A MIND FOR NUMBERS by Barbara Oakley

• Surprisingly, your brain can also work on a problem even while you are sleeping and are not aware of anything. But it does this only if you concentrate on trying to solve the problem before falling asleep.
• What’s surprising is that a lot of learners use ineffective and inefficient strategies.
• When we read material over and over, the material becomes familiar and fluent, meaning it is easy for our minds to process. We then think that this easy processing is a sign that we have learned something well, even though we have not.
• You’ll be surprised at how spending a minute or two glancing ahead before you read in depth will help you organize your thoughts.
• 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.
• 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.
• (Metaphors are powerful tools for learning in math and science.)
• 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,
• As long as we are consciously focusing on a problem, we are blocking the diffuse mode.
• “Befuddlement is a healthy part of the learning process.
• The learning process is all about working your way out of confusion.
• Articulating your question is 80 percent of the battle.
• Don’t worry about finishing the task—just worry about working on it.
• 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.
• For most people, shifting from focused to diffuse mode happens naturally if you distract yourself and then allow a little time to pass.
• The key is to do something else until your brain is consciously free of any thought of the problem.
• 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.
• Enlisting the diffuse mode helps you learn at a deep and creative level.
• 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.
• 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.
• Mistakes are inevitable.
• Keep your working sessions short.
• Remember, when you take breaks, your diffuse mode is still working away in the background.
• Sleep is probably the most effective and important factor in allowing your diffuse mode to tackle a difficult problem.
• According to recent research, 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.
• The resting times between your focused-mode efforts should be long enough to get your conscious mind completely off the problem you’re working on.
• A good rule of thumb, when you are first learning new concepts, is not to let things go untouched for longer than a day.
• Rising frustration is usually a good time-out signal for you, signaling that you need to shift to diffuse mode.
• Learning often means making sense of what we’ve ingested, and for that, we need to have ingested something.
• Success is important, but critically, so is failure. Persistence is key—but misplaced persistence causes needless frustration.
• Working memory is the part of memory that has to do with what you are immediately and consciously processing in your mind.
• It’s now widely believed that the working memory holds only about four chunks of information.
• We tend to automatically group memory items into chunks, so it seems our working memory is bigger than it actually is.
• Generally, you can hold about four items in your working memory,
• When you master a technique or concept in math or science, it occupies less space in your working memory. This frees your mental thinking space so that it can more easily grapple with other ideas, as shown on the right.
• Research has shown that when your brain first puts an item of information in long-term memory, you need to revisit it a few times to increase the chances you’ll later be able to find it when you need it.
• It takes time to move information from working memory to long-term memory. To help with this process, use a technique called spaced repetition.
• Studies have shown that sleep is a vital part of memory and learning.
• If 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.
• If you’re tired, it’s often best to just go to sleep and get up a little earlier the next day, so your reading is done with a better-rested brain.
• A sleep-deprived brain simply can’t make the usual connections you make during normal thinking processes.
• Use the focused mode to first start grappling with concepts and problems in math and science.
• After you’ve done your first hard focused work, allow the diffuse mode to take over. Relax and do something different!
• When frustration arises, it’s time to switch your attention to allow the diffuse mode to begin working in the background.
• It’s best to work at math and science in small doses—a little every day. This gives both the focused and diffuse modes the time they need to do their thing so you can understand what you are learning. That’s how solid neural structures are built.
• If procrastination is an issue, try setting a timer for twenty-five minutes and focusing intently on your task without allowing yourself to be drawn aside by text messages, web surfing, or other attractive distractions.
• There are two major memory systems: Working memory—like a juggler who can keep only four items in the air. Long-term memory—like a storage warehouse that can hold large amounts of material, but needs to be revisited occasionally to keep the memories accessible.
• Spaced repetition helps move items from working memory to long-term memory.
• Sleep is a critical part of the learning process. It helps you: Make the neural connections needed for normal thinking processes—which is why sleep the night before a test is so important. Figure out tough problems and find meaning in what you are learning. Strengthen and rehearse the important parts of what you are learning and prune away trivialities.
• Creativity is a numbers game: The best predictor of how many creative works we produce in our lifetime is . . . the number of works we produce.
• Criticism makes us better.
• Focusing your attention to connect parts of the brain is an important part of the focused mode of learning.
• Chunks are pieces of information that are bound together through meaning.
• One of the first steps toward gaining expertise in math and science is to create conceptual chunks—mental leaps that unite separate bits of information through meaning.
