education books

Highlights from the book “A Mind for Numbers”

On Habits

Procrastination is a single, monumentally important “keystone” bad habit.

Procrastination is like addiction.

In the insightful book The Power of Habit, author Charles Duhigg describes a lost soul—Lisa Allen, a middle-aged woman who had always struggled with her weight, who had begun drinking and smoking when she was sixteen, and whose husband had left her for another woman. Lisa had never held a job for more than a year and had fallen deeply into debt. But in a four-year span, Lisa turned her life around completely. She lost sixty pounds, was working toward a master’s degree, stopped drinking and smoking, and was so fit that she ran a marathon.

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. 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.

On finding beauty in Math and Science

“What, after all, is mathematics but the poetry of the mind, and what is poetry but the mathematics of the heart?” —David Eugene Smith, American mathematician and educator

Mandelbrot’s in heaven At least he will be when he’s dead Right now he’s still alive and teaching math at Yale

Gave us order out of chaos He gave us hope where there was none His geometry succeeds where other fail

If you ever lose your way A butterfly will flap its wings From a million miles away A little miracle will come to take you home

from Mandelbrot Set, song by Jonathan Coulton

There are hidden meanings in equations, just as there are in poetry. If you are a novice looking at an equation in physics, and you’re not taught how to see the life underlying the symbols, the lines will look dead to you. It is when you begin to learn and supply the hidden text that the meaning slips, slides, then finally leaps to life.

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.

“A mathematician who is not at the same time something of a poet will never be a full mathematician.” —German mathematician Karl Weierstrass

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.

On being a active learner

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

“What a wonderful stimulant it would be for the beginner if his instructor, instead of amazing and dismaying him with the sublimity of great past achievements, would reveal instead the origin of each scientific discovery, the series of errors and missteps that preceded it—information that, from a human perspective, is essential to an accurate explanation of the discovery.”

Inventor and author William Kamkwamba, born in 1987 in Africa, could not afford to attend school. So he began teaching himself by going to his village’s library, where he stumbled across a book titled Using Energy. But Kamkwamba didn’t just read the book. When he was only fifteen years old, he used the book to guide him in active learning: He built his own windmill. His neighbors called him misala—crazy—but his creation helped begin generating electricity and running water for his village and sparked the growth of grassroots technological innovation in Africa.

Taking responsibility for your own learning is one of the most important things you can do.

Celso Batalha, for example, a renowned professor of physics at Evergreen Valley College, has set up a popular reading group for his students about learning how to learn. And many professors use “active” and “collaborative teaching” techniques in the classroom that give students a chance to actively engage with the material and with each other.

The Man Who Knew Infinity: A Life of the Genius Ramanujan", by Robert Kanigel. This book tells the unbelievable, rags to intellectual riches story of the Indian mathematical genius Srinivasa Ramanujan and his friend British mathematician G. H. Hardy. My favorite episode is this:

Men of Mathematics, by E. T. Bell. This is an old classic that’s a show-stopping read for anyone who’s interested in how fascinating people think. Who could forget brilliant, doomed Évariste Galois who spent the night before he knew he was to die ‘feverishly dashing off his last will and testament, writing against time to glean a few of the great things in his teeming mind before the death which he foresaw could overtake him. Time after time he broke off to scribble in the margin “I have not time; I have not time,” and passed on to the next frantically scrawled outline.’ Truth be told, this is one of the few exciting stories that Professor Bell perhaps exaggerated, although Galois unquestionably spent that last evening putting the final polish on his life’s work. But this brilliant book has inspired generations of both men and women.”

On Focused and Diffuse mode

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 pre-frontal cortex, located right behind your forehead.

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.

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,

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

Evidence suggests that to grapple with a difficult problem, we must first put hard, focused-mode effort into it. (We learned that in grade school!) 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.

One remarkably inventive chemist of the mid-1800s, Alexander Williamson, observed that a solitary walk was worth a week in the laboratory in helping him progress in his work. (Lucky for him there were no smartphones then.) 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.

In other words, just using your diffuse mode doesn’t mean you can lollygag around and expect to get anywhere. 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.

General Diffuse-Mode Activators

  • Go to the gym
  • Play a sport like soccer or basketball
  • Jog, walk, or swim
  • Dance
  • Go for a drive (or tag along for the ride)
  • Draw or paint
  • Take a bath or shower
  • Listen to music, especially without words
  • Play songs you know well on a musical instrument
  • Meditate or pray
  • Sleep (the ultimate diffuse mode!)

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. Each interlude in which you are not directly focused on the problem allows your diffuse mode to look at it in a fresh way. When you turn your focused attention back to the problem, you consolidate new ideas and patterns that the diffuse mode has delivered.

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!

Be sure to schedule free time to nurture your diffuse mode.


Some feel that diffuse, intuitive ways of thinking are more in tune with our spirituality. The creativity that diffuse thinking promotes sometimes seems beyond human understanding. As Albert Einstein noted, “There are only two ways to live your life. One is as though nothing is a miracle. The other is as if everything is.”

