A Mind for Numbers

This is the greatest book I've ever read on learning. It has helped me immensely in my Engineering studies.

15 min read

Chapter 2: Easy does it

  • Prime your Mental Pump
    • Look though a chapter before reading it. Check out graphs, graphics, questions, and headlines.
  • The Two Modes
    • Focused-mode
      • You are actively focusing on something. Turn your attention to something, and your focused mode is on. Almost like a flashlight.
    • Diffuse-mode
      • Not actively focusing on anything, just letting the mind wander. It allows different areas of the brain to hook up and return valuable insights. Diffuse-mode is connected to focused-mode because it can only flow from preliminary thinking that has been done in focused mode (the diffuse mode must have clay to make bricks).
      • 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.
      • P19 - "Ultimately, this means that relaxation is an important part of hard work - and good work, for that matter".
    • Learning is more complicated than just switching between the two modes.
    • You switch between both modes - it is not possible to use both at the same time.
    • To learn about and be creative in math and science, we need to strengthen and use both the focused and diffuse modes.
      • Evidence suggests that to grapple with a difficult problem, we must first put hard, focused-mode work into it. The diffuse-mode is also an important part of solving the problem, but as long as we are consciously focusing on a problem, we are blocking the diffuse mode.
      • Usually, when you first focus on a problem, you will be stumped. To figure it out, you focus on it initially, and then turn your focus away.
  • The Einstellung Effect
    • In this phenomenon, an idea you already have in mind, or your simple initial thought, prevents a better ida or solution from being found.
    • In science, this wrong approach is easy to do because sometimes your initial intuition about what's happening is misleading. 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.
      • Blindly start working on homework without reading the textbook, attending lectures, viewing online lessons or speaking with someone knowledgeable.
    • To solve this problem, you switch between the modes.
  • Procrastination
    • When you procrastinate, you are leaving yourself only enough time to do superficial focused-mode learning.

Chapter 3: Learning is creating

  • Use the focused mode to first start grappling with concepts and problems in math an science.
  • After you've done your first hard focused work, allow the diffuse mode to take over. Relax and do something different.
    • General diffuse-mode activators:
      • Go to the gym
      • Play a sport like soccer or basketball
      • Jog, walk, or swim
      • Dance
      • Go for a drive
      • Draw or peint
      • Take a bath or shower
      • Listen to music, especially without words
      • Play songs you know well on a musical instrument
      • Meditate or pray
      • Sleep
    • Best used briefly
      • Play video games
      • Surf the web
      • Talk to friends
      • Volunteer to help someone with a simple task
      • Read a relaxing book
      • Text friends
      • Go to a movie or play
      • Television
    • You can't just rest all the time: Just using your diffuse mode doesn't mean you can lollygag around and expect to get anywhere.
  • 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 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 pomodoro - and remove distractions.
  • The two major memory systems:
    • Working memory - like a juggler who can only keep 4 items in the air. (Think RAM).
    • Long-term memory - like a storage house that can hold a large amount of material, but needs to be revisited occasionally to keep memories accessible. (Think HDD).
  • Spaced repetition helps move items from working memory to long-term memory. (Useful program: Anki).
  • Sleep is a critical part of the learning progress. 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.

Chapter 4: Chunking and avoiding illusions of competence

  • Chunking
    • Practice helps build strong neural patters - that is, conceptual chunks of understanding.
    • 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.
    • First step to chunking
      • Focused attention
      • Simply focus your attention on the information you want to chunk.
      • You can't do two things at once. Remove distractions.
    • Second step to chunking
      • Understanding of the basic idea
      • Understand the basic idea you are trying to chunk
      • Just understanding how a problem was solved does not necessarily create a chunk that you can easily call to mind later.
    • Third step to chunking
      • Practice to help you gain big-picture context
      • Gaining context, so you see not just how, but also when to use this chunk.
        • Context means going beyond the initial problem and seeing more broadly, repeating and practicing with both related and unrelated problems, so that you know when to use the chunk and when not to.
          • It's like a tool. It's useful if you know how to use it, but useless if you don't.
  • Importance of recall and illusions of competence
    • Simple recall - trying to remember the key points while looking at the page - is one of the best ways to help the chunking process along.
    • Attempting to recall the material you are trying to learn - retrieval practice - is far more effective than simply rereading the material.
    • Using recall - mental retrieval of the key ideas - rather than passive rereading will make your study time more focused and effective.
    • 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.
  • Practice makes permanent
    • 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 ofd 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.
  • Recall material while outside your usual place of study: the value of walking
    • Doing something physically active is especially helpful when you have trouble grasping a key idea.
    • 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 - doing a mixture of different kinds of problems - versus overlearning
    • Interleaving means practice by doing a mixture of different kinds of problems requiring different strategies.
    • You want your brain to become used to the idea that just knowing how to use a particular problem-solving technique isn't enough - you also need to know when to use it.
      • Consider index cards with the problem question on one side, and the question and solution steps on the other. (Anki).
    • Long study sessions are fine, but don't just focus on one problem.
  • Bottom-up, top-down
    • Bottom-up chunking process = practice and repetition
    • Top-down "big picture" process = Going beyond the initial problem, repeating and practicing with both related and unrelated problems.

Chapter 5: Preventing Procrastination

  • 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.
  • Imagine how your calf muscles would scream if you prepared for a big race till midnight before your first marathon to do your first practice run. In just the way way, you can't compete in math and science if you just cram at the last minute.
  • Procrastination is a single, monumentally important "keystone" bad habit.
  • Procrastination is like addiction.
  • How procrastination works:
      1. Unhappy feeling (in diffuse mode)
      • You know you have to do something that makes you uncomfortable.. math, science, reading, etc.
      1. You funnel attention onto a more pleasant task (focused mode)
      1. Feel happy (temporarily)
  • Procrastinators put off just that one little thing. Then they do it again, and again, gradually growing used to it. They can even look healthy. But the long term effects? Not so good.
    • 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.

