Index
A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)
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.
“What surprises me is that we all know that mathematical equations are encrypted messages, for which you need to know the key if you want to crack the code and know what is represented. Nevertheless, we wonder why higher math is difficult to teach, and often blame the educational system or bad teachers. I think that it is similarly a bit misplaced to blame evolution
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.
You have to unlearn your erroneous older ideas even while you’re learning new ones.10
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.
The harder you push your brain to come up with something creative, the less creative your ideas will be.
But be wary of the idea that some people are “left-brain” or “right-brain” dominant—research indicates that is simply not true.
“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!”
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.
Understanding how your mind works helps you better understand the creative nature of some of your thoughts.
We learn a great deal from our failures in math and science.9 Know that you are making progress with each mistake you catch when trying to solve a problem—finding errors should give you a sense of satisfaction. Edison himself is said to have noted “I have not failed. I’ve just found 10,000 ways that won’t work.”10
Consistency over time is key!
blinking is a vital activity that provides another means of reevaluating a situation. Closing our eyes seems to provide a micropause that momentarily deactivates our attention and allows us, for the briefest of moments, to refresh and renew our consciousness and perspective.
“sleeping on it” before making major decisions is generally a good idea,18 and why taking vacations is important.
Learning is often paradoxical. The very thing we need in order to learn impedes our ability to learn. We need to focus intently to be able to solve problems—yet that focus can also block us from accessing the fresh approach we may need. Success is important, but critically, so is failure. Persistence is key—but misplaced persistence causes needless frustration.
It takes time to move information from working memory to long-term memory. 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.
you go even further and set it in mind that you want to dream about the material, it seems to improve your chances of dreaming about it still further.
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.
Chunking the information you deal with helps your brain run more efficiently. Once you chunk an idea or concept, you don’t need to remember all the little underlying details; you’ve got the main idea—the chunk—and that’s enough.
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. And there is a top-down “big picture” process that allows you to see where what you are learning fits in.9 Both processes are vital in gaining mastery over the material. Context is where bottom-up and top-down learning meet.
“Intention to learn is helpful only if it leads to the use of good learning strategies.”
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.
Underline Marking Sottolineare if so,how!?
You don’t want to wait too long for the recall practice, so that you have to start the reinforcement of the concept from scratch every time.
Merely glancing at the solution to a problem and thinking you truly know it yourself is one of the most common illusions of competence in learning.
A student is unlikely to make creative discoveries in any subject without a comprehensive set of facts and concepts at his or her command. There is no necessary conflict in learning concepts and facts and in thinking creatively; the two are symbiotic.”
Law of Serendipity comes to play: Lady Luck favors the one who tries.
What to Do If You Can’t Grasp It If you don’t understand a method presented in a course you are taking, stop and work backward. Go to the Internet and discover who first figured out the method or some of the earliest people to use it. Try to understand how the creative inventor arrived at the idea and why the idea is used—you can often find a simple explanation that gives a basic sense of why a method is being taught and why you would want to use it.
In the same amount of time, by simply practicing and recalling the material, students learned far more and at a much deeper level than they did using any other approach,
the retrieval process itself enhances deep learning and helps us begin forming chunks
when learning any new skill or discipline, you need plenty of varied practice with different contexts. This helps build the neural patterns you need to make the new skill a comfortable part of your way of thinking.
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.
You want your brain to become used to the idea that just knowing how to use a particular problem-solving technique isn’t enough—you also need to know when to use it.
There’s evidence that writing by hand helps get the ideas into mind more easily than if you type the answer.
“The dread of doing a task uses up more time and energy than doing the task itself.”
the better you get at something, the more you’ll find you enjoy it.
Procrastination is like addiction. It offers temporary excitement and relief from boring reality. It’s easy to delude yourself that the most profitable use of any given moment is surfing the web for information instead of reading the textbook or doing the assigned problems. You start to tell yourself stories.
“I often find that when I cannot bring myself to start something, if I go for a quick run or do something active first, when I come back to it, it is much easier to start.”
remember that the better you get at something, the more enjoyable it can become.
You can change your reality.
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.
Multitasking is like constantly pulling up a plant. This kind of constant shifting of your attention means that new ideas and concepts have no chance to take root and flourish. When you multitask while doing schoolwork, you get tired more quickly. Each tiny shift back and forth of attention siphons off energy. Although each attention switch itself seems tiny, the cumulative result is that you accomplish far less for your effort. You also don’t remember as well, you make more mistakes, and you are less able to transfer what little you do learn into other contexts.
If you feel muzzy or featherbrained as you’re trying to look away and recall a key idea, or you find yourself rereading the same paragraphs over and over again, try doing a few situps, pushups, or jumping jacks. A little physical exertion can have a surprisingly positive effect on your ability to understand and recall. Try doing something active now, before recalling the ideas of this chapter.
Generating (that is, recalling) the material helps you learn it much more effectively than simply rereading it.
THE LAW OF SERENDIPITY Remember, Lady Luck favors the one who tries. So don’t feel overwhelmed with everything you need to learn about a new subject. Instead, focus on nailing down a few key ideas. You’ll be surprised at how much that simple framework can help.
Highlighting should be avoided because, at least in my experience, it provides only an illusion of competence. Retrieval practice is far more powerful. Try to get the main ideas of each page you are reading cemented in your mind before you turn the page.
It’s normal to sit down with a few negative feelings about beginning your work. It’s how you handle those feelings that matters.
Writing the list before you go to sleep enlists your zombies to help you accomplish the items on the list the next day.
