A Mind For Numbers Read online

  But in the most unlikely situation I could have ever imagined, I eventually found myself commissioned as a second lieutenant in the U.S. Army Signal Corps. I was suddenly expected to become an expert in radio, cable, and telephone switching systems. What a turning point! I went from being on top of the world, an expert linguist, in control of my destiny, to being thrown into a new technological world where I was as stunted as a stump.

  Yikes!

  I was made to enroll in mathematically oriented electronics training (I finished at the bottom of the class), and then off I went to West Germany, where I became a pitiable communications platoon leader. I saw that the officers and enlisted members who were technically competent were in demand. They were problem solvers of the first order, and their work helped everyone accomplish the mission.

  I reflected on the progress of my career and realized that I’d followed my inner passions without also being open to developing new ones. As a consequence, I’d inadvertently pigeonholed myself. If I stayed in the army, my poor technical know-how would always leave me a second-class citizen.

  On the other hand, if I left the service, what could I do with a degree in Slavic languages and literature? There aren’t a lot of jobs for Russian linguists. Basically, I’d be competing for entry-level secretarial-type jobs with millions of others who also had bachelor’s of arts degrees. A purist might argue that I’d distinguished myself in both my studies and my service and could find much better work, but that purist would be unaware of how tough the job market can sometimes be.

  Fortunately there was another unusual option. One of the great benefits of my service was that I had GI Bill money to offset the costs of future schooling. What if I used that support to do the unthinkable and try to retrain myself? Could I retool my brain from mathphobe to math lover? From technophobe to technogeek?

  I’d never heard of anyone doing anything like that before, and certainly not coming from the phobic depths I’d sunk to. There couldn’t possibly be anything more foreign to my personality than mastering math and science. But my colleagues in the service had shown me the concrete benefits of doing so.

  It became a challenge—an irresistible challenge.

  I decided to retrain my brain.

  It wasn’t easy. The first semesters were filled with frightening frustration. I felt like I was wearing a blindfold. The younger students around me mostly seemed to have a natural knack for seeing the solutions, while I was stumbling into walls.

  But I began to catch on. Part of my original problem, I found, was that I had been putting my effort forth in the wrong way—like trying to lift a piece of lumber when you’re standing on it. I began to pick up little tricks about not only how to study but when to quit. I learned that internalizing certain concepts and techniques could be a powerful tool. I also learned not to take on too much at once, allowing myself plenty of time to practice even if it meant my classmates would sometimes graduate ahead of me because I wasn’t taking as many courses each semester as they were.

  As I gradually learned how to learn math and science, things became easier. Surprisingly, just as with studying language, the better I got, the more I enjoyed what I was doing. This former Queen of the Confused in math went on to earn a bachelor’s degree in electrical engineering and then a master’s in electrical and computer engineering. Finally, I earned a doctorate in systems engineering, with a broad background that included thermodynamics, electromagnetics, acoustics, and physical chemistry. The higher I went, the better I did. By the time I reached my doctoral studies, I was breezing by with perfect grades. (Well, perhaps not quite breezing. Good grades still took work. But the work I needed to do was clear.)

  Now as a professor of engineering, I have become interested in the inner workings of the brain. My interest grew naturally from the fact that engineering lies at the heart of the medical images that allow us to tease out how the brain functions. I can now more clearly see how and why I was able to change my brain. I also see how I can help you learn more effectively without the frustration and struggle I experienced.1 And as a researcher whose work straddles engineering, the social sciences, and the humanities, I’m also aware of the essential creativity underlying not just art and literature, but also math and science.

  If you don’t (yet) consider yourself naturally good at math and science, you may be surprised to learn that the brain is designed to do extraordinary mental calculations. We do them every time we catch a ball, or rock our body to the beat of a song, or maneuver our car around a pothole in the road. We often do complex calculations, solving complex equations unconsciously, unaware that we sometimes already know the solution as we slowly work toward it.2 In fact, we all have a natural feel and flair for math and science. Basically, we just need to master the lingo and culture.

  In writing this book, I connected with hundreds of the world’s leading professor-teachers of mathematics, physics, chemistry, biology, and engineering, as well as education, psychology, neuroscience, and professional disciplines such as business and the health sciences. It was startling to hear how often these world-class experts had used precisely the approaches outlined in the book when they themselves were learning their disciplines. These techniques were also what the experts asked their students to use—but since the methods sometimes seem counterintuitive, and even irrational, instructors have often found it hard to convey their simple essence. In fact, because some of these learning and teaching methods are derided by ordinary instructors, superstar teachers sometimes divulged their teaching and learning secrets to me with embarrassment, unaware that many other top instructors shared similar approaches. By collecting many of these rewarding insights in one place, you too can easily learn and apply practical techniques gleaned in part from these “best of the best” teachers and professors. These techniques are especially valuable for helping you learn more deeply and effectively in limited time frames. You’ll also gain insight from students and other fellow learners—people who share your constraints and considerations.

