A Mind For Numbers by Barbara Oakley: Summary

May 15, 2020


What it's about?

A Mind For Numbers by Barbara Oakley: Summary

Summary Notes

“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.”

“You may want your learning to progress more quickly—to somehow command your diffuse mode to assimilate new ideas faster. But compare it to exercise. Constantly lifting weights won’t make your muscles any bigger—your muscles need time to rest and grow.”

“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.”

The Two Modes of Thinking

There are two modes of thinking, focused mode and diffused mode. The focus mode is how it sounds, it’s when you are focusing on what you are learning. The diffused mode is when you allow your mind to drift or relax from what you are trying to learn.

Oakley uses a great analogy to describe the difference between the two modes: “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.”

We need to understand these modes to help us understand how we learn best. It is a mix of focus and effort with relaxation that helps us create chunks of information that we can store in our long-term memories.

Focused Mode

“(Your) 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.”

“You can use the focused mode to multiply numbers—if you already know how to multiply, that is.”

“If you’re studying a language, you might use the focused mode to become more fluent with the Spanish verb conjugation you learned last week.”

Diffused Mode

“This mode of thinking allows the brain to look at the world from a much broader perspective.”

“Diffuse-mode insights often flow from preliminary thinking that’s been done in the focused mode.(The diffuse mode must have clay to make bricks!)”

How they work together: “these modes are highly important for learning...You’re in either one mode or the other—not consciously in both at the same time.”

It is important to understand the difference between the two modes as you need to use both when learning. “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.”

Example: “Here’s a cognitive exercise that can help you feel the shift from focused to diffuse mode. See whether you can form a new triangle that points down by moving only three coins. When you relax your mind, releasing your attention and focusing on nothing in particular, the solution can most easily come to you.”

A triangle of pennies for the pennie puzzle

Some ways to enter diffused mode:

  • Take a nap
  • Go to the gym
  • Walk, run or bike
  • Draw or paint
  • Meditate


“Chunking is what happens when you know something so well, like a song, or a scientific formula, or a verb conjugation,or a dance routine, that it is basically a snap to call it to mind and do it or use it. Creating neural patterns—“neural chunks”—underpins the development of all expertise.”

A mixed up puzzle to represent raw information, then a completed puzzel to represent a chunked piece of information.

“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. It’s like getting dressed in the morning. Usually you just think one simple thought—I’ll get dressed. But it’s amazing when you realize the complex swirl of underlying activities that take place with that one simple chunk of a thought."

Creating a Chunk

  1. The first step in chunking is to simply focus your attention on the information you want to chunk.

  2. The second step in chunking is to understand the basic idea you are trying to chunk, this step is a basic understanding—synthesizing the gist of what’s important.

    • “it’s important to realize that just understanding how a problem was solved does not necessarily create a chunk...Do not confuse the “aha!” of a breakthrough in understanding with solid expertise!”
  1. The third step to chunking is gaining context so you see not just how, but also when to use this chunk...context is where top down and bottom up meet.
    • **“**Context means going beyond the initial problem and seeing more broadly...so you see not only when to use the chunk, but when not to use it.”
      • “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.”

Memory and Enhancing It

There are two parts to our memory system, working memory and long-term memory. Understanding how both work is critical to improving how you learn.

Things you are learning move from the working memory to the long term memory.

“To move from working memory to long-term memory, two things should happen: the idea should be memorable...and it must be repeated.”

Working Memory

“Working memory is the part of memory that has to do with what you are immediately and consciously processing in your mind.” (Techie types sometimes equate short-term memory to random-access memory [RAM], and long-term memory to hard drive space.)

“Widely believed that the working memory holds only about four chunks of information.”

Knowing that you can only hold about 4 chunks of information in the working memory, is important for learning. It is why you need to work through some topics like programming methodically. When you first start learning you can’t process everything at once and if you try you become confused and frustrated.

Long-Term Memory

“Long-term memory might be thought of as a storage warehouse. Once items are in there, they generally stay put.” (Like Mom’s boxes in the garage once it goes to the garage it stays there forever.)

“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.”

Moving the things you are learning to long-term memory requires repetition, practice and a few other techniques.

Enhancing Memory

Analogies: “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.”

Writing: “Writing appears to help you to more deeply encode (that is, convert into neural memory structures) what you are trying to learn.”