10 Rules of Good Studying

  1. Use recall. After you read a page, look away and recall the main ideas. Highlight very little, and never highlight anything you haven’t put in your mind first by recalling. Try recalling main ideas when you are walking to class or in a different room from where you originally learned it. An ability to recall – to generate the ideas from inside yourself – is one of the key indicators of good learning.
  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. Pretend it’s a song and learn to play it over and over again in your mind, so the information combines into one smooth chunk you can pull up whenever you want.
  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. (Books generally are not set up this way, so you’ll need to do this on your own.) After every assignment and test, go over your errors, make sure you understand why you made them, and then rework your solutions. To study most effectively, handwrite (don’t type) a problem on one side of a flash card and the solution on the other. (Handwriting builds stronger neural structures in memory than typing). You might also photograph the card if you want to load it into a study app on your smartphone. Quiz yourself randomly on different type of problems. Another way to do this is to randomly flip through your book, pick out a problem, and see whether you can solve it cold.
  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. When you get frustrated with a math or science problem, take a break so that another part of your mind can take over and work in the background.
  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? Using an analogy really helps, like saying that the flow of electricity is like the flow of water. Don’t just think your explanation – say it out loud or put it in writing. The additional effect of speaking and writing allows you to more deeply encode (that is, covert into neural memory structures) what you are learning.
  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. Try to set up time and places where studying, not glancing at your computer or phone – is just something you naturally do.
  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. Post a picture or words in your workspace to remind you of your dream. Look at that when you find your motivation lagging. This work will pay off both for you and those you love!

Ten Rules of Bad Studying

  1. Passive rereading—sitting passively and running your eyes back over a page. 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.
  2. Letting highlights overwhelm you. 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. A little highlighting here and there is okay—sometimes it can be helpful in flagging important points. But if you are using highlighting as a memory tool, make sure that what you mark is also going into your brain.
  3. Merely glancing at a problem’s solution and thinking you know how to do it. This is one of the worst errors students make while studying. You need to be able to solve a problem step-by-step, without looking at the solution.
  4. Waiting until the last minute to study. Would you cram at the last minute if you were practicing for a track meet? Your brain is like a muscle—it can handle only a limited amount of exercise on one subject at a time.
  5. Repeatedly solving problems of the same type that you already know how to solve. If you just sit around solving similar problems during your practice, you’re not actually preparing for a test—it’s like preparing for a big basketball game by just practicing your dribbling.
  6. Letting study sessions with friends turn into chat sessions. Checking your problem solving with friends, and quizzing one another on what you know, can make learning more enjoyable, expose flaws in your thinking, and deepen your learning. But if your joint study sessions turn to fun before the work is done, you’re wasting your time and should find another study group.
  7. Neglecting to read the textbook before you start working problems. Would you dive into a pool before you knew how to swim? The textbook is your swimming instructor—it guides you toward the answers. You will flounder and waste your time if you don’t bother to read it. Before you begin to read, however, take a quick glance over the chapter or section to get a sense of what it’s about.
  8. Not checking with your instructors or classmates to clear up points of confusion. Professors are used to lost students coming in for guidance—it’s our job to help you. The students we worry about are the ones who don’t come in. Don’t be one of those students.
  9. Thinking you can learn deeply when you are being constantly distracted. Every tiny pull toward an instant message or conversation means you have less brain power to devote to learning. Every tug of interrupted attention pulls out tiny neural roots before they can grow.
  10. Not getting enough sleep. 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. Prolonged fatigue allows toxins to build up in the brain that disrupt the neural connections you need to think quickly and well. If you don’t get a good sleep before a test, NOTHING ELSE YOU HAVE DONE WILL MATTER.

More insights

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.

“Many people will tell you that they can’t nap. The one thing I learned from a single yoga class I took many years ago was to slow down my breathing. I just keep breathing slowly in and out and don’t think I must fall asleep. Instead, I think things like, Sleepytime! and just focus on my breathing. I also make sure it’s dark in the room, or I cover my eyes with one of those airplane sleep masks. Also, I set my phone alarm for twenty-one minutes because turning a short power nap into a longer sleep can leave you groggy. This amount of time gives me what’s basically a cognitive reboot.”

Spaced repetition helps move items from working memory to long-term memory.

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.

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

Imagine how your calf muscles would scream if you prepared for a big race by waiting till midnight the night before your first marathon to do your first practice run. In just the same way, you can’t compete in math and science if you just cram at the last minute.

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.

A short, helpful guide to getting started with meditation is “Buddha in Blue Jeans” by Tai Sheridan.

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

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.

“Deficiencies of innate ability may be compensated for through persistent hard work and concentration. One might say that work substitutes for talent, or better yet that it creates talent.”

I’d like to point educators toward the book Redirect, by psychology professor Timothy Wilson, which describes the seminal importance of failure-to-success stories (Wilson 2011). Helping students change their inner narratives forms one of the important goals of this book. A leader in describing the importance of change and growth in mindset is Carol Dweck (Dweck 2006).

Differentiation is effectively recursive division, and integration is effectively recursive multiplication, each carried out indefinitely, i.e., to infinitesimal values (which is possible because they depend on convergent series, which themselves are only known by inference, not direct inspection). This ability to project what an operation entails when carried out infinitely is what solves Zeno’s paradox, which seems impossible when stated in words. But in addition to this difficulty, the Leibnizian formalism we now use collapses this infinite recursion into a single character or the integral sign) because one can’t actually keep writing operations forever. This makes the character manipulation of calculus even less iconic of the corresponding physical referent.

  1. 10 Geary 2011. See also the landmark documentary A Private Universe, available at, which led to much research into misconceptions in understanding science.

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