“Eat your frogs first thing in the morning.”
like pulling weeds or sweeping the kitchen. These aren’t generally my favorite kinds of tasks, but somehow, because I’m using them as diffuse-mode breaks, I often look forward to them. Mixing other tasks up with your learning seems to make everything more enjoyable and keeps you from prolonged and unhealthy bouts of sitting.
Planning your quitting time is as important as planning your working time.
It’s important to transform distant deadlines into daily ones. Attack them bit by bit. Big tasks need to be translated into smaller ones that show up on your daily task list. The only way to walk a journey of a thousand miles is to take one step at a time.
“When I was younger, I thought that if I didn’t get something immediately, it meant I would never be able to get it, or I wasn’t smart. That isn’t true at all, of course. Now I understand that it’s really important to get started on something early, leaving time for it to digest. This leads to stress-free understanding that makes learning a lot more enjoyable.”
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.
(I’m not kidding about the “close your eyes” part—remember, that can help disengage you from your previous thought patterns.9
It’s worth reemphasizing that world-class experts in a variety of disciplines reveal that their path to expertise wasn’t easy. They slogged through some tedious, difficult times to get to their current level of expertise where they can glide by and make it all look easy.
But memorization of key facts is essential since it is these facts that form the seeds for the creative process of chunking! The important lesson is that we must continue jiggling and playing mentally with things we have memorized in order to form chunks.”
See Joshua Foer’s masterful TED talk for a demonstration of the memory palace technique for remembering speeches.6
They help remind you that meaning is important for remembering, even if the initial meaning is wacky. In short, memorization techniques remind you to make what you learn in your life meaningful, memorable, and fun.
“Theories describe and deal with the world on its own terms and must stand on their own two feet. Models stand on someone else’s feet. They are metaphors that compare the object of their attention to something else that it resembles. Resemblance is always partial, and so models necessarily simplify things and reduce the dimensions of the world. . . . In a nutshell, theories tell you what something is; models tell you merely what something is like”
Metaphors are never perfect. But then, all scientific models are just metaphors, which means they also break down at some point.2 But never mind that—metaphors (and models!) are vitally important in giving a physical understanding of the central idea behind the mathematical or scientific process or concept that you are trying to understand. Interestingly, metaphors and analogies are useful for getting people out of Einstellung—being blocked by thinking about a problem in the wrong way. For
The more you memorize using these innovative techniques, the more creative you become.
Just before you to go sleep, review something mentally that you are trying to learn. To boost this process, review it yet again when you first wake up.
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.”)
As Albert Einstein noted, “There are only two ways to live your life. One is as though nothing is a miracle. The other is as if everything is.”
we can make significant changes in our brain by changing how we think.
A synthesis—an abstraction, chunk, or gist idea—is a neural pattern. 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.
sometimes, things that look rough and messy—like clouds and shorelines—have a degree of order to them. Visual complexity can be created from simple rules,
physicist Jeffrey Prentis compares how a brand-new student of physics and a mature physicist look at equations.6 The equation is seen by the novice as just one more thing to memorize in a vast collection of unrelated equations. More advanced students and physicists, however, see with their mind’s eye the meaning beneath the equation, including how it fits into the big picture, and even a sense of how the parts of the equation feel.
Symbols and equations, in other words, have a hidden text that lies beneath them—a meaning that becomes clear once you are more familiar with the ideas.
Einstein’s theories of relativity arose not from his mathematical skills (he often needed to collaborate with mathematicians to make progress) but from his ability to pretend. He imagined himself as a photon moving at the speed of light, then imagined how a second photon might perceive him. What would that second photon see and feel?
It may seem silly to stage a play in your mind’s eye and imagine the elements and mechanisms you are studying as living creatures, with their own feelings and thoughts. But it is a method that works—it brings them to life and helps you see and understand phenomena that you couldn’t intuit when looking at dry numbers and formulas.
Multitasking during the learning process means you don’t learn as deeply—this can inhibit your ability to transfer what you are learning.
Persistence is often more important than intelligence.
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.
students learn best when they themselves are actively engaged in the subject instead of simply listening to someone else speak.
Failures are better teachers than successes because they cause you to rethink your approach.
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.
“the left hemisphere maintains our current beliefs while the right hemisphere evaluates and updates those beliefs when appropriate. Belief evaluation is thus dependent on interhemispheric interaction.”
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.
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
As Alan Baddeley and colleagues (2009, pp. 148–149) note: “We are not lacking in ways of defending ourselves against challenges to our self-esteem. We readily accept praise but tend to be skeptical of criticism, often attributing criticism to prejudice on the part of the critic. We are inclined to take credit for success when it occurs but deny responsibility for failure. If this stratagem fails, we are rather good at selectively forgetting failure and remembering success and praise.”
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.
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.
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.
there are ways to work with others that require only minimal interaction if you don’t like working in groups.”
equations are NOT merely expressions you plug numbers into to get other numbers. Equations tell a story about how the physical world works. For me, the key to understanding an equation in physics is to see the underlying story. A qualitative understanding of an equation is more important than getting quantitatively correct numbers out of it.
This approach works for some people, mostly because anything works for some people.
When the test is handed out to you, first take a quick look to get a sense of what it involves. (You should do this in any case.) Keep your eye out for what appears to be the hardest problem. Then 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.
“With my students, I talk about good worry and bad worry. Good worry helps provide motivation and focus while bad worry simply wastes energy.”
If you shift your thinking from “this test has made me afraid” to “this test has got me excited to do my best!” it can make a significant improvement in your performance.
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.
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.
Reshaping your brain is under your control. The key is patient persistence—working knowledgeably with your brain’s strengths and weaknesses.
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.
This work will pay off both for you and those you love!