• Chunking the information you deal with helps your brain run more efficiently.
• Chunking is the mental leap that helps you unite bits of information together through meaning.
• The first step in chunking, then, is to simply focus your attention on the information you want to chunk.
• When you first begin to learn something, you are making new neural patterns and connecting them with preexisting patterns that are spread through many areas of the brain.
• The second step in chunking is to understand the basic idea you are trying to chunk,
• Do not confuse the “aha!” of a breakthrough in understanding with solid expertise!
• The third step to chunking is gaining context so you see not just how, but also when to use this chunk.
• Learn the major concepts or points first.
• Attempting to recall the material you are trying to learn—retrieval practice—is far more effective than simply rereading the material.
• When you have the book (or Google!) open right in front of you, it provides the illusion that the material is also in your brain. But it’s not.
• 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.
• Using recall—mental retrieval of the key ideas—rather than passive rereading will make your study time more focused and effective.
• Try to touch again on something you’re learning within a day, especially if it’s new and rather challenging.
• It’s worth knowing that well-designed electronic flash card systems, such as Anki, have built into them the appropriate spaced repetition time to optimize the rate of learning new material.
• 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.
• The ability to combine chunks in novel ways underlies much of historical innovation.
• Lady Luck favors the one who tries.
• Most difficult problems are solved through intuition, because they make a leap away from what you are familiar with.
• If you don’t understand a method presented in a course you are taking, stop and work backward.
• Strengthening an initial learning pattern within a day after you first begin forming it is important.
• 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.
• 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, including simply rereading the text a number of times or drawing concept maps that supposedly enriched the relationships in the materials under study.
• The retrieval process itself enhances deep learning and helps us begin forming chunks.
• If you try to build connections between chunks before the basic chunks are embedded in the brain, it doesn’t work as well.
• The fact is, when learning any new skill or discipline, you need plenty of varied practice with different contexts.
• 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.
• Continuing the study or practice after it is well understood is called overlearning.
• 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.
• Just knowing how to use a particular problem-solving technique isn’t enough—you also need to know when to use it.
• EMPHASIZE INTERLEAVING INSTEAD OF OVERLEARNING
• Doing too many problems of the same kind in immediate succession provides diminishing returns.
• There’s evidence that writing by hand helps get the ideas into mind more easily than if you type the answer.
• Beware—a common illusion of competence is to continue practicing a technique you know, simply because it’s easy and it feels good to successfully solve problems.
• Practice helps build strong neural patterns—that is, conceptual chunks of understanding.
• Practice gives you the mental fluidity and agility you need for tests.
• Chunks are best built with: Focused attention. Understanding of the basic idea. Practice to help you gain big-picture context.
• Simple recall—trying to remember the key points without looking at the page—is one of the best ways to help the chunking process along.
• To remember the key points without looking at the page—is one of the best ways to help the chunking process along.
• Unlike procrastination, which is easy to fall into, willpower is hard to come by because it uses a lot of neural resources.
• You shouldn’t waste willpower on procrastination except when absolutely necessary!
• For most people, learning math and science depends on two things: brief study sessions where the neural “bricks” are laid, and time in between for the mental mortar to dry.
• We procrastinate about things that make us feel uncomfortable.
• The dread of doing a task uses up more time and energy than doing the task itself.
• Procrastination is a single, monumentally important “keystone” bad habit.
• 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.
• We procrastinate about things that make us feel uncomfortable. But what makes us feel good temporarily isn’t necessarily good for us in the long run.
• Procrastination can be like taking tiny amounts of poison. It may not seem harmful at the time. But the long-term effects can be very damaging.
• Habit is an energy saver for us. It allows us to free our mind for other types of activities.
• Habits have four parts:
• The Cue: This is the trigger that launches you into “zombie mode.”
• The Routine: This is your zombie mode—the routine, habitual response your brain is used to falling into when it receives the cue.
• The Reward: Habits develop and continue because they reward us—give us a dollop of pleasure.
• The Belief: Habits have power because of your belief in them.
• To change a habit, you’ll need to change your underlying belief.
• The trick to overwriting a habit is to look for the pressure point—your reaction to a cue. The only place you need to apply willpower is to change your reaction to the cue.
• If you protect your routine, eventually it will protect you.
• Don’t try to change everything at once.
• The Pomodoro technique—the twenty-five-minute timer—can be especially helpful in shifting your reaction to cues.
• Remember, habits are powerful because they create neurological cravings. It helps to add a new reward if you want to overcome your previous cravings. Only once your brain starts expecting the reward will the important rewiring take place that will allow you to create new habits.