A big reason for me starting this section of my blog is to help me store what I am learning from books in my long term memory.

Flash Cards: Old school flash cards or newer digital versions like ANKI are a great spaced repetition tool.

“Briefly repeat what you want to remember over several days; perhaps for a few minutes each morning or each evening, change the order of your cards sometimes. Gradually extend the times between repetitions as the material firms itself into your mind.”

Exercise:Regular exercise can make a substantive improvement in your memory and learning abilities. Exercise, it seems, helps create new neurons in areas that relate to memory. It also creates new signaling pathways.”

Exercise is a big one for me I am a massive believer in a healthy body, health mind.

Practice Makes Permanent

“The challenge with repetition and practice, which lie behind the mind’s creation of solid chunks, is that it can be boring.”

Practice and recall has been proven to increase retention. I think about this all the time now with programming - I am better of getting out of tutorials as soon as possible.

**“**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.”

Don’t fall into the trap of just doing the work/study to get it done: “(don’t just) finish a problem, check their answer in the back of the text, smile, and go on to the next problem. I suggest (you) insert a step between the smile and going on to the next problem—ask (yourself) this question: How would I know how to do the problem this way if I saw it on a test mixed together with other problems and I didn’t know it was from this section of the text? (You) need to think of every homework problem in terms of test preparation and not as part of a task they are trying to complete.”

Spaced Repetition vs. Cramming

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.”

This helps you move items from working memory to long-term memory.

“A good rule of thumb, when you are first learning new concepts, is not to let things go untouched for longer than a day...you can then increase the space of the repetitions as you progress.” This can simply be rewriting your notes or doing practice questions but it is best to be done within a day.

The main point here is why consistent effort is better than large study sessions. “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.”

Tools For Practice

Flash cards: “Consider creating index cards with the problem question on one side, and the question and solution steps on the other. That way you can easily shuffle the cards and be faced with a random variety of techniques you must call to mind.”

Interleaving: “Practice by doing a mixture of different kinds of problems requiring different strategies.”

“Focusing on one technique is a little like learning carpentry by only practicing with a hammer. After a while, you think you can fix anything by just bashing it.”

“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.”

Sleep: You ever have a problem you just couldn’t solve no matter how hard you try? Then you go to bed and bang the answer pops up in your mind as you dream. It turns out for practice sleep is just as important as the practice itself.

Sleep is a vital part of memory and learning. 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.

Recall: The simple act of trying to recall what you have been learning has been shown to help store information in long-term memory super effectively.

Research has shown that the more effort you put into recalling material, the deeper it embeds itself into your memory.”

“Pick a mathematical or scientific concept from your notes or from a page in the book. Read it over, then look away and see what you can recall—working toward understanding what you are recalling at the same time. Then glance back, reread the concept, and try it again.”


Procrastination in this book is pitched as the largest reason why you don’t learn what you should be learning. We unconsciously procrastinate all the time. Procrastination can come in many forms. It comes in sneaky hidden forms like doing easy problems or admin tasks that give the illusion of being productive and learning. It is not just surfing the web, watching YouTube or playing games.

“We procrastinate about things that make us feel uncomfortable...like a new topic we are unfamiliar with...The pain centers of their brains light up when they contemplate working on math.

It is the thought of the activity not the actual activity that causes this. “It was the anticipation that was painful. When the math-phobes actually did math, the pain disappeared.” So if you just sit down and do the work, the pain disappears and you might actually find you enjoy it.

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

Overcoming Procrastination

The books mentions a few ways to overcoming procrastination. Below are the tactics the books gives. The more powerful way to avoid procrastination is the formation of strong habits that help you skip past the “pain centers” in your brain going off. I’ll give my notes on that in the habits section, below are the tactics.

Create a to-do list the night before: “Why the day before? Research has shown this helps your subconscious to grapple with the tasks on the list so you figure out how to accomplish them.”

Persistence will make the topic more pleasurable in the long run: “ Keep this front of mind when you are getting ready to do a task you don’t want: 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.”

Pomodoro technique: "Set a timer for twenty five minutes and commit to doing a focused a twenty-five-minute timed work session—not on completing a task.”

Find your optimal work environment: “Different people function better in certain environments—some need a busy coffee shop, while others need a quiet library. You need to figure out what’s best for you.”

