Your brain has amazing abilities, but it did not come with an instruction manual.

Henri Poincaré was a nineteenth-century mathematician who once described how he cracked a difficult mathematical problem that he had been intensively working on for weeks without success.
He took a vacation. As he was getting on a bus in the south of France, the answer to the problem suddenly came to him, unbidden, from a part of his brain that had continued to work on the problem while he was enjoying his vacation. He knew he had the right solution even though he did not write down the details until he later returned to Paris.

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.
The intense effort before a vacation or falling asleep is important for priming your brain; otherwise it will work on some other problem. your brain will work just as hard at solving social problems as on math and science problems.

This wonderful guide will enrich both your learning and your life.

—Terrence J. Sejnowski


You will learn the simplest, most effective, and most efficient techniques researchers know about how to learn. And you’ll have fun while you’re doing it.

What’s surprising is that a lot of learners use ineffective and inefficient strategies.
we have surveyed college students about their learning. They most commonly use the strategy of repeated reading—simply reading through books or notes over and over. We and other researchers have found that this passive and shallow strategy often produces minimal or no learning. We call this “labor in vain”—students are putting in labor but not getting anywhere.

We don’t engage in passive rereading because we are dumb or lazy. We do it because we fall prey to a cognitive illusion. 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.

We’re on the edge of an explosion in knowledge about how to learn most effectively. In this new world of insight, you’ll find A Mind for Numbers to be an indispensable guide.

—Jeffrey D. Karpicke

Note to the reader

People who work professionally with math and science often spend years discovering effective learning techniques.
I’ve written this book to lay out these simple techniques so that you can immediately begin using them. What takes years for practitioners to discover is now at your fingertips.
Using these approaches, no matter what your skill levels in math and science, you can change your thinking and change your life.

It’s for parents whose children are either falling off the math track or trying to rocket toward math and science stardom.
It’s for the frazzled nine-to-five worker who hasn’t been able to pass an important certification test, and for the night-shift convenience store clerk who has dreamed of becoming a nurse—or even a doctor.
It’s for the growing army of homeschoolers.
It’s for teachers and professors—not only in math, science, engineering, and technology, but also in fields such as education, psychology, and business.
It’s for the retiree who finally has the time to embrace new knowledge in computing, for example, or the intricacies of great cooking.
And it’s for readers of all ages who love to learn a little about everything.

—Barbara Oakley,

{ 1 } Open the door

What are the odds you’d open your refrigerator door and find a zombie in there, knitting socks?
The odds are about the same that a touchy-feely, language-oriented person like me would end up as a professor of engineering.
Growing up, I hated math and science. I flunked my way through high school math and science courses, and only started studying trigonometry—remedial trigonometry—when I was twenty-six years old.

As a youngster, even the simple concept of reading a clock face didn’t seem to make sense to me. Why should the little hand point toward the hour? Shouldn’t it be the big hand, since the hour was more important than the minute?

I didn’t realize then that there were simple mental tricks that could have brought math into focus for me, tricks that are helpful not only for people who are bad at math, but also for those who are already good at it. I didn’t understand that my type of thinking is typical of people who believe they can’t do math and science.

I enlisted in the army because they would actually pay me to learn another language. I did so well in studying Russian (a language I’d selected on a whim) that an ROTC scholarship came my way. I graduated with honors. Russian flowed like warm syrup—my accent was so good that I found myself on occasion mistakenly taken for a native speaker.
I spent lots of time gaining this expertise—the better I got, the more I enjoyed what I was doing. And the more I enjoyed what I was doing, the more time I spent on it.

Could I retool my brain from mathphobe to math lover? From technophobe to technogeek?
There couldn’t possibly be anything more foreign to my personality than mastering math and science.
It became a challenge—an irresistible challenge.
It wasn’t easy.
The younger students around me mostly seemed to have a natural knack for seeing the solutions, while I was stumbling into walls.

But I began to catch on. Part of my original problem, I found, was that I had been putting my effort forth in the wrong way—like trying to lift a piece of lumber when you’re standing on it. I began to pick up little tricks about not only how to study but when to quit.
The higher I went, the better I did. By the time I reached my doctoral studies, I was breezing by with perfect grades.

the brain is designed to do extraordinary mental calculations. We do them every time we catch a ball, or rock our body to the beat of a song, or maneuver our car around a pothole in the road.