• The most important part of changing your procrastination habit is the belief that you can do it.
• 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.
• The essential idea here is that the zombie, habitual part of your brain likes processes, because it can march mindlessly along.
• If you learn under mild stress, you can handle greater stress much more easily.
• Focusing on process, not product, is important in avoiding procrastination.
• It is the consistent, daily time you spend getting into the flow of your studies that matters most.
• Focus on doing a Pomodoro—a twenty-five-minute timed work session—not on completing a task.
• One helpful tip is to just get started.
• One of the single most important pieces of advice I can give you on dealing with procrastination is to ignore distractions!
• Each tiny shift back and forth of attention siphons off energy.
• Practice ignoring distractions.
• A little bit of work on something that feels painful can ultimately be very beneficial.
• Habits such as procrastination have four parts: The cue The routine The reward The belief
• Change a habit by responding differently to a cue, or even avoiding that cue altogether. Reward and belief make the change long-lasting.
• A habit by responding differently to a cue, or even avoiding that cue altogether.
• Focus on the process (the way you spend your time) instead of the product (what you want to accomplish).
• Use the twenty-five-minute Pomodoro to stay productive for brief periods. Then reward yourself after each successful period of focused attention.
• Be sure to schedule free time to nurture your diffuse mode.
• Mental contrasting is a powerful motivating technique—think about the worst aspects of your present or past experiences and contrast these with the upbeat vision of your future.
• Multitasking means that you are not able to make full, rich connections in your thinking, because the part of your brain that helps make connections is constantly being pulled away before neural connections can be firmed up.
• A little physical exertion can have a surprisingly positive effect on your ability to understand and recall.
• Procrastination is the death of success.
• Preparation is key to success.
• Practice is where you are supposed to fail.
• Learning fundamental concepts of math and science can be a lot easier than learning subjects that require a lot of rote memorization.
• STEPS TO BUILDING A POWERFUL CHUNK
• 1. Work a key problem all the way through on paper.
• 2. Do another repetition of the problem, paying attention to the key processes.
• 3. Take a break.
• 4. Sleep.
• 5. Do another repetition.
• 6. Add a new problem.
• 7. Do “active” repetitions.
• Keep your focus on the parts of the problem that are the most difficult for you. This continued focus on the hard stuff is called “deliberate practice.”
• Generating (that is, recalling) the material helps you learn it much more effectively than simply rereading it.
• Research has shown that the more effort you put into recalling material, the deeper it embeds itself into your memory.
• Recall, not simple rereading, is the best form of deliberate practice in study.
• Remember, retrieval practice is one of the most powerful forms of learning. It is far more productive than simply rereading material.
• Chunking compresses your knowledge and makes room in your working memory for those pieces so you don’t go into mental overload so easily.
• Testing in itself is a powerful learning experience.
• A key to building mental flexibility and expertise is to build your library of chunked solution patterns.
• Chunking means integrating a concept into one smoothly connected neural thought pattern.
• Chunking helps increase the amount of working memory you have available.
• Building a chunked library of concepts and solutions helps build intuition in problem solving.
• When you are building a chunked library, it’s important to keep deliberate focus on some of the toughest concepts and aspects of problem solving.
• Occasionally you can study hard and fate deals a bad hand. But remember the Law of Serendipity: If you prepare well by practicing and building a good mental library, you will find that luck will be increasingly on your side. In other words, you guarantee failure if you don’t try, but those who consistently give it a good effort will experience many more successes.
• Research has confirmed that a special place devoted just to working is particularly helpful.
• Another trick involves using meditation to help you learn to ignore distracting thoughts.
• Self-experimentation can be used by non-experts to (a) see if the experts are right and (b) learn something they don’t know.
• The best way for you to gain control of your habits is simple: Once a week, write a brief weekly list of key tasks. Then, each day, write a list of the tasks that you can reasonably work on or accomplish. Try to write this daily task list the evening before.
• Do the most important and most disliked jobs first, as soon as you wake up. This is incredibly effective.
• Planning your quitting time is as important as planning your working time.
• It’s important to transform distant deadlines into daily ones.
• Big tasks need to be translated into smaller ones that show up on your daily task list.
• 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.
• A little procrastination here and there is unavoidable. But to be effective in learning math and science, you must master your habits.
• But remember the Law of Serendipity: Lady Luck favors the one who tries. Planning well is part of trying. Keep your eye on the goal, and try not to get too unsettled by occasional roadblocks.
