A Mind For Numbers

by Barbara Oakley, PhD

  We’re often told that empathy is universally beneficial, but it’s not.9 It’s important to learn to switch on an occasional cool dispassion that helps you to not only focus on what you are trying to learn, but also to tune people out if you discover their interests lie in undercutting you. Such undercutting is all-too-common, as people are often just as competitive as they are cooperative. When you’re a young person, mastering such dispassion can be difficult. We are naturally excited about what we are working on, and we like to believe that everyone can be reasoned with and that almost everyone is naturally good-hearted toward us.

  Like Cajal, you can take pride in aiming for success because of the very things that make other people say you can’t do it. Take pride in who you are, especially in the qualities that make you “different,” and use them as a secret talisman for success. Use your natural contrariness to defy the always-present prejudices from others about what you can do.

  NOW YOU TRY!

  Understanding the Value of “Bad”

  Pick a seemingly bad trait and describe how it could be beneficial in helping you learn or think creatively or independently. Could you think of a way to diminish the negative aspects of that trait, even as you enhance the positive aspects?

  SUMMING IT UP

  Learning on your own is one of the deepest, most effective ways to approach learning:

  It improves your ability to think independently.

  It can help you answer the strange questions that teachers sometimes throw at you on tests.

  In learning, persistence is often far more important than intelligence.

  Train yourself to occasionally reach out to people you admire. You can gain wise new mentors who, with a simple sentence, can change the course of your future. But use your teachers’ and mentors’ time sparingly.

  If you aren’t very fast at grasping the essentials of whatever you are studying, don’t despair. Surprisingly often, “slower” students are grappling with fundamentally important issues that quicker students miss. When you finally get what’s going on, you can get it at a deeper level.

  People are competitive as well as cooperative. There will always be those who criticize or attempt to undermine any effort or achievement you make. Learn to deal dispassionately with these issues.

  PAUSE AND RECALL

  Close the book and look away. What were the main ideas of this chapter? Which idea is most important—or are there several competing equally important ideas?

  ENHANCE YOUR LEARNING

  1. What are the advantages and disadvantages of learning on your own, without being guided by a formal program of study?

  2. Look up the phrase List of autodidacts on Wikipedia. Which of the many autodidacts there would you most like to emulate? Why?

  3. Choose a person among your own acquaintances (that is, not a celebrity) whom you admire but to whom you have never really spoken. Formulate a plan to say hello and introduce yourself—then carry it out.

  NEW YORK TIMES SCIENCE WRITER NICHOLAS WADE ON AN INDEPENDENT MIND

  Nicholas Wade writes for the Science Times section of the New York Times. Always an independent thinker, Wade owes his very existence to the similar independent thinking of his grandfather—one of the few male survivors of the Titanic. When most men followed a rumor and moved to the port side, Wade’s grandfather followed his intuition and deliberately moved the other way, to starboard.

  Here, Nicholas gives his insight on what he thinks are the most interesting books about scientists and mathematicians.

  “The Man Who Knew Infinity: A Life of the Genius Ramanujan, by Robert Kanigel. This book tells the unbelievable, rags to intellectual riches story of the Indian mathematical genius Srinivasa Ramanujan and his friend British mathematician G. H. Hardy. My favorite episode is this:

  ‘Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, “rather a dull number,” adding that he hoped that wasn’t a bad omen.

  ‘“No, Hardy,” said Ramanujan. “It is a very interesting number. It is the smallest number expressible as the sum of two cubes in different ways.”‘

  “Noble Savages, by Napoleon Chagnon. This beautifully written adventure story gives a sense of what it’s like to learn to survive and thrive in an utterly alien culture. Chagnon was originally trained as an engineer. His scientific research has shifted our understanding of how cultures develop.

  “Men of Mathematics, by E. T. Bell. This is an old classic that’s a show-stopping read for anyone who’s interested in how fascinating people think. Who could forget brilliant, doomed Évariste Galois who spent the night before he knew he was to die ‘feverishly dashing off his last will and testament, writing against time to glean a few of the great things in his teeming mind before the death which he foresaw could overtake him. Time after time he broke off to scribble in the margin “I have not time; I have not time,” and passed on to the next frantically scrawled outline.’ Truth be told, this is one of the few exciting stories that Professor Bell perhaps exaggerated, although Galois unquestionably spent that last evening putting the final polish on his life’s work. But this brilliant book has inspired generations of both men and women.”

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  avoiding overconfidence

  The Power of Teamwork

  Fred had a problem. He couldn’t move his left hand. This wasn’t surprising. While singing in the shower, Fred had suffered a nearly lethal right-hemisphere ischemic stroke a month before. The brain’s right hemisphere controls the left side of the body, which was why Fred’s left hand was now lifeless.

  Fred’s real problem, though, was worse. Even though he couldn’t move his left hand, Fred insisted—and truly believed—that he could. Sometimes he would excuse the lack of motion by saying he was just too tired to lift a finger. Or he’d insist that his left hand had moved. It was just that people hadn’t been watching. Fred would even covertly move his left hand with his right, and then loudly proclaim that his hand had moved on its own.

