content to life. The vast majority of students think about math as a set of numbers and methods and about English as books and words. When we approach math, English, science, or other subjects as opportunities for creativity and seeing things in multiple ways, it changes everything, stimulating vital brain growth and neural connections. Additionally, as teachers diversify the curriculum, moving from a simple list of numerical answers, pages of text, or scientific equations to visuals, models, words, videos, music, data, and drawings, the classroom changes from a place where all the work looks the same to one where the variety is enticing and creativity can be celebrated.

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This new openness to challenges and uncertainty seems to be a common reaction to becoming unlocked—people realize that it is good to struggle, that it is not a sign of a brain weakness, but of brain growth. This leads to more confidence in times of struggle and a willingness to share ideas that they are unsure of. One of the saddest, most central characteristics of fixed-brain thinking is the fear of being wrong. People’s minds are literally locked, immobilized, by their fear, which is why an approach to life that values multidimensionality, growth, and struggle is so liberating. Holly said: “I have so many more ideas because I let myself have ideas.”

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Another core benefit of working and living with a multidimensional approach is that when roadblocks appear, you know there are alternate routes. Many of the adults I interviewed for this book said they would no longer stop when they met challenges or roadblocks; they simply would find another strategy, another approach. A multidimensional approach to knowledge reveals that there isn’t only one way to do anything; there are always multiple ways forward.

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In many classrooms students are given problems they do not know how to access—which causes them to think negatively about themselves and their learning. When problems are changed to become “low floor and high ceiling”—problems that are accessible by all but lead to more challenging work—everybody can access them and take them to different places.

You do not have to live your life as an “expert”

 

Part of the process of change and of becoming limitless involves letting go of the idea that your past failures came about because there was something wrong with you. A similarly important change is realizing that you do not have to live your life as an “expert,” that you can go into situations and proudly share uncertainty. Jesse Melgares told me about these two aspects of the change he went through as he became unlocked. Jesse is an assistant principal in east LA, but in earlier years he taught mathematics and was, as he said, “extremely self-conscious,” thinking he did not know enough and nothing could change. When Jesse became an assistant principal, he needed to coach math teachers, but he was fearful that others would find out he was a fraud:

To be honest I would get a lot of paralyzing stress when someone asked me a math-related question. . . . It was terrible. It was like a boot on my chest. It’s what I woke up with in the morning, wondering, “Am I gonna be asked something that I don’t know the answer to? And will I be discovered as some sort of fraud?”

The feeling of paralyzing stress Jesse described, the fear of being asked something he could not answer, is a feeling shared by millions of people in different situations and jobs, and it is a feeling that I hope this book can change. For Jesse the change began when he took one of my online courses and realized: “Everything that I had been taught as a student of math when I was in the K–12 system and as a math educator was wrong.”

For Jesse, the first step in becoming unlocked was realizing that any trouble he had had learning in the past was not due to some deficit in him, but to the faulty system in place. This is a shift I have seen others make, and it is vital for those who have had bad learning experiences.

Jesse not only started feeling better about mathematics; he started a new “journey” discovering that mathematics was his passion. He shifted from feeling defeatist about math to seeing it as an exciting challenge. Jesse is now the director of mathematics for twenty-five schools—quite a change for a person who used to feel paralyzing stress when he thought about math. New knowledge about the brain allowed him to shift his perspective, his mindset, and his belief in himself. Jesse still meets questions he cannot answer, but instead of being afraid, he thinks: “Well, I don’t know what the answers are but, you know, we’ll figure it out. This is a challenge.” This shift in perspective is typical for people who have become unlocked. When people change their mindset and become aware of the positive benefits of struggle, they take a new and much more positive approach to challenge and uncertainty. They let go of the need to be the expert and replace it with curiosity and the desire to collaborate.

The Power of Yet

 

When students become disillusioned because others are ahead of them or complain that they don’t understand something, a word that Carol Dweck champions using with them is “yet.” When I ask adults to visually represent an idea, I often hear them say, “I am terrible at drawing.” I tell them, “You mean you have not learned to draw well yet.” This may seem like a small linguistic change, but it is an important one. It moves the focus from the perceived personal lack to the process of learning.

Beliefs vs Brains

 

In order to study the impact of our beliefs on our health, Stanford researchers Alia Crum and Octavia Zahrt collected data from 61,141 people over an extensive time span, twenty-one years. The researchers found that those people who thought they were doing more exercise were actually healthier than those who thought they were doing less, even when the amount of exercise they were doing was the same.

