A Mind For Numbers

date Jul 6, 2021
authors Barbara Oakley
reading time 13 mins

Table of Contents


Diffuse-mode thinking

Diffuse-mode thinking is also essential for learning math and science. It allows us to suddenly gain a new insight on a problem we’ve been struggling with and is associated with “big-picture” perspectives. Diffuse-mode thinking is what happens when you relax your attention and just let your mind wander.

Focused mode thinking

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.


(The German word Einstellung means “mindset”—basically you can remember Einstellung as installing a roadblock because of the way you are initially looking at something.)… You have to unlearn your erroneous older ideas even while you’re learning new ones.

Working memory

Working memory is the part of memory that has to do with what you are immediately and consciously processing in your mind. It used to be thought that our working memory could hold around seven items, or “chunks,” but it’s now widely believed that the working memory holds only about four chunks of information.


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


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.


Transfer is the ability to take what you learn in one context and apply it to something else. For example, you may learn one foreign language and then find that you can pick up a second foreign language more easily than the first.

How to learn

Repeated reading provides no 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.

2 states of mind: focused vs relaxed

neuroscientists have been making profound advances in understanding the two different types of networks that the brain switches between — highly attentive states and more relaxed resting state networks.

When to use difussed mode thinking

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.

What is the question?

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!”

How to trigger diffused mode of thinking

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.

Spaced repetition

Research has shown that if you try to glue things into your memory by repeating something twenty times in one evening, for example, it won’t stick nearly as well as it will if you practice it the same number of times over several days or weeks.


Part of what this special sleep-time tidying does is erase trivial aspects of memories and simultaneously strengthen areas of importance. During sleep, your brain also rehearses some of the tougher parts of whatever you are trying to learn—going over and over neural patterns to deepen and strengthen them.

Dreaming about difficult parts

It seems that if you go over the material right before taking a nap or going to sleep for the evening, you have an increased chance of dreaming about it. 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. Dreaming about what you are studying can substantially enhance your ability to understand — it somehow consolidates your memories into easier-to-grasp chunks.

Learn retrieve practise

Attempting to recall the material you are trying to learn—retrieval practice—is far more effective than simply rereading the material. Psychologist Jeffrey Karpicke and his colleagues have shown that many students experience illusions of competence when they are studying.

Classroom vs real-world application

Getting a concept in class versus being able to apply it to a genuine physical problem is the difference between a simple student and a full-blown scientist or engineer.

Related concepts

You’ll find that once you put the first problem or concept in your library, whatever it is, then the second concept will go in a bit more easily. And the third more easily still. Not that all of this is a snap, but it does get easier.

Varied practise with contexts

The fact is, 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.

  1. Read (but don’t yet solve) assigned homework and practice exams/quizzes.
  2. Review lecture notes
  3. Rework example problems presented in lecture notes.
  4. Work assigned homework and practice exam/quiz questions.

Working on hard problems

Take a break to find the solution

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.

Practise with focussed and diffused mode

As the days and weeks pass, it’s the distributed practice—the back and forth between focused-mode attention and diffuse-mode relaxation—that does the trick… Constantly lifting weights won’t make your muscles any bigger—your muscles need time to rest and grow before you use them again. Taking time off between weight sessions helps build strong muscles in the long run. Consistency over time is key!

Long and difficult problem

Figuring out a difficult problem or learning a new concept almost always requires one or more periods when you aren’t consciously working on the problem.

Work on it

Usually a few hours is long enough for the diffuse mode to make significant progress but not so long that its insights fade away before being passed on to the focused mode. A good rule of thumb, when you are first learning new concepts, is not to let things go untouched for longer than a day.

Sleep and vacations

This idea of looking from fresh perspectives also gives us insight on why “sleeping on it” before making major decisions is generally a good idea, and why taking vacations is important.



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. I sometimes find it excruciating to pull the trigger and expose my work to other people, but every time I do, it turns out for the best.


Criticism makes us better: By exposing our work to others, and by externalizing it so we can inspect it ourselves, we gain unique perspective and insight and develop new and improved plans for the next version.

Slow hunch

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. Key to the slow hunch is simply having mental access to aspects of an idea.

Problem solving

Logical or by intuition

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.

Go backwards in history

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


Like addiction

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.

Temporary pleasure can become longterm stress

Over time, your habitual, zombielike response in obtaining those temporary dollops of pleasure can gradually lower your self-confidence, leaving you with even less of a desire to learn how to work effectively. Procrastinators report higher stress, worse health, and lower grades.


Habits have four parts:

  1. The Cue: This is the trigger that launches you into “zombie mode.”
  2. The Routine: This is your zombie mode—the routine, habitual response your brain is used to falling into when it receives the cue.
  3. The Reward: Habits develop and continue because they reward us—give us a dollop of pleasure.
  4. The Belief: Habits have power because of your belief in them.

Flow state

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.

Process, not outcome

Learn to focus on process, not product. Process means the flow of time and the habits and actions associated with that flow of time—as in, “I’m going to spend twenty minutes working.” Product is an outcome—for example, a homework assignment that you need to finish.

Negative feeling about the beginning of your work

It’s normal to sit down with a few negative feelings about beginning your work. It’s how you handle those feelings that matters. Researchers have found that the difference between slow and fast starters is that the nonprocrastinating fast starters put their negative thinking aside, saying things to themselves like, “Quit wasting time and just get on with it. Once you get it going, you’ll feel better about it.”

Take notes on when and how you procrastinate

There are many different ways to monitor your behavior. The most important idea here is that keeping a written history over several weeks appears to be critical in helping you make changes.

Connecting to other subjects are not possible

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. Students who interrupt their work constantly not only don’t learn as deeply, but also aren’t able to transfer what little they do learn as easily to other topics.

Learning maths and science

Fitting in the big picture

In a classic paper, physicist Jeffrey Prentis compares how a brand-new student of physics and a mature physicist look at equations. 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.


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.

Combining traditional and nontraditional learning

We will continue to see fascinating innovations from people who are able to combine the best aspects of traditional and nontraditional learning with their own self-taught approaches.

Not intimidated by authority

Feynman was at that time just a youngster in the crowd of hundreds of prominent physicists at Los Alamos. But he was singled out by Bohr to do private brainstorming together before Bohr would meet with the other physicists. Why? Feynman was the only one who wasn’t intimidated by Bohr and who would tell Bohr that some of his ideas were foolish.

Increase your weak ties

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.

Starting hard

“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.


  1. Use recall.
  2. Test yourself.
  3. Chunk your problems.
  4. Space your repetition.
  5. Alternate different problem-solving techniques during your practice.
  6. Take breaks.
  7. Use explanatory questioning and simple analogies.
  8. Focus.
  9. Eat your frogs first. Do the hardest thing earliest in the day, when you are fresh.
  10. Make a mental contrast.


  1. Passive rereading—sitting
  2. Letting highlights overwhelm you.
  3. Merely glancing at a problem’s solution and thinking you know how to do it.
  4. Waiting until the last minute to study.
  5. Repeatedly solving problems of the same type that you already know how to solve.
  6. Letting study sessions with friends turn into chat sessions.
  7. Neglecting to read the textbook before you start working problems.
  8. Not checking with your instructors or classmates to clear up points of confusion.
  9. Thinking you can learn deeply when you are being constantly distracted.
  10. Not getting enough sleep.