  Fortunately, as the months went by, Fred’s left hand gradually regained its function. Fred laughed with his doctor about how he’d tricked himself into believing that he could move his hand in the weeks immediately following the stroke; he spoke cheerfully about returning to his work as an accountant.

  But there were signs that Fred wasn’t returning to business as usual. He used to be a caring, considerate guy, but the new Fred was dogmatic and self-righteous.

  There were other changes. Fred used to be a keen practical joker, but now he just nodded along without understanding the punch lines to others’ jokes. Fred’s skill at investing also evaporated, and his cautiousness was replaced by naive optimism and overconfidence.

  Even worse, Fred seemed to have become emotionally tone-deaf. He tried to sell his wife’s car without asking her permission and was surprised when she became upset. When their beloved old family dog died, Fred sat placidly eating popcorn, watching his wife and children cry as if it were a scene out of a movie.

  What made these changes more difficult to understand was that Fred seemed to have retained his intelligence—even his formidable way with numbers. He could still quickly work up a business profit-and-loss statement and solve complex algebra problems. One interesting anomaly, however, was that if Fred made a mistake in his calculations, concluding something nonsensical, such as that a hot dog stand had a loss of nearly a billion dollars, it didn’t bother him. There was no big-picture “click” that said, “Wait a minute, that answer doesn’t make sense.”

  It turns out that Fred is a typical victim of “broad-perspective perceptual disorder of the right hemisphere.”1 Fred’s stroke had incapacitated broad areas of the right hemisphere of his brain. He could still fun
ction, but only partially.

  Although we need to be careful about faulty and superficial “left brain/right brain” assumptions, we also don’t want to throw the baby out with the bathwater and ignore worthwhile research that gives intriguing hints about hemispheric differences.2 Fred reminds us of the dangers of not using our full cognitive abilities, which involve many areas of our brain. Not using some of our abilities isn’t as devastating for us as it is for Fred. But even subtle avoidance of some of our capabilities can have a surprisingly negative impact on our work.

  The arrow on this CT scan of the brain points toward the shadowed damage caused by a right-hemisphere ischemic stroke.

  Avoiding Overconfidence

  There’s a great deal of evidence from research that the right hemisphere helps us step back and put our work into big-picture perspective.3 People with damage to the right hemisphere are often unable to gain “aha!” insights. That’s why Fred wasn’t able to catch the punch lines of jokes. The right hemisphere, it turns out, is vitally important in getting onto the right track and doing “reality checks.”4

  In some sense, when you whiz through a homework or test problem and don’t go back to check your work, you are acting a little like a person who is refusing to use parts of your brain. You’re not stopping to take a mental breath and then revisit what you’ve done with the bigger picture in mind, to see whether it makes sense.5 As leading neuroscientist V. S. Ramachandran has noted, the right hemisphere serves as a sort of “‘Devil’s Advocate,’ to question the status quo and look for global inconsistencies,” while “the left hemisphere always tries to cling tenaciously to the way things were.”6 This echoes the pioneering work of psychologist Michael Gazzaniga, who posited that the left hemisphere interprets the world for us—and will go to great lengths to keep those interpretations unchanging.7

  When you work in focused mode, it is easy to make minor mistakes in your assumptions or calculations. If you go off track early on, it doesn’t matter if the rest of your work is correct—your answer is still wrong. Sometimes it’s even laughably wrong—the equivalent of calculating a circumference of the earth that is only 21/2 feet around. Yet these nonsensical results just don’t matter to you, because the more left-centered focused mode has associated with it a desire to cling to what you’ve done.

  That’s the problem with the focused, left-hemisphere-leaning mode of analysis. It provides for an analytical and upbeat approach. But abundant research evidence suggests that there is a potential for rigidity, dogmatism, and egocentricity.

  When you are absolutely certain that what you’ve done on a homework or test is fine—thank you very much—be aware that this feeling may be based on overly confident perspectives arising in part from the left hemisphere. When you step back and recheck, you are allowing for more interaction between hemispheres—taking advantage of the special perspectives and abilities of each.

  People who haven’t felt comfortable with math often fall into the trap of “equation sheet bingo.” They desperately try to find a pattern in what the teacher or book did and fit their equations to that pattern. Good learners vet their work to ensure that it makes sense. They ask themselves what the equations mean and where they come from.

  “The first principle is that you must not fool yourself—and you are the easiest person to fool.”8

  —Physicist Richard Feynman, advising how to avoid pseudo-science that masquerades as science

  The Value of Brainstorming with Others

  Niels Bohr was heavily involved in the Manhattan Project—the U.S. race during World War II to build the nuclear bomb before the Nazis. He was also one of the greatest physicists who ever lived—which ultimately made it difficult for him to think intelligently about physics.