• A simple timer plus pen and paper are often the most straightforward tools to avoid procrastination, but you can also make use of technology.
• Mental tricks can be powerful tools.
• Put yourself in a place with few interruptions, such as a library, to help with procrastination.
• Practice ignoring distracting thoughts by simply letting them drift past.
• If your attitude is troubled, reframe your focus to shift attention from the negative to the positive.
• Realize it’s perfectly normal to sit down with a few negative feelings about beginning your work.
• Planning your life for “playtime” is one of the most important things you can do to prevent procrastination, and one of the most important reasons to avoid procrastination.
• At the heart of procrastination prevention is a reasonable daily to-do list, with a weekly once-over to ensure you’re on track from a big-picture perspective.
• Write your daily task list the evening before.
• Eat your frogs first.
• 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.
• A healthy form of procrastination entails learning to pause and reflect before jumping in and accomplishing something. You are learning to wait wisely. There is always something to be done. Prioritizing allows you to gain big-picture context for your decision making. Sometimes waiting allows a situation to resolve itself.
• 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.
• 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.
• All you need to do is keep working to improve your process.
• Keep a planner-journal so you can easily track when you reach your goals and observe what does and doesn’t work.
• Commit yourself to certain routines and tasks each day.
• Write your planned tasks out the night before, so your brain has time to dwell on your goals to help ensure success.
• Arrange your work into a series of small challenges. Always make sure you (and your zombies!) get lots of rewards. Take a few minutes to savor the feelings of happiness and triumph.
• Deliberately delay rewards until you have finished a task.
• Watch for procrastination cues.
• Put yourself in new surroundings with few procrastination cues, such as the quiet section of a library.
• Obstacles arise, but don’t make a practice of blaming all your problems on external factors. If everything is always somebody else’s fault, it’s time to start looking in the mirror.
• Gain trust in your new system. You want to work hard during times of focused concentration—and also trust your system enough that when it comes time to relax, you actually relax without feelings of guilt.
• Have backup plans for when you still procrastinate. No one is perfect, after all.
• Eat your frogs first.
• This is a firm reminder that it’s best to focus without distraction if you really want to put something into memory.
• To begin tapping into your visual memory system, try making a very memorable visual image representing one key item you want to remember.
• The more neural hooks you can build by evoking the senses, the easier it will be for you to recall the concept and what it means.
• The memory palace technique involves calling to mind a familiar place—like the layout of your house—and using it as a sort of visual notepad where you can deposit concept-images that you want to remember.
• Songs that help cement ideas in your mind are related to the memory palace technique in that they also make preferential use of the brain’s right hemisphere.
• The memory palace technique—placing memorable nudges in a scene that is familiar to you—allows you to dip into the strength of your visual memory system.
• Learning to use your memory in a more disciplined, yet creative manner helps you learn to focus your attention, even as you create wild, diffuse connections that build stronger memories.
• By memorizing material you understand, you can internalize the material in a profound way. And you are reinforcing the mental library you need to become a genuine master of the material.
• One of the best things you can do to not only remember but understand concepts in math and science is to create a metaphor or analogy for it—often, the more visual, the better. A metaphor is just a way of realizing that one thing is somehow similar to another.
• The little book Calculus Made Easy, by Silvanus Thompson, has helped generations of students master the subject.
• Sometimes textbooks can get so focused on all the details that you lose sight of the most important, big-picture concepts.
• Metaphors are never perfect. But then, all scientific models are just metaphors, which means they also break down at some point. 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.
• Metaphors and visualization—being able to see something in your mind’s eye—have been uniquely powerful in helping the scientific and engineering world move forward.
• Focusing your attention brings something into your temporary working memory. But for that “something” to move from working memory to long-term memory, two things should happen: the idea should be memorable (there’s a gigantic flying mule braying f = ma on my couch!), and it must be repeated.
• Much of what goes on around you is basically trivial.
• Repetition is important; even when you make something memorable, repetition helps get that memorable item firmly lodged in long-term memory.
• Writing appears to help you to more deeply encode (that is, convert into neural memory structures) what you are trying to learn.
• Remember that sleep is when your mind repeats patterns and pieces together solutions.
• Interestingly, one of the best ways to remember people’s names is to simply try to retrieve the people’s names from memory at increasing time intervals after first learning the name.
• Another key to memorization is to create meaningful groups that simplify the material.
• It’s much easier to remember numbers by associating them with memorable events.
• If you’re memorizing something commonly used, see whether someone’s come up with a particularly memorable memory trick by searching it out online.