  Remember, this is a book for math experts and mathphobes alike. This book was written to make it easier for you to learn math and science, regardless of your past grades in those subjects or how good or bad you think you are at them. It is designed to expose your thought processes so you can understand how your mind learns—and also how your mind sometimes fools you into believing you’re learning, when you’re actually not. The book also includes plenty of skill-building exercises that you can apply directly to your current studies. If you’re already good at numbers or science, the insights in this book can help make you better. They will broaden your enjoyment, creativity, and equation-solving elegance.

  If you’re simply convinced you don’t have a knack for numbers or science, this book may change your mind. You may find it hard to believe, but there’s hope. When you follow these concrete tips based on how we actually learn, you’ll be amazed to see the changes within yourself, changes that can allow new passions to bloom.

  What you discover will help you be more effective and creative, not only in math and science, but in almost everything you do.

  Let’s begin!

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  easy does it:

  Why Trying Too Hard Can Sometimes Be Part of the Problem

  If you want to understand some of the most important secrets to learning math and science, look at the following picture.

  The man on the right is legendary chess grand master Garry Kasparov. The boy on the left is thirteen-year-old Magnus Carlsen. Carlsen has just wandered away from the board during the height of a speed chess game, where little time is given to think about moves or strategy. That’s a little like casually deciding to do a backflip while walking a tightrope across Niagara Falls.

  Yes, Carlsen was psyching out his opponent. Rather than obliterating the upstart youngster, the flustered Kasparov played to a draw. But the brilliant Carlsen, who went on to become the youngest top-
rated chess player in history, was doing something far beyond playing mind games with his older opponent. Gaining insight into Carlsen’s approach can help us understand how the mind learns math and science. Before we go into how Carlsen psyched out Kasparov, we need to cover a couple of important ideas about how people think. (But I promise, we’ll come back to Carlsen.)

  Thirteen-year-old Magnus Carlsen (left), and legendary genius Garry Kasparov playing speed chess at the “Reykjavík Rapid” in 2004. Kasparov’s shock is just beginning to become apparent.

  We’re going to be touching on some of the main themes of the book in this chapter, so don’t be surprised if you have to toggle around a bit in your thinking. Being able to toggle your thinking—getting a glimpse of what you are learning before returning later to more fully understand what’s going on, is itself one of the main ideas in the book!

  NOW YOU TRY!

  Prime Your Mental Pump

  As you first begin looking at a chapter or section of a book that teaches concepts of math or science, it helps to take a “picture walk” through the chapter, glancing not only at the graphics, diagrams, and photos, but also at the section headings, summary, and even questions at the end of the chapter, if the book has them. This seems counterintuitive—you haven’t actually read the chapter yet, but it helps prime your mental pump. So go ahead now and glance through this chapter and the questions at the end of the chapter.

  You’ll be surprised at how spending a minute or two glancing ahead before you read in depth will help you organize your thoughts. You’re creating little neural hooks to hang your thinking on, making it easier to grasp the concepts.

  Focused versus Diffuse Thinking

  Since the very beginning of the twenty-first century, neuroscientists have been making profound advances in understanding the two different types of networks that the brain switches between—highly attentive states and more relaxed resting state networks.1 We’ll call the thinking processes related to these two different types of networks the focused mode and diffuse mode, respectively—these modes are highly important for learning.2 It seems you frequently switch back and forth between these two modes in your day-to-day activities. You’re in either one mode or the other—not consciously in both at the same time. The diffuse mode does seem to be able to work quietly in the background on something you are not actively focusing on.3 Sometimes you may also flicker for a rapid moment to diffuse-mode thinking.

  Focused-mode thinking is essential for studying math and science. It involves a direct approach to solving problems using rational, sequential, analytical approaches. The focused mode is associated with the concentrating abilities of the brain’s prefrontal cortex, located right behind your forehead.4 Turn your attention to something and bam—the focused mode is on, like the tight, penetrating beam of a flashlight.

  The prefrontal cortex is the area right behind the forehead.

  Diffuse-mode thinking is also essential for learning math and science. It allows us to suddenly gain a new insight on a problem we’ve been struggling with and is associated with “big-picture” perspectives. Diffuse-mode thinking is what happens when you relax your attention and just let your mind wander. This relaxation can allow different areas of the brain to hook up and return valuable insights. Unlike the focused mode, the diffuse mode seems less affiliated with any one area of the brain—you can think of it as being “diffused” throughout the brain.5 Diffuse-mode insights often flow from preliminary thinking that’s been done in the focused mode. (The diffuse mode must have clay to make bricks!)