In fact, because some of these learning and teaching methods are derided by ordinary instructors, superstar teachers sometimes divulged their teaching and learning secrets to me with embarrassment, unaware that many other top instructors shared similar approaches.

If you’re already good at numbers or science, the insights in this book can help make you better. They will broaden your enjoyment, creativity, and equation-solving elegance.
If you’re simply convinced you don’t have a knack for numbers or science, this book may change your mind.

{ 2 } Easy does it: Why Trying Too Hard Can Sometimes Be Part of the Problem

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

The man on the right is legendary chess grand master Garry Kasparov. The boy on the left is thirteen-year-old Magnus Carlsen. Carlsen has just wandered away from the board during the height of a speed chess game, where little time is given to think about moves or strategy. That’s a little like casually deciding to do a backflip while walking a tightrope across Niagara Falls.
Yes, Carlsen was psyching out his opponent. Rather than obliterating the upstart youngster, the flustered Kasparov played to a draw.
Gaining insight into Carlsen’s approach can help us understand how the mind learns math and science.

Prime Your Mental Pump:
As you first begin looking at a chapter or section of a book that teaches concepts of math or science, it helps to take a “picture walk” through the chapter, glancing not only at the graphics, diagrams, and photos, but also at the section headings, summary, and even questions at the end of the chapter, if the book has them.
This seems counterintuitive—you haven’t actually read the chapter yet, but it helps prime your mental pump.
You’ll be surprised at how spending a minute or two glancing ahead before you read in depth will help you organize your thoughts. You’re creating little neural hooks to hang your thinking on, making it easier to grasp the concepts.

Focused versus Diffuse Thinking

highly attentive states and more relaxed resting state networks. the focused mode and diffuse mode.
It seems you frequently switch back and forth between these two modes in your day-to-day activities.
The diffuse mode does seem to be able to work quietly in the background on something you are not actively focusing on.

Learning involves a complex flickering of neural processing among different areas of the brain, as well as back and forth between hemispheres.
So this means that thinking and learning is more complicated than simply switching between the focused and diffuse modes.

The Focused Mode—A Tight Pinball Machine

Metaphors are powerful tools for learning in math and science.
In the old game of pinball, you pull back on a spring-loaded plunger and it whacks a ball, which ends up bouncing randomly around the circular rubber bumpers.
When you focus your attention on a problem, your mind pulls back the mental plunger and releases a thought.
that thought takes off, bumping around like the pinball.
the diffuse mode has its circular rubber bumpers farther apart.

The close bumpers of the focused mode mean that you can more easily think a precise thought. Basically, the focused mode is used to concentrate on something that’s already tightly connected in your mind, often because you are familiar and comfortable with the underlying concepts.

For example, you can use the focused mode to multiply numbers—if you already know how to multiply, that is.
When you focus on something, it's a little like an octopus that sends its tentacles to different areas of its surroundings to fiddle with whatever it’s working on.
The octopus has tentacles to make connections, just as your working memory has limitation with how many things it can hold at once.

Why Math and Science Can Be More Challenging

Focused problem solving in math and science is often more effortful than focused-mode thinking involving language and people.
This may be because humans haven’t evolved over the millennia to manipulate mathematical ideas.

the Einstellung effect: In this phenomenon, an idea you already have in mind, or your simple initial thought, prevents a better idea or solution from being found.
We saw this in the focused pinball picture, where your initial pinball thought went to the upper part of the brain, but the solution thought pattern was in the lower part of the image.
You have to unlearn your erroneous older ideas even while you’re learning new ones.

This is precisely why one significant mistake students sometimes make in learning math and science is jumping into the water before they learn to swim.
In other words, they blindly start working on homework without reading the textbook, attending lectures, viewing online lessons, or speaking with someone knowledgeable. This is a recipe for sinking.

The Diffuse Mode—A Spread-Out Pinball Machine

This mode of thinking allows the brain to look at the world from a much broader perspective.
If you are grappling with a new concept or trying to solve a new problem, you don’t have preexisting neural patterns to help guide your thoughts.

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

If you are trying to understand or figure out something new, your best bet is to turn off your precision-focused thinking and turn on your “big picture” diffuse mode, long enough to be able to latch on to a new, more fruitful approach.

