Cajal’s teachers, as Cajal later recollected, showed a sadly mistaken valuing of abilities. Quickness was taken as cleverness, memory for ability, and submissiveness for rightness.13 Cajal’s success despite his “flaws” shows us how even today, teachers can easily underestimate their students—and students can underestimate themselves.
Deep Chunking
Cajal worked his way fitfully through medical school. After adventures in Cuba as an army doctor and several failed attempts at competitive examinations to place as a professor, he finally obtained a position as a professor of histology, studying the microscopic anatomy of biological cells.
Each morning in his work in studying the cells of the brain and the nervous system, Cajal carefully prepared his microscope slides. Then he spent hours carefully viewing the cells that his stains had highlighted. In the afternoon, Cajal looked to the abstract picture of his mind’s eye—what he could remember from his morning’s viewings—and began to draw the cells. Once finished, Cajal compared his drawing with the image he saw in the microscope. Then Cajal went back to the drawing board and started again, redrawing, checking, and redrawing. Only after his drawing captured the synthesized essence, not of just a single slide, but of the entire collection of slides devoted to a particular type of cell, did Cajal rest.14
Cajal was a master photographer—he was even the first to write a book in Spanish on how to do color photography. But he never felt that photographs could capture the true essence of what he was seeing. Cajal could only do that through his art, which helped him abstract—chunk—reality in a way that was most useful for helping others see the essence of the chunks.
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.15 That’s why great art, poetry, music, and literature can be so compelling. When we grasp the chunk, it takes on a new life in our own minds—we form ideas that enhance and enlighten the neural patterns we already possess, allowing us to more readily see and develop other related patterns.
Once we have created a chunk as a neural pattern, we can more easily pass that chunked pattern to others, as Cajal and other great artists, poets, scientists, and writers have done for millennia. Once other people grasp that chunk, not only can they use it, but also they can more easily create similar chunks that apply to other areas in their lives—an important part of the creative process.
Here you can see that the chunk—the rippling neural ribbon—on the left is very similar to the chunk on the right. This symbolizes the idea that once you grasp a chunk in one subject, it is much easier for you to grasp or create a similar chunk in another subject. The same underlying mathematics, for example, echo throughout physics, chemistry, and engineering—and can sometimes also be seen in economics, business, and models of human behavior. This is why it can be easier for a physics or engineering major to earn a master’s in business administration than someone with a background in English or history.16
Metaphors and physical analogies also form chunks that can allow ideas even from very different areas to influence one another.17 This is why people who love math, science, and technology often also find surprising help from their activities or knowledge of sports, music, language, art, or literature. My own knowledge of how to learn a language helped me in learning how to learn math and science.
One important key to learning swiftly in math and science is to realize that virtually every concept you learn has an analogy—a comparison—with something you already know.18 Sometimes the analogy or metaphor is rough—such as the idea that blood vessels are like highways, or that a nuclear reaction is like falling dominoes. But these simple analogies and metaphors can be powerful tools to help you use an existing neural structure as a scaffold to help you more rapidly build a new, more complex neural structure. As you begin to use this new structure, you will discover that it has features that make it far more useful than your first simplistic structure. These new structures can in turn become sources of metaphor and analogy for still newer ideas in very different areas. (This, indeed, is why physicists and engineers have been sought after in the world of finance.) Physicist Emanual Derman, for example, who did brilliant research in theoretical particle physics, moved on to the company Goldman Sachs, eventually helping to develop the Black-Derman-Toy interest-rate model. Derman eventually took charge of the firm’s Quantitative Risk Strategies group.
SUMMING IT UP
Brains mature at different speeds. Many people do not develop maturity until their midtwenties.
Some of the 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 chunks to help you be smarter about your approach to all sorts of life and career challenges.
PAUSE AND RECALL
Close the book and look away. What were the main ideas of this chapter? You will find that you can recall these ideas more easily if you relate them to your own life and career goals.
ENHANCE YOUR LEARNING
1. In his career, Santiago Ramón y Cajal found a way to combine his passion for art with a passion for science. Do you know other people, either famous public figures or family friends or acquaintances, who have done something similar? Is such a confluence possible in your own life?
