A Mind For Numbers

by Barbara Oakley, PhD

  —Susan Sajna Hebert, Professor of Psychology, Lakehead University

  Final Thoughts on Testing

  The day before a test (or tests), have a quick look over the materials to brush up on them. You’ll need both your focused-mode and diffuse mode “muscles” the next day, so you don’t want to push your brain too hard. (You wouldn’t run a ten-mile race the day before running a marathon.) Don’t feel guilty if you can’t seem to get yourself to work too hard the day before a big examination. If you’ve prepared properly, this is a natural reaction: You are subconsciously pulling back to conserve mental energy.

  While taking a test, you should also remember how your mind can trick you into thinking what you’ve done is correct, even if it isn’t. This means that, whenever possible, you should blink, shift your attention, and then double-check your answers using a big-picture perspective, asking yourself, “Does this really make sense?” There is often more than one way to solve a problem, and checking your answers from a different perspective provides a golden opportunity for verifying what you’ve done.

  If there’s no other way to check except to step back through your logic, keep in mind that simple issues like missed minus signs, incorrectly added numbers, and “dropped atoms” have tripped up even the most advanced mathematics, science, and engineering students. Just do your best to catch them. In science classes, having your units of measurement match on each side of the equation can provide an important clue about whether you’ve done the problem correctly.

  The order in which you work tests is also important. Students generally work tests from front to back. When you are checking your work, if you start more toward the back and work toward the front, it sometimes seems to give your brain a fresher perspective that can allow you to more easily catch errors.

  Nothing is ever certain. Occasionally you can study hard and the test gods simply don’t cooperate. But if you prepare well by practicing and by building a strong mental library of problem-solving techniques, and approach test taking wisely, you will find that luck will increasingly be on your side.

  SUMMING IT UP

  Not getting enough sleep the night before a test can negate any other preparation you’ve done.

  Taking a test is serious business. Just as fighter pilots and doctors go through checklists, going through your own test preparation checklist can vastly improve your chances of success.

  Counterintuitive strategies such as the hard-start–jump-to-easy technique can give your brain a chance to reflect on harder challenges even as you’re focusing on other, more straightforward problems.

  The body puts out chemicals when it is under stress. How you interpret your body’s reaction to these chemicals makes all the difference. If you shift your thinking from “This test has made me afraid” to “This test has got me excited to do my best!” it helps improve your performance.

  If you are panicked on a test, momentarily turn your attention to your breathing. Relax your stomach, place your hand on it, and slowly draw a deep breath. Your hand should move outward, and your whole chest should expand like a barrel.

  Your mind can trick you into thinking that what you’ve done is correct, even if it isn’t. This means that, whenever possible, you should blink, shift your attention, and then double-check your answers using a big-picture perspective, asking yourself, “Does this really make sense?”

  PAUSE AND RECALL

  Close the book and look away. What were the main ideas of this chapter? What new ideas will be particularly important for you to try related to testing?

  ENHANCE YOUR LEARNING

  1. What is the one extraordinarily important preparation step for taking a test? (Hint: If you don’t take this step, nothing else you do to prepare for the test matters.)

  2. Explain how you would determine whether it is time to pull yourself off a difficult problem on a test when you are using the hard-start–jump-to-easy technique.

  3. A deep-breathing technique was suggested to help with feelings of panic. Why do you think the discussion emphasized breathing so that the belly rises, rather than just the upper chest?

  4. Why would you want to try to shift your attention momentarily before rechecking your answers on a test?

  PSYCHOLOGIST SIAN BEILOCK ON HOW TO PREVENT THE DREADED “CHOKE”

  Sian Beilock is a psychology professor at the University of Chicago. She is one of the world’s leading experts on how to reduce feelings of panic under high-stakes conditions, and is the author of the book Choke: What the Secrets of the Brain Reveal about Getting It Right When You Have To.7

  “High-stakes learning and performance situations can put you under a lot of stress. However, there is a growing body of research showing that fairly simple psychological interventions can lower your anxiety about tests and boost what you learn in the classroom. These interventions don’t teach academic content; they target your attitudes.

  “My research team has found that if you write about your thoughts and feelings about an upcoming test immediately before you take the test, it can lessen the negative impact of pressure on performance. We think that writing helps to release negative thoughts from mind, making them less likely to pop up and distract you in the heat of the moment.

  “The minor stress of many self-tests as you master the material can also prepare you for the more intense stress of real tests. As you’ve learned in this book, testing yourself while you are learning is a great way to commit information to mind, making it easier to fish out in the heat of a high-stakes exam.

  “It’s also true that negative self-talk—that is, negative thoughts arising from your own mind—can really hurt your performance, so make sure that what you say and think about yourself as you are preparing for tests is always upbeat. Cut yourself off in midthought if need be to prevent negativity, even if you feel the dragons of doom await you. If you flub a problem, or even many problems, keep your spirits up and turn your focus to the next problem.