As we’ll see, the diffuse mode has a mind of its own—you can’t simply command it to turn on. But we’ll soon get to some tricks that can help you transition between modes.


I realized my best guitar riffs always came to me when I was ‘just messing around’ as opposed to when I sat down intent on creating a musical masterpiece.

The harder you push your brain to come up with something creative, the less creative your ideas will be.

Ultimately, this means that relaxation is an important part of hard work.

Why Are There Two Modes of Thinking?

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

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


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!”

One mode will process the information it receives and then send the result back to the other mode. This volleying of information back and forth as the brain works its way toward a conscious solution appears essential for understanding and solving all but trivial problems and concepts.

Procrastination Prelude

when you procrastinate, you are leaving yourself only enough time to do superficial focused-mode learning.
You are also increasing your stress level because you know you have to complete what feels like an unpleasant task.
you’ll be left with a shaky foundation.

Focusing Intently but Briefly:
Turn off your phone and any sounds or sights (or websites) that might signal an interruption. Then set a timer for twenty-five minutes and put yourself toward doing a twenty-five-minute interlude of work focused on a task—any task.
Don’t worry about finishing the task—just worry about working on it.
Once the twenty-five minutes is up, reward yourself with web surfing, checking your phone, or whatever you like to do.
This reward is as important as the work itself.
You’ll be amazed at how productive a focused twenty-five-minute stint can be—especially when you’re just focusing on the work itself, not on finishing.

imagine that at the end of the day, you are reflecting on the one most important task that you accomplished that day. What would that task be? Write it down.
Then work on it. Try to complete at least three of these twenty-five-minute sessions that day, on whatever task or tasks you think are most important.
At the end of your workday, look at what you crossed off your list and savor the feeling of accomplishment.
Then write a few key things that you would like to work on the next day. This early preparation will help your diffuse mode begin to think about how you will get those tasks done the next day.


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. Your initial ideas about problem solving can sometimes be very misleading.


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

{ 3 } Learning is creating: Lessons from Thomas Edison’s Frying Pan

Thomas Edison was one of the most prolific inventors in history, with more than one thousand patents to his name.
Even as his lab was burning to the ground in a horrific accidental fire, Edison was excitedly sketching up plans for a new lab, even bigger and better than before. How could Edison be so phenomenally creative? The answer, as you’ll see, relates to his unusual tricks for shifting his mode of thinking.

Shifting between the Focused and Diffuse Modes

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

Creativity expert Howard Gruber has suggested that one of the three B’s usually seems to do the trick: the bed, the bath, or the bus.
Walking spurs creativity in many fields; a number of famous writers, such as Jane Austen, Carl Sandburg, and Charles Dickens, found inspiration during their frequent long walks.

Once you are distracted from the problem at hand, the diffuse mode has access and can begin pinging about in its big-picture way to settle on a solution.

After your break, when you return to the problem at hand, you will often be surprised at how easily the solution pops into place. Even if the solution doesn’t appear, you will often be further along in your understanding. It can take a lot of hard focused-mode work beforehand, but the sudden, unexpected solution that emerges from the diffuse mode can make it feel almost like the “aha!” mode.

That whispered, intuitive solution to whatever puzzle you are attempting to deal with is one of the most elusively cool feelings of math and science—and art, literature, and anything else creative.

When faced with a difficult problem, instead of focusing intently on it, Edison, according to legend, took a nap. But he did so while sitting in a lounge chair, holding a ball bearing in his hand above a plate on the floor. As he relaxed, his thoughts moved toward free and open diffuse-mode thinking. (This is a reminder that falling asleep is a good way to get the brain thinking loosely about a problem you want to solve, or anything you are working on creatively.) When Edison fell asleep, the ball bearing fell from his hands. The clatter woke him so he could grasp the fragments of his diffuse-mode thinking to create new approaches.

Creativity Is about Harnessing and Extending Your Abilities

Dalí called this “sleeping without sleeping.”

Many people think they are not creative, when that is simply untrue. We all have the ability to make new neural connections and pull from memory something that was never put there in the first place.

Understanding how your mind works helps you better understand the creative nature of some of your thoughts.

From Focused to Diffuse:
Read the following sentence and identify how many errors it contains:
Thiss sentence contains threee errors.
The first two errors are easily discovered using a focused-mode approach. The third, paradoxical error becomes obvious only when you change perspectives and adopt a more diffuse approach.