A Mind for Numbers: How to Excel at Math and Science (Even if You Flunked Algebra)

Author: Barbara Oakley

• 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.

• 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.

• 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.

• 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.

• when you procrastinate, you are leaving yourself only enough time to do superficial focused-mode learning.

• For example, there are more than three hundred different known proofs of the Pythagorean theorem. As we will soon learn, technical problems and their solutions may be considered a form of poetry.

• One way to think of the diffuse mode is as a base station when you are mountain climbing. Base stations are essential resting spots in the long journey to difficult mountaintops. You use them to pause, reflect, check your gear, and make sure you’ve got the right route picked out. But you would never confuse resting at a base station with the hard work of getting to the top of a mountain.

• Remember, accepting the first idea that comes to mind when you are working on an assignment or test problem can prevent you from finding a better solution.

• According to recent research, blinking is a vital activity that provides another means of reevaluating a situation.

• Working memory is the part of memory that has to do with what you are immediately and consciously processing in your mind.

• To help with this process, use a technique called spaced repetition. As you may have guessed, this technique involves repeating what you are trying to retain, like a new vocabulary word or a new problem-solving technique, but spacing this repetition out over a number of days.

• Dreaming about what you are studying can substantially enhance your ability to understand—it somehow consolidates your memories into easier-to-grasp chunks.

• Going without sleep the night before an examination can mean that even if you are perfectly prepared, your mind is simply unable to function properly, so you do poorly on the test.

• Focused practice and repetition—the creation of memory traces—are also at the heart of an impeccably played golf stroke, a master chef’s practiced flip of an omelet, or a basketball free throw.

• But that path to expertise is built bit by bit. Small memorized free spins, heel turns, and kicks become incorporated into larger, more creative interpretations.

• just understanding how a problem was solved does not necessarily create a chunk that you can easily call to mind later. Do not confuse the “aha!” of a breakthrough in understanding with solid expertise!

• As you can see from the following “top-down, bottom-up” illustration, learning takes place in two ways. There is a bottom-up chunking process where practice and repetition can help you both build and strengthen each chunk, so you can easily gain access to it when needed.

• Attempting to recall the material you are trying to learn—retrieval practice—is far more effective than simply rereading the material.

• You may be surprised to learn that highlighting and underlining must be done carefully—otherwise they can be not only ineffective but also misleading. It’s as if the motion of your hand can fool you into thinking you’ve placed the concept in your brain.

• When marking up the text, train yourself to look for main ideas before making any marks, and keep your text markings to a minimum—one sentence or less per paragraph.13 Words or notes in a margin that synthesize key concepts are a good idea.

• Using recall—mental retrieval of the key ideas—rather than passive rereading will make your study time more focused and effective.

• The only time rereading text seems to be effective is if you let time pass between rereadings so that it becomes more of an exercise in spaced repetition.

• This is why many professors recommend that, if at all possible, you rewrite your notes during the evening after a lecture. This helps to solidify newly forming chunks and reveals the holes in your understanding that professors just love to target on tests.

• If you work a problem by just looking at the solution, and then tell yourself, “Oh yeah, I see why they did that,” then the solution is not really yours—you’ve done almost nothing to knit the concepts into your underlying neurocircuitry.

• The ability to combine chunks in novel ways underlies much of historical innovation. Steven Johnson, in his brilliant book Where Good Ideas Come From, describes the “slow hunch”—the gentle, years-long simmering of focused and diffuse processes that has resulted in creative breakthroughs ranging from Darwin’s evolutionary theory to the creation of the World Wide Web.

• (An important side note here is that a key difference between creative scientists and technically competent but nonimaginative ones is their breadth of interest.

• There are two ways to solve problems—first, through sequential, step-by-step reasoning, and second, through more holistic intuition. Sequential thinking, where each small step leads deliberately toward the solution, involves the focused mode. Intuition, on the other hand, often seems to require a creative, diffuse mode linking of several seemingly different focused mode thoughts.

• In building a chunked library, you are training your brain to recognize not only a specific problem, but different types ˙and classes of problems so that you can automatically know how to quickly solve whatever you encounter. You’ll start to see patterns that simplify problem solving for you and will soon find that different solution techniques are lurking at the edge of your memory.

• We love creativity and the idea of being able to learn by seeing the big picture. But you can’t learn mathematics or science without also including a healthy dose of practice and repetition to help you build the chunks that will underpin your expertise.

• This reinforces an idea we’ve alluded to already. When we retrieve knowledge, we’re not being mindless robots—the retrieval process itself enhances deep learning and helps us begin forming chunks.29 Even more of a surprise to researchers was that the students themselves predicted that simply reading and recalling the materials wasn’t the best way to learn.