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If you enter a challenging situation believing in yourself, but then mess up, your brain will react more positively than if you go into a situation thinking, “I don’t think I can do this.” If we have a difficult job or a problematic situation at home, this result should prompt us to go into those situations believing in ourselves. If we enter difficult situations with positive beliefs, our brains will become more resilient and adaptative when we make errors than if we are doubting ourselves. This change in belief alters the physical structures of the brain and creates avenues for higher-level thinking and creative problem solving. Just as those who believed they were engaging in healthy exercise became healthier, those who believe they are learning more productively actually learn more.

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One of Dweck’s studies revealed the immediate impact of the word “smart.” Two groups of students were given a challenging task. On completion, one group was praised for being “really smart,” and the other was praised for working hard. Both groups were then offered a choice between two follow-up tasks, one that was described as easy and one that was described as challenging. Ninety percent of the students praised for working hard chose the harder task, whereas the majority of the students praised for being “smart” chose the easy task. When students are praised for being smart, they want to keep the label; they choose an easy follow-up task, so they can continue to look “smart.”

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It is just as important to take on ideas about social comparison with students as it is to make them aware of the value of struggle. I have had many conversations with learners of all ages who argue that brains must be fixed, because some people appear to get ideas faster and to be naturally “gifted” at certain subject areas. What they do not realize is that brains are growing and changing every day. Every moment is an opportunity for brain growth and development. Some have simply developed stronger pathways on a different time line. It is critical that students understand that they too can develop those pathways at any time—they can catch up with other students if they take the right approach to learning.

Making mistakes are the best times for brain growth

Most of us have grown up with the idea that mistakes are bad, especially if we attended test-driven schools, where we were frequently marked down for making mistakes, or our parents punished mistakes with harsh words and actions. This is unfortunate, and this is why. 

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... [In a workshop, Carol Dweck] announced that every time we make mistakes, synapses fire in the brain, indicating brain growth. All the teachers in the room were shocked, as they had all been working under the premise that mistakes are to be avoided. Carol was drawing from work that has researched the brain’s response when we make mistakes, particularly investigating the different ways brains respond when people have a growth or a fixed mindset.

Jason Moser and his colleagues extended Carol’s work investigating the brain’s response when we make mistakes. Moser and his team found something stunning. They had asked participants to take tests while they monitored the participants’ brains with MRI technology. They looked at the scans when people got questions correct and when they got them incorrect. The researchers found that when people made mistakes, brains were more active, producing strengthening and growth, than when people got work correct. Neuroscientists now agree that mistakes positively contribute to the strengthening of neural pathways.

This learning key is particularly significant because most teachers design classes so that everyone is successful. Curricula and textbooks are designed with trivial, unchallenging questions, so that students will get a high percentage of answers correct. The common belief is that getting most answers correct will motivate students toward greater 

success. Here’s the problem, though. Getting questions right is not a good brain exercise.

For students to experience growth, they need to be working on questions that challenge them, questions that are at the edge of their understanding. And they need to be working on them in an environment that encourages mistakes and makes students aware of the benefits of mistakes. This point is critical. Not only should the work be challenging to foster mistakes; the environment must also be encouraging, so that the students do not experience challenge or struggle as a deterrent. Both components need to work together. 

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One of the significant characteristics of the highly effective learning described is the presence of mistakes and the role of struggle and error in transforming people from beginners into experts. This is consistent with the brain research showing increased brain activity when people struggle and make mistakes and decreased activity when they get work correct. Unfortunately, most learners think they should always be getting work correct, and many feel that if they make mistakes or struggle, they are not good learners—when this is the very best thing they can be doing.

Practice is important for the development of any knowledge or skill. Anders Ericsson helped the world understand the nature of expert performance and found that most world-class experts—pianists, chess players, novelists, athletes—practiced for around ten thousand hours over twenty years. He also found that their success was not related to tests of intelligence but to the amount of “deliberate practice” they undertook. Importantly, although people succeed because they are trying hard, the people who become experts are trying hard in the right way. A range of different researchers describe effective practice in the same way—people pushing at the edge of their understanding, making mistakes, correcting them, and making more.

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Japan has always scored well in mathematics—it has always finished in one of the top-five TIMSS positions—and was one of the countries visited in the study. The researchers found that Japanese students spent 44 percent of their time “inventing, thinking, and struggling with underlying concepts,” whereas students in the US engaged in this kind of behavior less than 1 percent of the time.

Jim Stigler, one of the authors of the study, writes that the Japanese teachers want the students to struggle—and recalls the times when they would purposely give the wrong answer so that students would go back and work with foundational concepts. In my thousands of observations of classrooms over many years in the US and the UK, I have never seen this kind of practice; more typically I have seen teachers who seem to want to save students from struggle. Many times I have observed students asking for help and teachers structuring the work for students, breaking down questions and converting them into small easy steps. In doing so they empty the work of challenge and opportunities for struggle. Students complete the work and feel good, but often learn little.