2. How can you avoid falling into the trap of thinking that quicker people are automatically more clever?
3. Doing what you are told to do can have benefits and drawbacks. Compare Cajal’s life with your own. When has doing what you were told been beneficial? When has it inadvertently created problems?
4. Compared to Cajal’s handicaps, how do your own limitations stack up? Can you find ways to turn your disadvantages into advantages?
{ 14 }
developing the mind’s eye through equation poems
Learn to Write an Equation Poem—Unfolding Lines That Provide a Sense of What Lies Beneath a Standard Equation
Poet Sylvia Plath once wrote: “The day I went into physics class it was death.”1 She continued:
A short dark man with a high, lisping voice, named Mr. Manzi, stood in front of the class in a tight blue suit holding a little wooden ball. He put the ball on a steep grooved slide and let it run down to the bottom. Then he started talking about let a equal acceleration and let t equal time and suddenly he was scribbling letters and numbers and equals signs all over the blackboard and my mind went dead.
Mr. Manzi had, at least in this semiautobiographical retelling of Plath’s life, written a four-hundred-page book with no drawings or photographs, only diagrams and formulas. An equivalent would be trying to appreciate Plath’s poetry by being told about it, rather than being able to read it for yourself. Plath was, in her version of the story, the only student to get an A, but she was left with a dread for physics.
“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
Physicist Richard Feynman’s introductory physics classes were entirely different. Feynman, a Nobel Prize winner, was an exuberant guy who played the bongos for fun and talked more like a down-to-earth taxi driver than a pointy-headed intellectual.
When Feynman was about eleven years old, an off-the-cuff remark had a transformative impact on him. He remarked to a fri
end that thinking is nothing more than talking to yourself inside.
“Oh yeah?” said Feynman’s friend. “Do you know the crazy shape of the crankshaft in a car?”
“Yeah, what of it?”
“Good. Now tell me: How did you describe it when you were talking to yourself?”
It was then that Feynman realized that thoughts can be visual as well as verbal.2
He later wrote about how, when he was a student, he had struggled to imagine and visualize concepts such as electromagnetic waves, the invisible streams of energy that carry everything from sunlight to cell phone signals. He had difficulty describing what he saw in his mind’s eye.3 If even one of the world’s greatest physicists had trouble imagining how to see some (admittedly difficult-to-imagine) physical concepts, where does that leave us normal folks?
We can find encouragement and inspiration in the realm of poetry.4 Let’s take a few poetic lines from a song by American singer-songwriter Jonathan Coulton, called “Mandelbrot Set,”5 about a famous mathematician, Benoit Mandelbrot.
Mandelbrot’s in heaven
He gave us order out of chaos, he gave us hope where there was none
His geometry succeeds where others fail
So 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
The essence of Mandelbrot’s extraordinary mathematics is captured in Coulton’s emotionally resonant phrases, which form images that we can see in our own mind’s eye—the gentle flap of a butterfly’s wings that spreads and has effects even a million miles away.
Mandelbrot’s work in creating a new geometry allowed us to understand that sometimes, things that look rough and messy—like clouds and shorelines—have a degree of order to them. Visual complexity can be created from simple rules, as evidenced in modern animated movie-making magic. Coulton’s poetry also alludes to the idea, embedded in Mandelbrot’s work, that tiny, subtle shifts in one part of the universe ultimately affect everything else.
The more you examine Coulton’s words, the more ways you can see it applied to various aspects of life—these meanings become clearer the more you know and understand Mandelbrot’s work.
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.6 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
When you see the letter a, for acceleration, you might feel a sense of pressing on the accelerator in a car. Zounds! Feel the car’s acceleration pressing you back against the seat.
Do you need to bring these feelings to mind every time you look at the letter a? Of course not; you don’t want to drive yourself crazy remembering every little detail underlying your learning. But that sense of pressing acceleration should hover as a chunk in the back of your mind, ready to slip into working memory if you’re trying to analyze the meaning of a when you see it roaming around in an equation.