• recalling material when you are outside your usual place of study helps you strengthen your grasp of the material by viewing it from a different perspective.

• Interleaving means practice by doing a mixture of different kinds of problems requiring different strategies.

• Overlearning can have its place—it can help produce an automaticity that is important when you are executing a serve in tennis or playing a perfect piano concerto. But be wary of repetitive overlearning during a single session in math and science learning—research has shown it can be a waste of valuable learning time.

• Once you have the basic idea of a technique down during your study session (sort of like learning to ride a bike with training wheels), start interleaving your practice with problems of different types.

• Rather than devote a long session to the study or practice of the same skill or concept so that overlearning occurs, students should divide their effort across several shorter sessions. This doesn’t mean that long study sessions are necessarily a bad idea. Long sessions are fine as long as students don’t devote too much time to any one skill or concept. Once they understand ‘X,’ they should move on to something else and return to ‘X’ on another day.”

• We procrastinate about things that make us feel uncomfortable.

• Procrastination is a single, monumentally important “keystone” bad habit.7 A habit, in other words, that influences many important areas of your life. Change it, and a myriad of other positive changes will gradually begin to unfold.

• It’s easy to feel distaste for something you’re not good at. But the better you get at something, the more you’ll find you enjoy it.

• This is a typical procrastination pattern. You think about something you don’t particularly like, and the pain centers of your brain light up. So you shift and narrow your focus of attention to something more enjoyable.8 This causes you to feel better, at least temporarily.

• Researchers have found that procrastination can even become a source of pride as well as an excuse for doing poorly. “I crammed for the quiz last night after finishing the lab report and the marketing interview. Of course I could have done better. But with so many things on my plate, what do you expect?”

• Procrastinators report higher stress, worse health, and lower grades.

• You may even start telling yourself that procrastination is an innate characteristic—a trait that is as much a part of you as your height or the color of your hair. After all, if procrastination were easily fixable, wouldn’t you have fixed it by now?

• The only place you need to apply willpower is to change your reaction to the cue.

• Cues usually fall into one of the following categories: location, time, how you feel, reactions to other people, or something that just happened.

• Students often find that developing new cues, such as starting homework as soon as they get home from school or right after their first break from class, are helpful.

• As procrastination expert Piers Steel, author of The Procrastination Equation, points out, “If you protect your routine, eventually it will protect you.”

• a public commitment to study is made in the very act of removing temptation. Friends and family can be helpful if you enlist them.

• The Pomodoro technique—the twenty-five-minute timer—can be especially helpful in shifting your reaction to cues.

• Also, it helps to have something in your stomach when starting particularly difficult tasks. This ensures that you have mental energy for that momentary dollop of willpower as you are getting started.

• Why are you procrastinating? Can you substitute in an emotional payoff?

• It’s particularly important to realize that giving yourself even a small “attaboy” or “attagirl” jump-starts the process of rewiring your brain. This rewiring, sometimes called learned industriousness, helps brighten tasks you once thought were boring and uninteresting.

• Don’t feel bad if you find that you have trouble getting into a “flow” state at first. I sometimes find it takes a few days of drudgery through a few cycles of the Pomodoro technique before flow begins to unfold and I find myself starting to enjoy work in a very new area.

• Belief that your new system works is what can get you through.

• A powerful approach is mental contrasting.6 In this technique, you think about where you are now and contrast it with what you want to achieve.

• In mental contrasting, it’s the contrast of where you want to be with where you are now, or where you have been, that makes the difference.

• If you find yourself avoiding certain tasks because they make you uncomfortable, there is a great way to reframe things: Learn to focus on process, not product.

• To prevent procrastination, you want to avoid concentrating on product. Instead, your attention should be on building processes—habits—that coincidentally allow you to do the unpleasant tasks that need to be done. For example, let’s say you don’t like doing your math homework. So you put off working on the homework. It’s only five problems, you think. How hard could that be?

• The essential idea here is that the zombie, habitual part of your brain likes processes, because it can march mindlessly along.

• You may object that it is stressful being under the timer. But researchers have found something fascinating and counterintuitive. If you learn under mild stress, you can handle greater stress much more easily.

• Practice ignoring distractions. It is a far more powerful technique than trying to will yourself to not feel those distractions in the first place.

• “We trick ourselves into doing what we ought to be doing… . To a great degree, the highest-performing people I know are those who have installed the best tricks in their lives … The smart part of us sets up things for us to do that the not-so-smart part responds to almost automatically, creating behavior that produces high-performance results.”

