A Mind For Numbers

by Barbara Oakley, PhD

  Related to these difficulties in math and science is another challenge. It’s called the Einstellung effect (pronounced EYE-nshtellung). In this phenomenon, an idea you already have in mind, or your simple initial thought, prevents a better idea or solution from being found.9 We saw this in the focused pinball picture, where your initial pinball thought went to the upper part of the brain, but the solution thought pattern was in the lower part of the image. (The German word Einstellung means “mindset”—basically you can remember Einstellung as installing a roadblock because of the way you are initially looking at something.)

  This kind of wrong approach is especially easy to do in science because sometimes your initial intuition about what’s happening is misleading. You have to unlearn your erroneous older ideas even while you’re learning new ones.10

  The Einstellung effect is a frequent stumbling block for students. It’s not just that sometimes your natural intuitions need to be retrained—it’s that sometimes it is tough even figuring out where to begin, as when tackling a homework problem. You bumble about—your thoughts far from the actual solution—because the crowded bumpers of the focused mode prevent you from springing to a new place where the solution might be found.

  This is precisely why one significant mistake students sometimes make in learning math and science is jumping into the water before they learn to swim.11 In other words, they blindly start working on homework without reading the textbook, attending lectures, viewing online lessons, or speaking with someone knowledgeable. This is a recipe for sinking. It’s like randomly allowing a thought to pop off in the focused-mode pinball machine without paying any real attention to where the solution truly lies.

  Understanding how to obtain real solutions is important, not only in math and science problem solving, but for life in general. For example, a little research, self-awareness, and even self-experimentation can prevent you from being parted with your money—or even your good health—on products that come with bogus “scientific” claims.12 And just having a little knowledge of the relevant math can help prevent you from defaulting on your mortgage—a situation that can have a major negative impact on your life.13

  The Diffuse Mode—A Spread-Out Pinball Machine

  Think back several pages to the illustration of the diffuse-mode pinball machine brain, where the bumpers were spread far apart. This mode of thinking allows the brain to look at the world from a much broader perspective. Can you see how a thought can travel much further before it runs into a bumper? The connections are further apart—you can quickly zoom from one clump of thought to another that’s quite far away. (Of course, it’s hard to think precise, complex thoughts while in this mode.)

  If you are grappling with a new concept or trying to solve a new problem, you don’t have preexisting neural patterns to help guide your thoughts—there’s no fuzzy underlying pathway to help guide you. You may need to range widely to encounter a potential solution. For this, diffuse mode is just the ticket!

  Another way to think of the difference between focused and diffuse modes is to think of a flashlight. You can set a flashlight so it has a tightly focused beam that can penetrate deeply into a small area. Or you can set the flashlight onto a more diffuse setting where it casts its light broadly, but not very strongly in any one area.

  If you are trying to understand or figure out something new, your best bet is to turn off your precision-focused thinking and turn on your “big picture” diffuse mode, long enough to be able to latch on to a new, more fruitful approach. As we’ll see, the diffuse mode has a mind of its own—you can’t simply command it to turn on. But we’ll soon get to some tricks that can help you transition between modes.

  COUNTERINTUITIVE CREATIVITY

  “When I was learning about the diffuse mode, I began to notice it in my daily life. For instance, I realized my best guitar riffs always came to me when I was ‘just messing around’ as opposed to when I sat down intent on creating a musical masterpiece (in which case my songs were often clichéd and uninspiring). Similar things happened when I was writing a school paper, trying to come up with an idea for a school project, or trying to solve a difficult math problem. I now follow the rule of thumb that is basically: The harder you push your brain to come up with something creative, the less creative your ideas will be. So far, I have not found a single situation where this does not apply. Ultimately, this means that relaxation is an important part of hard work—and good work, for that matter.”

  —Shaun Wassell, freshman, computer engineering

  Why Are There Two Modes of Thinking?

  Why do we have these two different thinking modes? The answer may be related to two major problems that vertebrates have had in staying alive and passing their genes on to their offspring. A bird, for example, needs to focus carefully so it can pick up tiny pieces of grain as it pecks the ground for food, and at the same time, it must scan the horizon for predators such as hawks. What’s the best way to carry out those two very different tasks? Split things up, of course. You can have one hemisphere of the brain more oriented toward the focused attention needed to peck at food and the other oriented toward scanning the horizon for danger. When each hemisphere tends toward a particular type of perception, it may increase the chance of survival.14 If you watch birds, they’ll first peck, and then pause to scan the horizon—almost as if they are alternating between focused and diffuse modes.