Similarly, when you see m, for mass, you might feel the inertial laziness of a fifty-pound boulder—it takes a lot to get it moving. When you see the letter f, for force, you might see with your mind’s eye what lies underneath force—that it depends on both mass and acceleration: m·a, as in the equation f = m·a. Perhaps you can feel what’s behind the f as well. Force has built into it a heaving oomph (acceleration), against the lazy mass of the boulder.
Let’s build on that just a wee bit more. The term work in physics means energy. We do work (that is, we supply energy) when we push (force) something through a distance. We can encrypt that with poetic simplicity: w = f×d. Once we see w for work, then we can imagine with our mind’s eye, and even our body’s feelings, what’s behind it. Ultimately, we can distill a line of equation poetry that looks like this:
w
w = f·d
w = (ma)·d
Symbols and equations, in other words, have a hidden text that lies beneath them—a meaning that becomes clear once you are more familiar with the ideas. Although they may not phrase it this way, scientists often see equations as a form of poetry, a shorthand way to symbolize what they are trying to see and understand. Observant people recognize the depth of a piece of poetry—it can have many possible meanings. In just the same way, maturing students gradually learn to see the hidden meaning of an equation with their mind’s eye and even to intuit different interpretations. It’s no surprise to learn that graphs, tables, and other visuals also contain hidden meaning—meaning that can be even more richly represented in the mind’s eye than on the page.
Simplify and Personalize Whatever You Are Studying
We’ve alluded to this before, but it’s worth revisiting now that we’ve got better insight into how to imagine the ideas that underlie equations. 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. Santiago Ramón y Cajal, for example, treated the microscopic scenes before him as if they were inhabited by living creatures that hoped and dreamed just as people themselves do.7 Cajal’s colleague and friend, Sir Charles Sherrington, who coined the word synapse, told friends that he had never met another scientist who had this intense ability to breathe life into his work. Sherrington wondered whether this might have been a key contributing factor to Cajal’s level of success.
Einstein was able to imagine himself as a photon.8 We can gain a sense of what Einstein saw by looking at this beautiful vision by Italian physicist Marco Bellini of an intense laser pulse (the one in front), being used to measure the shape of a single photon (the one in the back).
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. What would that second photon see and feel?
Barbara McClintock, who won the Nobel Prize for her discovery of genetic transposition (“jumping genes” that can change their place on the DNA strand), wrote about how she imagined the corn plants she studied: “I even was able to see the internal parts of the chromosomes—actually everything was there. It surprised me because I actually felt as if I were right down there and these were my friends.”9
Pioneering geneticist Barbara McClintock imagined gigantic versions of the molecular elements she was dealing with. Like other Nobel Prize winners, she personalized—even made friends with—the elements she was studying.
It may seem silly to stage a play in your mind’s eye and imagine the elements and mechanisms you are studying as living creatures, with their own feelings and thoughts. But it is a method that works—it brings them to life and helps you see and understand phenomena that you couldn’t intuit when looking at dry numbers and formulas.
Simplifying is also important. Richard Feynman, the bongo-playing physicist we met earlier in this chapter, was famous for asking scientists and mathematicians to explain their ideas in a simple way so that he could grasp them. Surprisingly, simple explanations are possible for almost any concept, no matter how complex. When you cultivate simple explan
ations by breaking down complicated material to its key elements, the result is that you have a deeper understanding of the material.10 Learning expert Scott Young has developed this idea in what he calls the Feynman technique, which asks people to find a simple metaphor or analogy to help them grasp the essence of an idea.11
The legendary Charles Darwin would do much the same thing. When trying to explain a concept, he imagined someone had just walked into his study. He would put his pen down and try to explain the idea in the simplest terms. That helped him figure out how he would describe the concept in print. Along those lines, the website Reddit.com has a section called “Explain like I’m 5” where anyone can make a post asking for a simple explanation of a complex topic.12
You may think you really have to understand something in order to explain it. But 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. This is why teachers often say that the first time they ever really understood the material was when they had to teach it.
IT’S NICE TO GET TO KNOW YOU!
“Learning organic chemistry is not any more challenging than getting to know some new characters. The elements each have their own unique personalities. The more you understand those personalities, the more you will be able to read their situations and predict the outcomes of reactions.”
A Mind For Numbers Page 17