• Although there is little research in this area,11 many educators have observed that there seems to be a muscle memory related to writing by hand.
• If you really want to boost your memory as well as your general ability to learn, it seems one of the best ways to do it is to exercise.
• By using mental pictures instead of words to remember things, you can leap more easily into expert status.
• Learning to process ideas visually in math and science is a powerful way to become a master of the material.
• Memory tricks allow people to expand their working memory with easy access to long term memory.
• Understanding also helps a lot with the memorization process.
• Your childlike creativity is still there—you just need to reach out to it.
• Metaphors can help you learn difficult ideas more quickly.
• Repetition is critical in allowing you to firm up what you want to remember before the ideas fade away.
• Meaningful groups and abbreviations can allow you to simplify and chunk what you are trying to learn so you can store it more easily in memory.
• Stories—even if they are just used as silly memory tricks—can allow you to more easily retain what you are trying to learn.
• Writing and saying what you are trying to learn seems to enhance retention.
• Exercise is powerfully important in helping your neurons to grow and make new connections.
• 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.
• 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.
• At some point, self-consciously “understanding” why you do what you do just slows you down and interrupts flow, resulting in worse decisions.
• Teachers and professors can inadvertently get too caught up in following rules.
• Once you understand why you do something in math and science, you shouldn’t keep reexplaining the how.
• Being smarter often equates to having a larger working memory.
• Your ability to solve complex problems may make you overthink simple problems, going for the convoluted answer and overlooking the simple, more obvious solution.
• Research has shown that smart people can have more of a tendency to lose themselves in the weeds of complexity.
• It is the practice—particularly deliberate practice on the toughest aspects of the material—that can help lift average brains into the realm of those with more “natural” gifts.
• Smarter people also sometimes struggle because they can so easily imagine every complexity, good and bad.
• 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.
• At some point, after you’ve got chunked material well in hand (and in brain), you start to let go of conscious awareness of every little detail and do things automatically.
• It may seem intimidating to work alongside other students who grasp material more quickly than you do. But “average” students can sometimes have advantages when it comes to initiative, ability to get things done, and creativity.
• Part of the key to creativity is to be able to switch from full focused concentration to the relaxed, daydreamy diffuse mode.
• Focusing too intently can inhibit the solution you are seeking—like trying to hammer a screw because you think it’s a nail. When you are stuck, sometimes it’s best to get away from a problem for a while and move on to something else, or to simply sleep on it.
• It seems people can enhance the development of their neuronal circuits by practicing thoughts that use those neurons.
• We can make significant changes in our brain by changing how we think.
• Metaphors and physical analogies also form chunks that can allow ideas even from very different areas to influence one another.
• One important key to learning swiftly in math and science is to realize that virtually every concept you learn has an analogy—a comparison—with something you already know.
• Brains mature at different speeds. Many people do not develop maturity until their midtwenties.
• Some of the most formidable heavyweights in science started out as apparently hopeless juvenile delinquents.
• One trait that successful professionals in science, math, and technology gradually learn is how to chunk—to abstract key ideas.
• Metaphors and physical analogies form chunks that can allow ideas from very different areas to influence one another.
• Regardless of your current or intended career path, keep your mind open and ensure that math and science are in your learning repertoire. This gives you a rich reserve of chunks to help you be smarter about your approach to all sorts of life and career challenges.
• Simplify and Personalize Whatever You Are Studying
• One of the most important things we can do when we are trying to learn math and science is to bring the abstract ideas to life in our minds.
• Simplifying is also important.
• Surprisingly, simple explanations are possible for almost any concept, no matter how complex. When you cultivate simple explanations by breaking down complicated material to its key elements, the result is that you have a deeper understanding of the material.
• Learning expert Scott Young has developed this idea in what he calls the Feynman technique, which asks people to find a simple metaphor or analogy to help them grasp the essence of an idea.11
• You’ll be surprised to see how often understanding arises as a consequence of attempts to explain to others and yourself, rather than the explanation arising out of your previous understanding.
• Transfer is the ability to take what you learn in one context and apply it to something else.
• Equations are just ways of abstracting and simplifying concepts. This means that equations contain deeper meaning, similar to the depth of meaning found in poetry.
• Equations are just ways of abstracting and simplifying concepts.
• Your “mind’s eye” is important because it can help you stage plays and personalize what you are learning about.
• It’s important to grasp the chunked essence of a mathematical concept, because then it’s easier to transfer and apply that idea in new and different ways.