Turn-off distractions: Put your phone in another room on silent, turn off notifications on your computer, block websites that aren’t productive or related to your study and get away from people that might distraction.

Work on your mindset:** Researchers have found that the difference between slow and fast starters is that the non-procrastinating 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.”

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

Start small and build : Don’t say I’m going to do 4 hours of study, instead start small commit to just doing one task or one problem._“_Write down three “microtasks” that you can do within a few minutes....At this point, close your eyes and tell your mind that you have nothing else to worry about, no other concerns, just your first microtask.”

Creating Good Learning/Working Habits

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

Habits in this book are pitched as a way to overcome the poor habits of procrastination: “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.”

“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.”

There are better books on how to form habits my favorite is Atomic Habits by James Clear. I found it a lot more practical than The Power of Habit. I will just list the steps on how you form a habit below and the talk more generally about how to use them in your learning.

  1. The Cue - I will start my study as soon as I get home or I will go for a run as soon as I wake up.
  2. The Routine - “Your brain wants to automatically go into this routine when you’ve gotten your cue, so this is the pressure point where you must actively focus on rewiring your old habit. The key to rewiring is to have a plan. Developing a new ritual can be helpful.”
  3. The Reward - I went for a run, I get to eat dessert or I did my study I'm allowed to play games for 1 hour.
    1. “It’s particularly important to realize that giving yourself even a small “attaboy” or “attagirl” jump-starts the process of rewiring your brain. This rewiring, sometimes called learned industriousness, helps brighten tasks you once thought were boring and uninteresting.”
  4. The Belief- “You may find that when the going gets stressful, you long to fall back into old, more comfortable habits. Belief that your new system works is what can get you through.”

Finally to help you form and retain these new habits: “Keeping a written history over several weeks appears to be critical in helping you make changes.”

Learning on your own

Okay here is why Universities can be a waste of money. While I see some value in University for networking, branding and streamlining hiring for HR. The truth is students learn best when they themselves engage in the subject.

Research has shown that students learn best when they themselves are actively engaged in the subject instead of simply listening to someone else speak.”

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

“Often, no matter how good your teacher and textbook are, it’s only when you sneak off and look at other (materials) that you begin to see that what you learn through a single teacher or book is a partial version of the full...reality of the subject.”

“Good learners vet their work to ensure that it makes sense. They ask themselves what the equations mean and where they come from.”

Fellow students are also important: “A student’s ability to grapple personally with the material, sometimes bouncing it off fellow learners, is key.” This sentence is why I am considering a boot camp for coding.

General Tips

Experiment until you find what works: “By its nature, self-experimentation involves making sharp changes in your life...(We need to) monitor ourselves in a hundred ways, makes it easy for self-experimentation to reveal unexpected side effects...Moreover, daily measurements of acne, sleep, or anything else, supply a baseline that makes it even easier to see unexpected changes.”

Don’t look for what you are passionate about, you’ll develop passion for what you become good at: “Over the past decades, students who have blindly followed their passion, without rational analysis of whether their choice of career truly was wise, have been more unhappy with their job choices than those who coupled passion with rationality.” This has also shown up in one of my favorite books ‘so good they can’t ignore you.’

“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.

Leaving work until the last minute, or doing large binge sessions is not productive: “I do my best work under deadlines,” is simply not true.

“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.”

Find your ideal environment: “Different people function better in certain environments some need a busy coffee shop, while others need a quiet library. You need to figure out what’s best for you.”

Learn at your own pace don’t compare yourself to others: “Take a step back and look dispassionately at your strengths and weaknesses. If you need more time to learn math and science, that’s simply the reality.”

“You may be surprised to discover that learning slowly can mean you learn more deeply than your fast-thinking classmates.”

Don’t always persevere when you’re stuck: “Rising frustration is usually a good time-out signal for you, signaling that you need to shift to diffuse mode.”

Be careful what you highlight and only pick the most important parts: “Highlighting and underlining must be done carefully—otherwise they can be not only ineffective but also misleading.”

Skim the chapter first: “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.”

It helps as: “You’re creating little neural hooks to hang your thinking on, making it easier to grasp the concepts.”

Avoid the illusion of competence: “(You) have an illusion of competence because the solution is already there.”

Explain to others what you are learning : "Simplifying is also important...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.”

“Observe what happens when you are talking to other people about what you are studying. 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.”

See the subreddit explain to me like I’m five.