  “Finally, one reason students sometimes choke on a test is that they frantically dive right in to solving a problem before they’ve really thought about what they are facing. Learning to pause for a few seconds before you start solving a problem or when you hit a roadblock can help you see the best solution path—this can help prevent the ultimate choking feeling when you suddenly realize you’ve spent a lot of time pursuing a dead end.

  “You can definitely learn to keep your stress within bounds. Surprisingly, you wouldn’t want to eliminate stress altogether, because a little stress can help you perform at your best when it matters most.

  “Good luck!”

  { 18 }

  unlock your potential

  Richard Feynman, the bongo-playing, Nobel Prize–winning physicist, was a happy-go-lucky guy. But there were a few years—the best and worst of his life—when his exuberance was challenged.

  In the early 1940s, Feynman’s beloved wife, Arlene, lay in a distant hospital, deathly ill with tuberculosis. He only rarely could get away to see her because he was in the isolated New Mexico town of Los Alamos, working on one of the most important projects of World War II—the top-secret Manhattan Project. Back then, Feynman was nobody famous. No special privileges were afforded him.

  To help keep his mind occupied when his workday ended and anxiety or boredom reared its head, Feynman began a focused effort to peer into people’s deepest, darkest secrets: He began figuring out how to open safes.

  Becoming an accomplished safecracker isn’t easy. Feynman developed his intuition, mastering the internal structures of the locks, practicing like a concert pianist so his fingers could swiftly run through remaining permutations if he could discover the first numbers of a combination.

  Eventually, Feynman happened to learn of a professional locksmith who had recently been hired at Los Alamos—a real expert who could open a safe in seconds.

  An expert, right
at hand! Feynman realized if only he could befriend this man, the deepest secrets of safecracking would be his.

  IN THIS BOOK we’ve explored new ways of looking at how you learn. Sometimes, as we’ve discovered, your desire to figure things out right now is what prevents you from being able to figure things out. It’s almost as if, when you reach too quickly with your right hand, your left hand automatically latches on and holds you back.

  Great artists, scientists, engineers, and chess masters like Magnus Carlsen tap into the natural rhythm of their brains by first intently focusing their attention, working hard to get the problem well in mind. Then they switch their attention elsewhere. This alternation between focused and diffuse methods of thinking allows thought clouds to drift more easily into new areas of the brain. Eventually, snippets of these clouds—refined, refluffed—can return with useful parts of a solution.

  Reshaping your brain is under your control. The key is patient persistence—working knowledgeably with your brain’s strengths and weaknesses.

  You can improve your focusing ability by gently redirecting your responses to interrupting cues like your phone’s ring or the beep of a text message. The Pomodoro—a brief, timed period of focused attention—is a powerful tool in diverting the well-meaning zombies of your habitual responses. Once you’ve done a bout of hard, focused work, you can then really savor the mental relaxation that follows.

  The result of weeks and months of gradual effort? Sturdy neural structures with well-cured mortar laid between each new learning period. Learning in this way, with regular periods of relaxation between times of focused attention, not only allows us to have more fun, but also allows us to learn more deeply. The relaxation periods provide time to gain perspective—to synthesize both the context and the big picture of what we are doing.

  Be mindful that parts of our brain are wired to believe that whatever we’ve done, no matter how glaringly wrong it might be, is just fine, thank you very much. Indeed, our ability to fool ourselves is part of why we check back—Does this really make sense?—before turning in an examination. By ensuring that we step back and take fresh perspectives on our work, by testing ourselves through recall, and by allowing our friends to question us, we can better catch our illusions of competence in learning. It is these illusions, as much as any real lack of understanding, that can trip us up en route to success in studying math and science.

  Rote memorization, often at the last minute, has given many lower-level learners the illusory sense that they understand math and science. As they climb to higher levels, their weak understanding eventually crumbles. But our growing understanding of how the mind truly learns is helping us move past the simplistic idea that memorization is always bad. We now know that deep, practiced internalization of well-understood chunks is essential to mastering math and science. We also know that, just as athletes can’t properly develop their muscles if they train in last-minute cramming sessions, students in math and science can’t develop solid neural chunks if they procrastinate in their studies.

  No matter what our age and degree of sophistication, parts of our brain remain childlike. This means that we sometimes can become frustrated, a signal to us to take a breather. But our ever-present inner child also gives us the potential to let go and use our creativity to help us visualize, remember, make friends with, and truly understand concepts in math and science that at first can seem terribly difficult.

  In a similar way, we’ve found that persistence can sometimes be misplaced—that relentless focus on a problem blocks our ability to solve that problem. At the same time, big-picture, long-term persistence is key to success in virtually any domain. This kind of long-term stick-to-it-iveness is what can help get us past the inevitable naysayers or unfortunate vicissitudes of life that can temporarily make our goals and dreams seem too far to grasp.