Chapter 6: Zombies Everywhere

  • Habits can be good and bad.
    • Habits are when your brain launches into a preprogrammed "zombie" mode.
  • A little bit of work on something that feels painful can ultimately be very beneficial.
  • Chunking, that automatically connected neural pattern that arises from frequent practice, is intimately related to habit.
  • Habit is an energy saver for us. It allows us to free our mind for other types of activities.
    • Think that if you had to be hyper-altert every time you did something, as if you did it for the first time. Habits remove this energy-expenditure. They "automate" the task.
  • Habits have four parts.
    • The Cue: The trigger that launches you into zombie mode.
    • The Routine: The actual zombie mode. The routine is a habitual response your brain is used to falling into when it receives the cue. These zombie responses can be harmless, useful, or in the worst case, so destructive that they defy common sense.
    • The Reward: Habits develop and continue because they reward us.
      • Procrastination is an easy habit to develop because the reward, moving your mind's fucus to something more pleasant, happens so quickly.
      • However, good habits can also be rewarded - finding ways to reward good study habits in math and science is vital to escaping procrastination.
    • The Belief:Habits have power because of your belief in them. Perhaps you might feel that you'll never be able to change your habit of putting off your studies until late in the day. To change a habit, you'll need to change your underlying belief.
    • Changing a habit by responding differently to a cue, or even avoiding that cue altogether. Reward and belief make the change long-lasting.
  • 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.
  • Mental contrasting is when you think of some bright future, and then contrast to your "bad" current self. This gives you hope, and inspires you.
    • 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.
  • If you find yourself avoiding certain tasks because they make you uncomfortable, there is a great way to reframe things.
    • Learn to focus on process, not product. This means that you focus on the flow of time and the habits and actions associated with the flow of time, as in "I'm going to spend twenty minutes working". The product is the finished outcome - for example a homework assignment that needs to be finished.
    • By focusing on the process rather than product, you allow yourself to back away from judging yourself (Am I getting closer to finishing?) and allow yourself to relax into the flow of the work.
    • Multitasking is like constantly pulling up a plant. This kind of constant shifting of your attention means that new ideas and concepts have no chance to take root and flourish.
  • Use pomodoro to stay productive - then reward yourself after each successful period of focused attention.
  • 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.

Chapter 7: Chunking versus choking

  • Chunking means integrating a concept into one smoothly connected neural though 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. 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 works, you guarantee failure if you don't try, but those who consistently give it a good effort you will experience many more successes.
  • 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 as soon as you can the next day.
  6. Add a new problem, and repeat the steps.
  7. Do "active" repetitions - mentally review key problem steps in your mind while doing something else (walking, etc).
  • Generating (recalling) the material helps you learn it much more effectively than simply rereading it.

The Law of Serendipity

Lady luck favors the one who tries.

Hitting a wall - knowledge collapse

Sometimes you hit a wall in constructing your understanding.

This happens when your mind is restructuring its understanding, building a more solid foundation.

It feels as if you are taking a step back, but when you emerge from the period, you will have taken a leap forward.

Test taking

Choking - panicking top the point where you freeze - can happen when your working memory is filled to capacity, yet you still don't have enough room for additional critical pieces you need to solve a problem.

Taking mini tests constantly is a powerful learning experience. It changes and adds to what you know, also making dramatic improvements to your ability to retain the 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.

Chapter 8: tools, tips and tricks

  • It's normal to sit down with a few negative feelings about beginning your work. It's how you handle those feelings that matters.

The positive approach to procrastination

  1. No going onto the computer during their procrastination time. It's just too engrossing.
  2. Before procrastinating, identify the easiest homework problem. (No solving is necessary at this point).
  3. Copy the equation or equations that are needed to solve the problem onto a small piece of paper and carry that paper around until they are ready to quit procrastinating and get back to work.
  • "Eat your frogs first thing in the morning" - Do the most important and most disliked jobs first, as soon as you wake up. This is incredibly effective.
  • Use a task list and a planner.
  • It is very important to transform distant deadlines into daily ones - attacking them bit by bit. They need to be translated into smaller ones that show up on your daily task list. The only way to walk a journey of a thousand miles is to take one step at a time.
  • To be effective in learning math and science, you must master your habits.
  • Mental tricks can be powerful tools. The following are some of the most effective:
    • 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
      • Meditation
    • 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.

Chapter 9: Procrastination zombie wrap-up

Tips on overcoming procrastination:

  • 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 feeling 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 in the 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.

Chapter 10: Enhancing your memory

This chapter is about visually remembering something and memory palaces.

Chapter 11: More memory tricks

  • 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 jus 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.

Chapter 12: Learning to appreciate your talent

  • 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 the 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.

Chapter 13: Sculpting your brain

  • Brains mature at different speeds. Many people do not develop maturity until their mid twenties.
  • Some of them 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 chinks to help you be smarter about your approach to all sorts of life and career challenges.

Chapter 14: Developing the mind's eye

  • 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.
  • Your "mind's eye" is important because it can help you stage plays and personalize what you are learning about.
  • Transfer is the ability to take what you learn in one context and apply it to something else.
  • It is important to grasp the chunked essence to 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.

Chapter 15: Renaissance learning

  • 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.

Chapter 16: Avoiding overconfidence

  • 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 oni 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 towards 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.

Chapter 17: Test taking

  • 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 excite dto 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?".

The 10 rules of good studying

  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
  10. Make a mental contrast

The 10 rules of bad studying

  1. Passive rereading
  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

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