  In humans, we see a similar splitting of brain functions. The left side of the brain is somewhat more associated with careful, focused attention. It also seems more specialized for handling sequential information and logical thinking—the first step leads to the second step, and so on. The right seems more tied to diffuse scanning of the environment and interacting with other people, and seems more associated with processing emotions.15 It also is linked with handling simultaneous, big-picture processing.16

  The slight differences in the hemispheres give us a sense of why two different processing modes may have arisen. But be wary of the idea that some people are “left-brain” or “right-brain” dominant—research indicates that is simply not true.17 Instead it is clear that both hemispheres are involved in focused as well as diffuse modes of thinking. To learn about and be creative in math and science, we need to strengthen and use both the focused and diffuse modes.19

  Here’s a quick example that gives a sense of the difference between focused and diffuse thinking. If you are given two triangles to put together into a square shape, it’s easy to do, as shown on the left. If you are given two more triangles and told to form a square, your first tendency is to erroneously put them together to form a rectangle, as shown in the middle. This is because you’ve already laid down a focused-mode pattern that you have a tendency to follow. It takes an intuitive, diffuse leap to realize that you need to completely rearrange the pieces if you want to form another square, as shown on the right.18

  Evidence suggests that to grapple with a difficult problem, we must first put hard, focused-mode effort into it. (We learned that in grade school!) Here’s the interesting part: The diffuse mode is also often an important part of problem solving, especially when the problem is difficult. But as long as we are consciously focusing on a problem, we are blocking the diffuse mode.

  There’s a winner at Ping-Pong only if the ball is able to go back and forth.

  EMBRACE BEFUDDLEMENT!

  “Befuddlement is a healthy part of the learning process. When students approach a problem and don’t know how to do it, they’ll often decide they’re no good at the subject. Brighter students, in particular, can have difficulty in this way—their breezing through high school leaves them no reason to think that being confused is normal and necessary. But the learning process is all about working your way out of confusion. Articulating your question is 80 percent of the battle. By the time you’ve figured out what’s confusing, you’re likely to have answered the question yourself!”


  —Kenneth R. Leopold, Distinguished Teaching Professor, Department of Chemistry, University of Minnesota

  The bottom line is that problem solving in any discipline often involves an exchange between the two fundamentally different modes. One mode will process the information it receives and then send the result back to the other mode. This volleying of information back and forth as the brain works its way toward a conscious solution appears essential for understanding and solving all but trivial problems and concepts.20 The ideas presented here are extremely helpful for understanding learning in math and science. But as you are probably beginning to see, they can be just as helpful for many other subject areas, such as language, music, and creative writing.

  NOW YOU TRY!

  Shifting Modes

  Here’s a cognitive exercise that can help you feel the shift from focused to diffuse mode. See whether you can form a new triangle that points down by moving only three coins.

  When you relax your mind, releasing your attention and focusing on nothing in particular, the solution can most easily come to you.

  You should know that some children get this exercise instantly, while some highly intelligent professors finally just give up. To answer this question, it helps to summon your inner child. The solutions for this challenge and for all the “Now You Try!” challenges in the book can be found in the endnotes.21

  Procrastination Prelude

  Many people struggle with procrastination. We’ll have a lot to say later in this book about how to deal effectively with procrastination. For now, keep in mind that when you procrastinate, you are leaving yourself only enough time to do superficial focused-mode learning. You are also increasing your stress level because you know you have to complete what feels like an unpleasant task. The resulting neural patterns will be faint and fragmented and will quickly disappear—you’ll be left with a shaky foundation. In math and science in particular, this can create severe problems. If you cram for a test at the last minute or quickly breeze through your homework, you won’t have time for either learning mode to help you tackle the tougher concepts and problems or to help you synthesize the connections in what you are learning.

  NOW YOU TRY!

  Focusing Intently but Briefly

  If you often find yourself procrastinating, as many of us do, here’s a tip. Turn off your phone and any sounds or sights (or websites) that might signal an interruption. Then set a timer for twenty-five minutes and put yourself toward doing a twenty-five-minute interlude of work focused on a task—any task. Don’t worry about finishing the task—just worry about working on it. Once the twenty-five minutes is up, reward yourself with web surfing, checking your phone, or whatever you like to do. This reward is as important as the work itself. You’ll be amazed at how productive a focused twenty-five-minute stint can be—especially when you’re just focusing on the work itself, not on finishing. (This method, known as the Pomodoro technique, will be discussed in more detail in chapter 6.)

  If you want to apply a more advanced version of this approach, imagine that at the end of the day, you are reflecting on the one most important task that you accomplished that day. What would that task be? Write it down. Then work on it. Try to complete at least three of these twenty-five-minute sessions that day, on whatever task or tasks you think are most important.

  At the end of your workday, look at what you crossed off your list and savor the feeling of accomplishment. Then write a few key things that you would like to work on the next day. This early preparation will help your diffuse mode begin to think about how you will get those tasks done the next day.

  SUMMING IT UP

  Our brain uses two very different processes for thinking—the focused and diffuse modes. It seems you toggle back and forth between these modes, using one or the other.

  It is typical to be stumped by new concepts and problems when we first focus on them.

  To figure out new ideas and solve problems, it’s important not only to focus initially, but also to subsequently turn our focus away from what we want to learn.