  A central theme of this book is the paradoxical nature of learning. Focused attention is indispensable for problem solving—yet it can also block our ability to solve problems. Persistence is key—but it can also leave us unnecessarily pounding our heads. Memorization is a critical aspect of acquiring expertise—but it can also keep us focused on the trees instead of the forest. Metaphor allows us to acquire new concepts—but it can also keep us wedded to faulty conceptions.

  Study in groups or alone, start hard or start easy, learn concretely or in abstract, success or failure . . . In the end, integrating the many paradoxes of learning adds value and meaning to everything we do.

  Part of the magic long used by the world’s best thinkers has been to simplify—to put things into terms that even a kid brother or sister can understand. This, indeed, was Richard Feynman’s approach; he challenged some of the most esoteric theoretical mathematicians he knew to put their complicated theories in simple terms.

  It turned out they could. You can, too. And like both Feynman and Santiago Ramón y Cajal, you can use the strengths of learning to help reach your dreams.

  AS FEYNMAN CONTINUED to refine his safecracking skills, he befriended the professional locksmith. Through time and talk, Feynman gradually swept away superficial pleasantries, digging deeper and deeper so that he could understand the nuance behind what he saw to be the locksmith’s utter mastery.

  Late one night, at long last, that most valuable of arcane knowledge became clear.

  The locksmith’s secret was that he was privy to the manufacturers’ default settings.

  By knowing the default settings, the locksmith was often able to slip into safes that had been left unchanged since they’d arrived from the manufacturer. Whereas everyone thought that safecracking wizardry was involved, it was a simple understanding of how the device arrived from the manufacturer that was fundamental.

  Like Feynman, you can achieve startling insights into how to understand more simply, easily, and with less frustration. By understanding your brain’s default settings—the natural way it learns and thinks—and taking advantage of this knowledge, you, too, can become an expert.

  In the beginning of the book, I mentioned that there are simple mental tricks that can bring math and science into focus, tricks that are helpful not only for people who are bad at math and science but also for those who already good at it. You’ve walked through all these tricks in the course of reading this book. But, as you now know, nothing beats grasping the chunked and simplified essence. So what follows sums up my final thoughts—the chunked essence of some of the central ideas in this book, distilled into the ten best and worst rules of studying.

  Remember—Lady Luck favors the one who tries. A little insight into learning how to learn best doesn’t hurt, either.

  TEN RULES OF GOOD STUDYING

  1. Use recall. After you read a page, look away and recall the main ideas. Highlight very little, and never highlight anything you haven’t put in your mind first by recalling. Try recalling main ideas when you are walking to class or in a different room from where you originally learned it. An ability to recall—to generate the ideas from inside yourself—is one of the key indicators of good learning.

  2. Test yourself. On everything. All the time. Flash cards are your friend.

  3. Chunk your problems. Chunking is understanding and practicing with a problem solution so that it can all come to mind in a flash. After you solve a problem, rehearse it. Make sure you can solve it cold—every step. Pretend it’s a song and learn to play it over and over again in your mind, so the information combines into one smooth chunk you can pull up whenever you want.

  4. Space your repetition. Spread out your learning in any subject a little every day, just like an athlete. Your brain is like a muscle—it can handle only a limited amount of exercise on one subject at a time.

  5. Alternate different problem-solving techniques during your practice. Never practice too long at any one session using only one problem-solving technique—after a while, you are just mimicking what you did on the previous problem. Mix i
t up and work on different types of problems. This teaches you both how and when to use a technique. (Books generally are not set up this way, so you’ll need to do this on your own.) After every assignment and test, go over your errors, make sure you understand why you made them, and then rework your solutions. To study most effectively, handwrite (don’t type) a problem on one side of a flash card and the solution on the other. (Handwriting builds stronger neural structures in memory than typing.) You might also photograph the card if you want to load it into a study app on your smartphone. Quiz yourself randomly on different types of problems. Another way to do this is to randomly flip through your book, pick out a problem, and see whether you can solve it cold.

  6. Take breaks. It is common to be unable to solve problems or figure out concepts in math or science the first time you encounter them. This is why a little study every day is much better than a lot of studying all at once. When you get frustrated with a math or science problem, take a break so that another part of your mind can take over and work in the background.

  7. Use explanatory questioning and simple analogies. Whenever you are struggling with a concept, think to yourself, How can I explain this so that a ten-year-old could understand it? Using an analogy really helps, like saying that the flow of electricity is like the flow of water. Don’t just think your explanation—say it out loud or put it in writing. The additional effort of speaking and writing allows you to more deeply encode (that is, convert into neural memory structures) what you are learning.

  8. Focus. Turn off all interrupting beeps and alarms on your phone and computer, and then turn on a timer for twenty-five minutes. Focus intently for those twenty-five minutes and try to work as diligently as you can. After the timer goes off, give yourself a small, fun reward. A few of these sessions in a day can really move your studies forward. Try to set up times and places where studying—not glancing at your computer or phone—is just something you naturally do.