• Research has confirmed that a special place devoted just to working is particularly helpful.

• A short, helpful guide to getting started with meditation is Buddha in Blue Jeans by Tai Sheridan.

• It’s normal to sit down with a few negative feelings about beginning your work. It’s how you handle those feelings that matters.

• Overall, Roberts has found that self-experimentation is extremely helpful in testing ideas as well as in generating and developing new hypotheses.

• Keep notes on when you don’t complete what you had intended to complete, what the cues are, and your zombie-mode habitual reaction to procrastination cues.

• Also, different people function better in certain environments—some need a busy coffee shop, while others need a quiet library.

• “Eat your frogs first thing in the morning.” Do the most important and most disliked jobs first, as soon as you wake up. This is incredibly effective.

• Notice my goal finish time for the day: 5:00 P.M. Doesn’t seem right, does it? But it is right, and it is one of the most important components of your daily planner-journal. Planning your quitting time is as important as planning your working time. Generally, I aim to quit at 5:00 P.M.,

• Chronic procrastinators, as it turns out, tend to see each act of procrastination as a unique, unusual act, a “just this one time” phenomenon that won’t be repeated again.

• These kinds of rare creative breakthroughs—relaxed moments of insight followed by mentally strenuous, all-out, late-night labor—are very different from a typical day of studying math and science.

• People who make a habit of getting their work done in binges are much less productive overall than those who generally do their work in reasonable, limited stints.

• Staying in the zone too long will send you toward burnout.3 An impending deadline can ratchet up stress levels, moving you into a zone where the stress hormones can kick in and assist in thinking. But relying on adrenaline can be a dangerous game, because once stress goes too high, the ability to think clearly can disappear.

• Especially when it comes to learning math and science, the bingeing excuse, “I do my best work under deadlines,” is simply not true.4

• Just as focused attention isn’t always good, seemingly nasty habits of procrastination aren’t always bad.

• You may be surprised to learn that the difference in the way that math experts (professors and graduate students) and math novices (undergraduate students) solve physics problems is that experts are slower to begin solving a problem.

• Pausing gives you time to access your library of chunks and allows your brain to make connections between a particular problem and the bigger picture.

• Emotions that goad you by saying, “Just do it, it feels right,” can be misleading in other ways.

• In choosing your career, for example, “Follow your passion” may be like deciding to marry your favorite movie star. It sounds great until reality rears its head. The proof is in the outcome: Over the past decades, students who have blindly followed their passion, without rational analysis of whether their choice of career truly was wise, have been more unhappy with their job choices than those who coupled passion with rationality.8

• We develop a passion for what we are good at. The mistake is thinking that if we aren’t good at something, we do not have and can never develop a passion for it.

• Many of my most creative and successful students have overcome ADHD and related attention difficulties using the types of tools I’ve outlined in this book. You can, too. If your attention is easily divided, you especially will benefit from tools that help keep you focused on a specific task for a short period of time. These tools include a planner-journal, a whiteboard by your door, a timer, and scheduling and timing apps and programs on your smartphone or computer.

• You’ve told me to use as little as possible of my willpower in dealing with procrastination. But shouldn’t I be using my willpower a lot so that I can strengthen it? Willpower is a lot like muscle. You have to use your muscles to strengthen and develop them over time. But at any given time, your muscles have only so much energy available. Developing and using willpower is a bit of a balancing act.10 This is why it’s often important to pick only one difficult thing at a time that requires self-discipline if you are trying to make changes.

• Our ancestors never needed a vast memory for names or numbers. But they did need a memory for how to get back home from the three-day deer hunt, or for the location of the plump blueberries on the rocky slopes to the south of camp. These evolutionary needs helped lock in superior “where things are and how they look” memory systems.

• The memory palace technique is useful for remembering unrelated items, such as a grocery list (milk, bread, eggs).

• See Joshua Foer’s masterful TED talk for a demonstration of the memory palace technique for remembering speeches.6 If you’d like to see how to apply these ideas directly to memorizing formulas, try out the SkillsToolbox .com website for a list of easy-to-remember visuals for mathematical symbols.7

• It’s one of those slo-mo camera shots where the runner is never quite able to reach the ribbon, just as we might not quite be able to get to the actual limit. Incidentally, the little book Calculus Made Easy, by Silvanus Thompson, has helped generations of students master the subject.

• Purists might sniff that using oddball memorization gimmicks isn’t really learning. But research has shown that students who use these types of tricks outperform those who don’t.

• The more you memorize using these innovative techniques, the more creative you become.