  Learning involves a complex flickering of neural processing among different areas of the brain, as well as back and forth between hemispheres.6 So this means that thinking and learning is more complicated than simply switching between the focused and diffuse modes. But fortunately, we don’t need to go deeper into the physical mechanisms. We’re going to take a different approach.

  The Focused Mode—A Tight Pinball Machine

  To understand focused and diffuse mental processes, we’re going to play some pinball. (Metaphors are powerful tools for learning in math and science.) In the old game of pinball, you pull back on a spring-loaded plunger and it whacks a ball, which ends up bouncing randomly around the circular rubber bumpers.

  This happy zombie is playing neural pinball.

  Look at the following illustration. When you focus your attention on a problem, your mind pulls back the mental plunger and releases a thought. Boom—that thought takes off, bumping around like the pinball in the head on the left. This is the focused mode of thinking.

  Notice how the round bumpers are very close together in the focused mode. In contrast, the diffuse mode on the right has its circular rubber bumpers farther apart. (If you want to pursue the metaphor still further, you can think of each bumper as a cluster of neurons.)

  The close bumpers of the focused mode mean that you can more easily think a precise thought. Basically, the focused mode is used to concentrate on something that’s already tightly connected in your mind, often because you are familiar and comfortable with the underlying concepts. If you look closely at the upper part of the focused-mode thought pattern, you’ll see a wider, “well-trodden” part of the line. That broader path shows how the focused-mode thought is following along a route you’ve already practiced or experienced.

  For example, you can use the focused mode to multiply numbers—if you already know how to multiply, that is. If you’re studying a language, you might use the focused mode to become more fluent with the Spanish verb conjugation you learned last week. If you’re a swimmer, you might use the focused mode to analyze your breaststroke as you practice staying low to allow more energy to go into your forward motion.

  When you focus on something, the consciously attentive prefrontal cortex automatically sends out signals along neural pathways. These signals link different areas of your brain related to what you’re thinking about. This process is a little like an octopus that sends its tentacles to different areas of its surroundings to fiddle with whatever it’s working on. The octopus has only so many tentacles to make connections, just as your working memory has only so many things it can hold at once. (We’ll talk more about the working memory later.)

  In the game “pinball,” a ball, which represents a thought, shoots up from the spring-loaded plunger to bounce randomly against rows of rubber bumpers. These two pinball machines represent focused (left) and diffuse (right) ways of thinking. The focused approach relates to intense concentration on a specific problem or concept. But while in focused mode, sometimes you inadvertently find yourself focusing intently and trying to solve a problem using erroneous thoughts that are in a different place in the brain from the “solution” thoughts you need to actually solve the problem.

  As an example of this, note the upper “thought” that your pinball first bounces around in on the left-hand image. It is very far away and completely unconnected from the lower pattern of thought in the same brain. You can see how part of the upper thought seems to have an underlying broad path. This is because you’ve thought something similar to that thought before. The lower thought is a new thought—it doesn’t have that underlying broad pattern.

  The diffuse approach on the right often involves a big-picture perspective. This thinking mode is useful when you are learning something new. As you can see, the diffuse mode doesn’t allow you to focus tightly and intently to solve a specific problem—but it can allow you to get closer to where that solution lies because you’re able to travel much farther before running into another bumper.

  You often first funnel a problem into your brain by focusing your attention on words—reading the book or looking at your notes from a lecture. Your attentional octopus activates your focused mode. As you do your initial focused noodling around with the problem, you are thinking tightly, using the pinball bumpers that are close together to follow alo
ng familiar neural pathways related to something you already know or are familiar with. Your thoughts rattle easily through the previously ingrained patterns and quickly settle on a solution. In math and science, however, it often doesn’t take much of a change for a problem to become quite different. Problem solving then grows more difficult.

  Why Math and Science Can Be More Challenging

  Focused problem solving in math and science is often more effortful than focused-mode thinking involving language and people.7 This may be because humans haven’t evolved over the millennia to manipulate mathematical ideas, which are frequently more abstractly encrypted than those of conventional language.8 Obviously, we can still think about math and science—it’s just that the abstractness and encryptedness adds a level—sometimes a number of levels—of complexity.

  What do I mean by abstractness? You can point to a real live cow chewing its cud in a pasture and equate it with the letters c-o-w on the page. But you can’t point to a real live plus sign that the symbol “+” is modeled after—the idea underlying the plus sign is more abstract. By encryptedness, I mean that one symbol can stand for a number of different operations or ideas, just as the multiplication sign symbolizes repeated addition. In our pinball analogy, it’s as if the abstractness and encryptedness of math can make the pinball bumpers a bit spongier—it takes extra practice for the bumpers to harden and the pinball to bounce properly. This is why dealing with procrastination, while important in studying any discipline, is particularly important in math and science. We’ll be talking more about this later.