• 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.
• Approaching material with a goal of learning it on your own gives you a unique path to mastery.
• Research has shown that students learn best when they themselves are actively engaged in the subject instead of simply listening to someone else speak.
• College isn’t the only way to learn.
• Taking responsibility for your own learning is one of the most important things you can do.
• Truly great teachers make the material seem both simple and profound, set up mechanisms for students to learn from each other, and inspire students to learn on their own.
• There will always be those who criticize or attempt to undermine any effort or achievement you make.
• The greater your achievement, the more other people will sometimes attack and demean your efforts.
• Failure is not so terrible. Analyze what you did wrong and use it to correct yourself to do better in the future. Failures are better teachers than successes because they cause you to rethink your approach.
• Many individuals are far more interested in affirming their own ideas and making themselves look good than they are in helping you.
• Use your natural contrariness to defy the always-present prejudices from others about what you can do.
• Learning on your own is one of the deepest, most effective ways to approach learning: It improves your ability to think independently. It can help you answer the strange questions that teachers sometimes throw at you on tests.
• In learning, persistence is often far more important than intelligence.
• Train yourself to occasionally reach out to people you admire. You can gain wise new mentors who, with a simple sentence, can change the course of your future. But use your teachers’ and mentors’ time sparingly.
• If you aren’t very fast at grasping the essentials of whatever you are studying, don’t despair. Surprisingly often, “slower” students are grappling with fundamentally important issues that quicker students miss. When you finally get what’s going on, you can get it at a deeper level.
• People are competitive as well as cooperative. There will always be those who criticize or attempt to undermine any effort or achievement you make. Learn to deal dispassionately with these issues.
• If you go off track early on, it doesn’t matter if the rest of your work is correct—your answer is still wrong.
• Good learners vet their work to ensure that it makes sense.
• If you or one of your study buddies thinks something is wrong in your understanding, it’s important to be able to plainly say so, and to hash out why it’s wrong without worrying about hurt feelings.
• The focused mode can allow you to make critical errors even though you feel confident you’ve done everything correctly. Rechecking your work can allow you to get a broader perspective on it, using slightly different neural processes that can allow you to catch blunders.
• Working with others who aren’t afraid to disagree can: Help you catch errors in your thinking. Make it easier for you to think on your feet and react well in stressful situations. Improve your learning by ensuring that you really understand what you are explaining to others and reinforcing what you know. Build important career connections and help steer you toward better choices.
• Criticism in your studies, whether you are giving or receiving it, shouldn’t be taken as being about you. It’s about what you are trying to understand.
• It is easiest of all to fool yourself.
• Testing is itself an extraordinarily powerful learning experience.
• If you compare how much you learn by spending one hour studying versus one hour taking a test on that same material, you will retain and learn far more as a result of the hour you spent taking a test.
• Testing, it seems, has a wonderful way of concentrating the mind.
• 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.
• 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.
• 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?”
• Sometimes, as we’ve discovered, 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.
• The Pomodoro—a brief, timed period of focused attention—is a powerful tool in diverting the well-meaning zombies of your habitual responses.
• We now know that deep, practiced internalization of well-understood chunks is essential to mastering math and science.
• 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.
• 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.
• 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.
• 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.
• 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 it up and work on different types of problems. This teaches you both how and when to use a technique.
• 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.
• 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?
• 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.
• 9. Eat your frogs first. Do the hardest thing earliest in the day, when you are fresh.
• 10. Make a mental contrast. Imagine where you’ve come from and contrast that with the dream of where your studies will take you.
• An ability to recall—to generate the ideas from inside yourself—is one of the key indicators of good learning.
• Handwriting builds stronger neural structures in memory than typing.
• 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?
• The additional effort of speaking and writing allows you to more deeply encode (that is, convert into neural memory structures) what you are learning.
• Unless you can prove that the material is moving into your brain by recalling the main ideas without looking at the page, rereading is a waste of time.
• Highlighting your text can fool your mind into thinking you are putting something in your brain, when all you’re really doing is moving your hand.
• You need to be able to solve a problem step-by-step, without looking at the solution.
• Your brain is like a muscle—it can handle only a limited amount of exercise on one subject at a time.
• Every tug of interrupted attention pulls out tiny neural roots before they can grow.
• Your brain pieces together problem-solving techniques when you sleep, and it also practices and repeats whatever you put in mind before you go to sleep.
• If you don’t get a good sleep before a test, NOTHING ELSE YOU HAVE DONE WILL MATTER.