  Bohr was so respected as the genius who had intuited quantum theory that his thinking was considered unassailable. This meant that he could no longer brainstorm with others. No matter what cockamamie idea Bohr might propose, the other physicists working on the bomb would ooh and ahh over it as if it were something sacred.

  Bohr handled this challenge in an intriguing way.

  Richard Feynman, as it turned out, was good at not being intimidated by other people—at simply doing physics, no matter who he was with. He was so good that he became Bohr’s ace in the hole. Feynman was at that time just a youngster in the crowd of hundreds of prominent physicists at Los Alamos. But he was singled out by Bohr to do private brainstorming together before Bohr would meet with the other physicists. Why? Feynman was the only one who wasn’t intimidated by Bohr and who would tell Bohr that some of his ideas were foolish.9

  Niels Bohr lounging with Albert Einstein in 1925.

  As Bohr knew, brainstorming and working with others—as long as they know the area—can be helpful. It’s sometimes just not enough to use more of your own neural horsepower—both modes and hemispheres—to analyze your work. After all, everyone has blind spots. Your naively upbeat focused mode can still skip right over errors, especially if you’re the one who committed the original errors.10 Worse yet, sometimes you can blindly believe you’ve got everything nailed down intellectually, but you haven’t. (This is the kind of thing that can leave you in shock when you discover you’ve flunked a test you’d thought you aced.)

  By making it a point to do some of your studying with friends, you can more easily catch where your thinking has gone astray. Friends and teammates can serve as a sort of ever-questioning, larger-scale diffuse mode, outside your own brain, that can catch what you missed, or what you just can’t see. And of course, as mentioned earlier, explaining to friends helps build your own understanding.

  The importance of working with others doesn’t just relate to problem solving—it’s also important in career building. A single small tip from a teammate to take a course from the outstanding Professor Passionate, or to check out a new job opening, can make an extraordinary difference in how your life unfolds. One of the most-cited papers in sociology, “The Strength of Weak Ties,” by sociologist Mark Granovetter, describes how the number of acquaintances you have—not the number of good friends—predicts your access to the latest ideas as well as your success on the job market.11 Your good friends, after all, tend to run in the same social circles that you do. But acquaintances such as class teammates tend to run in different circles—meaning that your access to the “outside your brain” interpersonal diffuse mode is exponentially larger.

  Those you study with should have, at least on occasion, an aggressively critical edge to them. Research on creativity in teams has shown that nonjudgmental, agreeable interactions are less productive than sessions where criticism is accepted and even solicited as part of the game.12 If you or one of your study buddies thinks something is wrong in your understanding, it’s important to be able to plainly say so, and to hash out why it’s wrong without worrying about hurt feelings. Of course, you don’t want to go about gratuitously bashing other people, but too much concern for creating a “safe environment” for criticism actually kills the ability to think constructively and creatively, because you’re focusing on the other people rather than the material at hand. Like Feynman, you want to remember that criticism, whether you are giving or receiving it, isn’t really about you. It’s about what you are trying to understand. In a related vein, people often don’t realize that competition can be a good thing—competition is an intense form of collaboration that can help bring out people’s best.

  Brainstorming buddies, friends, and teammates can help in another way. You often don’t mind looking stupid in front of friends. But you don’t want to look too stupid—at least, not too often. Studying with others, then, can be a little bit like practicing in front of an audience. Research has shown that such public practice makes it easier for you to think on your feet and react well in stressful situations such as those you encounter when you take tests or give a presentation.13 There is yet another value to study buddies—t
his relates to when credible sources are in error. Inevitably, no matter how good they are, your instructor—or the book—will make a mistake. Friends can help validate and untangle the resulting confusion and prevent hours of following false leads as you try to find a way to explain something that’s flat-out wrong.

  But a final word of warning: study groups can be powerfully effective for learning in math, science, engineering, and technology. If study sessions turn into socializing occasions, however, all bets are off. Keep small talk to a minimum, get your group on track, and finish your work.14 If you find that your group meetings start five to fifteen minutes late, members haven’t read the material, and the conversation consistently veers off topic, find yourself another group.

  TEAMWORK FOR INTROVERTS

  “I’m an introvert and I don’t like working with people. But when I wasn’t doing so well in my college engineering classes (back in the 1980s), I decided that I needed a second pair of eyes, although I still didn’t want to work with anyone. Since we didn’t have online chatting back then, we wrote notes on each other’s doors in the dorms. My classmate Jeff and I had a system: I would write ‘1) 1.7 m/s’—meaning that the answer to homework problem one was 1.7 meters per second. Then I’d get back from a shower and see that Jeff had written, ‘No, 1) 11 m/s.’ I’d desperately go through my own work and find a mistake, but now I had 8.45 m/s. I’d go down to Jeff’s room and we’d argue intensively with both our solutions out while he had a guitar slung around his shoulder. Then we’d both go back to our own work on our own time and I’d suddenly see that the answer was 9.37 m/s, and so would he, and we’d both get 100 percent on the homework assignment. As you can see, there are ways to work with others that require only minimal interaction if you don’t like working in groups.”