  The Einstellung effect refers to getting stuck in solving a problem or understanding a concept as a result of becoming fixated on a flawed approach. Switching modes from focused to diffuse can help free you from this effect. Keep in mind, then, that sometimes you will need to be flexible in your thinking. You may need to switch modes to solve a problem or understand a concept. Your initial ideas about problem solving can sometimes be very misleading.

  PAUSE AND RECALL

  Close the book and look away. What were the main ideas of this chapter? Don’t worry if you can’t recall very much when you first begin trying this. As you continue practicing this technique, you’ll begin noticing changes in how you read and how much you recall.

  ENHANCE YOUR LEARNING

  1. How would you recognize when you are in the diffuse mode? How does it feel to be in the diffuse mode?

  2. When you are consciously thinking of a problem, which mode is active and which is blocked? What can you do to escape this blocking?

  3. Recall an episode where you experienced the Einstellung effect. How were you able to change your thinking to get past the preconceived, but erroneous, notion?

  4. Explain how the focused and diffuse modes might be equated to an adjustable beam on a flashlight. When can you see farther? When can you see more broadly, but less far?

  5. Why is procrastination sometimes a special challenge for those who are studying math and science?

  SHIFTING OUT OF BEING STUCK: INSIGHT FROM NADIA NOUI-MEHIDI, A SENIOR STUDYING ECONOMICS

  “I took Calculus I in eleventh grade and it was a nightmare. It was so profoundly different from anything I had learned before that I didn’t even know how to learn it. I studied longer and harder than I ever had before, yet no matter how many problems I did or how long I stayed in the library I was learning nothing. I ultimately just stuck to what I could get by with through memorizing. Needless to say, I did not do well on the AP [advanced placement] exam.

  “I avoided math for the next two years, and then as a sophomore in college, I took Calculus I and got a 4.0. I don’t think I was any smarter two years later, but there was a complete shift in the way I was approaching the subject.

  “I think in high school I was stuck in the focused mode of thinking (Einstellung!) and felt that if I kept trying to approach problems in the same way it would eventually click.

  “I now tutor students in math and economics and the issues are almost always that they are fixated on looking at the details of the problem for clues on how to solve it, and not on understanding the problem itself. I don’t think you can tutor someone on how to think—it’s kind of a personal journey. But here are some things that have helped me understand a concept that at first seems complicated or confusing.

  1. I understand better when I read the book rather than listen to someone speak, so I always read the book. I skim first so I know basically what the chapter is trying to get at and then I read it in detail. I read the chapter more than once (but not in a row).

  2. If after reading the book, I still don’t fully understand what’s going on, I Google or look at YouTube videos on the subject. This isn’t because the book or professor isn’t thorough, but rather because sometimes hearing a slightly different way of phrasing something can make your mind look at the problem from a different angle and spark understanding.

  3. I think most clearly when I’m driving. Sometimes I’ll just take a break and drive around—this helps a lot. I have to be somewhat occupied because if I just sit down and think I end up getting bored or distracted and can’t concentrate.

  { 3 }

  learning is creating:

  Lessons from Thomas Edison’s Frying Pan

  Thomas Edison was one of the most prolific inventors in history, with more than one thousand patents to his name. No
thing got in the way of his creativity. Even as his lab was burning to the ground in a horrific accidental fire, Edison was excitedly sketching up plans for a new lab, even bigger and better than before. How could Edison be so phenomenally creative? The answer, as you’ll see, relates to his unusual tricks for shifting his mode of thinking.

  Shifting between the Focused and Diffuse Modes

  For most people, shifting from focused to diffuse mode happens naturally if you distract yourself and then allow a little time to pass. You can go for a walk, take a nap, or go to the gym. Or you can work on something that occupies other parts of your brain: listening to music, conjugating Spanish verbs, or cleaning your gerbil cage.1 The key is to do something else until your brain is consciously free of any thought of the problem. Unless other tricks are brought into play, this generally takes several hours. You may say, “I don’t have that kind of time.” You do, however, if you simply switch your focus to other things you need to do, and mix in a little relaxing break time.

  Creativity expert Howard Gruber has suggested that one of the three B’s usually seems to do the trick: the bed, the bath, or the bus.2 One remarkably inventive chemist of the mid-1800s, Alexander Williamson, observed that a solitary walk was worth a week in the laboratory in helping him progress in his work.3 (Lucky for him there were no smartphones then.) Walking spurs creativity in many fields; a number of famous writers, such as Jane Austen, Carl Sandburg, and Charles Dickens, found inspiration during their frequent long walks.

  Once you are distracted from the problem at hand, the diffuse mode has access and can begin pinging about in its big-picture way to settle on a solution.4 After your break, when you return to the problem at hand, you will often be surprised at how easily the solution pops into place. Even if the solution doesn’t appear, you will often be further along in your understanding. It can take a lot of hard focused-mode work beforehand, but the sudden, unexpected solution that emerges from the diffuse mode can make it feel almost like the “aha!” mode.