• One study of how actors memorize their scripts showed that they avoid verbatim memorization. Instead, they depend on an understanding of the characters’ needs and motivations in order to remember their lines.

• In baseball, for example, you don’t learn how to hit in one day. Instead, your body perfects your swing from plenty of repetition over a period of years.

• Remember—people learn by trying to make sense out of information they perceive. They rarely learn anything complex simply by having someone else tell it to them. (As math teachers say, “Math is not a spectator sport.”)

• Once you understand why you do something in math and science, you shouldn’t keep reexplaining the how.

• Such tightly controlled attention could use an occasional whiff of ADHD-like fresh air—the ability, in other words, to have your attention shift even if you don’t want it to shift.

• If you are one of those people who can’t hold a lot in mind at once—you lose focus and start daydreaming in lectures, and have to get to someplace quiet to focus so you can use your working memory to its maximum—well, welcome to the clan of the creative.

• Extremely smart people are more likely than people of normal intelligence to procrastinate because it always worked when they were growing up, which means they are less likely to learn certain critical life skills early on.

• There’s evidence that myelin sheaths, the fatty insulation that helps signals move more quickly along a neuron, often don’t finish developing until people are in their twenties. This may explain why teenagers often have trouble controlling their impulsive behavior—the wiring between intention and control areas isn’t completely formed.

• Good chunks form neural patterns that resonate, not only within the subject we’re working in, but with other subjects and areas of our lives. The abstraction helps you transfer ideas from one area to another.

• Once other people grasp that chunk, not only can they use it, but also they can more easily create similar chunks that apply to other areas in their lives—an important part of the creative process.

• But these simple analogies and metaphors can be powerful tools to help you use an existing neural structure as a scaffold to help you more rapidly build a new, more complex neural structure. As you begin to use this new structure, you will discover that it has features that make it far more useful than your first simplistic structure. These new structures can in turn become sources of metaphor and analogy for still newer ideas in very different areas.

• thoughts can be visual as well as verbal.

• “A mathematician who is not at the same time something of a poet will never be a full mathematician.”

• Simplifying is also important. Richard Feynman, the bongo-playing physicist we met earlier in this chapter, was famous for asking scientists and mathematicians to explain their ideas in a simple way so that he could grasp them. Surprisingly, simple explanations are possible for almost any concept, no matter how complex.

• Mathematicians feel that if you learn math the way they teach it, which centers on the abstract, chunked essence without a specific application in mind, you’ve captured skills that are easy for you to transfer to a variety of applications.

• In contrast, engineering, business, and many other professions all naturally gravitate toward math that focuses on their specific areas to help build student engagement and avoid the complaint of “When am I ever going to use this?” Concretely applied math also gets around the issue that many “real-world” word problems in mathematics textbooks are simply thinly disguised exercises.

• One of the most problematic aspects of procrastination—constantly interrupting your focus to check your phone messages, e-mails, or other updates—is that it interferes with transfer.

• Transfer is the ability to take what you learn in one context and apply it to something else.

• Approaching material with a goal of learning it on your own gives you a unique path to mastery. Often, no matter how good your teacher and textbook are, it’s only when you sneak off and look at other books or videos that you begin to see that what you learn through a single teacher or book is a partial version of the full, three-dimensional reality of the subject, which has links to still other fascinating topics that are of your choosing.

• Teacher-centered approaches, where the teacher is considered to be the one with the answers, may sometimes inadvertently foster a sense of helplessness about learning among students.

• Train yourself to get past the gulp stage and force yourself to reach out and ask questions—real and to-the-point questions, not questions meant to show off what you know.

• Be wary, however, of falling into “sticky student” syndrome. Kind teachers, in particular, can become magnets for students whose true needs involve desire for the ego-boosting attention of the instructor far more than answers to the actual questions being posed.

• As you will find when you reach the work world (if you haven’t already), many individuals are far more interested in affirming their own ideas and making themselves look good than they are in helping you.

• 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.

• 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

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

• 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.)

• 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.

• 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.

• when you start working problems, start first with what appears to be the hardest one. But steel yourself to pull away within the first minute or two if you get stuck or get a sense that you might not be on the right track. This does something exceptionally helpful. “Starting hard” loads the first, most difficult problem in mind, and then switches attention away from it. Both these activities can help allow the diffuse mode to begin its work.

• Another good tip for panicky test takers is to momentarily turn your attention to your breathing.

• 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?”

• 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.

• 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.

• 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.

• My advice to these students has always been the same advice I gave my daughter: “Get good at it, and then see if you still want to quit.”