I saw a very similar teaching approach, focused on struggle, in a visit to classrooms in China, another country that scores highly in mathematics. I had been asked to visit China to give a talk at a conference and managed, as I like to do, to sneak away and visit some classrooms. In a number of high-school math classrooms, lessons were approximately one hour long, but at no time did I see students working on more than three questions in one hour. This contrasts strongly with a typical US high-school math classroom, where students chug through about thirty questions in an hour—about ten times more. The questions worked on in Chinese classrooms were deeper and more involved than the ones in US classrooms. Teachers would ask provocative questions, deliberately making incorrect statements that students would be challenged to argue against.

One of the lessons I watched was on a topic that is often uninspiring in US classrooms—complementary and supplementary angles. The teacher in China asked the students to define a complementary angle, and the students gave their own ideas for a definition. Often the teacher would push the students’ definition to a place that made it incorrect and playfully ask, “Is this right, then?” The students would groan and try to make the definition more correct. The teacher bantered with the students, playfully extending and sometimes twisting their ideas to push the students to deeper thinking. The students probed, extended, clarified, and justified for a long time, reaching depths that were impressive.

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"I am going to figure this out if it kills me."

When I talk with teachers, they often say this sort of persistence is missing in the students they teach. One of the most common complaints I hear from teachers is that students don’t want to struggle; they want to be told what to do. To the teachers it seems as though students just can’t be bothered with struggling, which is probably what it looks like. The truth is, however, that when students don’t want to struggle, it is because they have a fixed mindset; at some point in their lives they have been given the idea that they cannot be successful and that struggle is an indication that they are not doing well.

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As an academic, I experience a lot of failure. To keep our youcubed center at Stanford running, supporting staff salaries and providing free materials for teachers and parents, we have to apply for lots of grants—most of which are rejected. I also have to submit our papers to journals, where rejection is part of the process. If they are not rejected, they are subject to reviewers’ comments. I have had reviewers dismiss my work entirely, saying that it is “not research, just a story.” It is nearly impossible to keep going as an academic without viewing “failure” as an opportunity to improve. A wise professor named Paul Black, my PhD advisor, once said to me: “Whenever you send a paper to a journal, have in mind the next journal you will send it to when the paper is rejected.” I have used his advice a number of times.

Taking a limitless approach—particularly when embracing challenge and struggle—also helps when we encounter difficult people. In today’s world of social media, it seems impossible to make a statement about anything without getting pushback, some of it aggressive. I have experienced extreme and aggressive pushback many times, and I now know that it is important to stay strong in those moments and to look for something positive. Instead of dismissing a challenge or beating yourself up, think, “I will take something from this situation and use it to improve.”

Your brain is constantly reorganizing, growing, and changing

The first step in living a limitless, unlocked life is to know brains are constantly reorganizing, growing, and changing. Remembering that every day of our lives, we wake up with a changed brain. In every moment of our lives our brains have opportunities to make connections, to strengthen pathways, and to form new pathways. When we face a challenging situation, rather than turn away because of fear of not being good enough, we should dive in, knowing that the situation presents opportunities for brain growth. As we start to recognize the huge implications of the adaptability of our brains, we will start to open our minds, and live differently.

Giftedness??

 

 

The idea of giftedness is not only inaccurate and damaging; it is gender and racially biased. We have many different forms of evidence showing that those who believe in fixed brains and giftedness also believe that boys, men, and certain racial groups are gifted and girls, women, and other racial groups are not.

One of the forms of evidence that shows this clearly was collected by Seth Stephens-Davidowitz, who focused his attention on google searches. His study revealed something very interesting and disturbing. He found that the most commonly googled word following “Is my two-year-old son . . .” is “gifted.” He also found that parents search the words “Is my son gifted?” two and a half times more than the words “Is my daughter gifted?” This is despite the fact that young children of different genders have equal potential.

Sadly, the problem is not limited to parents. Daniel Storage and his colleagues conducted analyses of anonymous reviews on RateMyProfessors.com, and they found that students were twice as likely to call male rather than female professors “brilliant” and three times as likely to call male rather than female professors “geniuses.”20 These and other studies show that ideas of giftedness and genius are intertwined with racist and sexist assumptions.

I am convinced that the majority of people who have gender or racial biases do not think about them consciously or perhaps even realize they have them. I also contend that if we were to dispel the idea that some people are “naturally” gifted and instead recognize that everyone is on a growth journey and can achieve amazing things, some of the most insidious biases against women and people of color would disappear. This is needed in the STEM fields more than anywhere else; it is no coincidence that STEM subjects evidence the strongest fixed thinking and the starkest inequities in participation. 

Part of the reason so many students are dissuaded from thinking they are capable of learning math is the attitudes of the teachers and professors who teach them.