فصل 14

دوره: ذهنی برای اعداد / درس 15

فصل 14

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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 friend 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 explanations 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.” —Kathleen Nolta, Ph.D., Senior Lecturer in Chemistry and recipient of the Golden Apple Award, recognizing excellence in teaching at the University of Michigan NOW YOU TRY!

Stage a Mental Play

Imagine yourself within the realm of something you are studying—looking at the world from the perspective of the cell, or the electron, or even a mathematical concept. Try staging a mental play with your new friends, imagining how they feel and react.

Transfer—Applying What You’ve Learned in New Contexts

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. That’s because when you learned the first foreign language, you also acquired general language-learning skills, and potentially similar new words and grammatical structures, that transferred to your learning the second foreign language.13 Learning math by applying it only to problems within a specific discipline, such as accounting, engineering, or economics, can be a little like deciding that you are not really going to learn a foreign language after all—you’re just going to stick to one language and just learn a few extra English vocabulary words. Many mathematicians feel that learning math through entirely discipline-specific approaches makes it more difficult for you to use mathematics in a flexible and creative way.

Mathematicians feel that if you learn math the way they teach it, which centers on the abstract, chunked essence without a specific application in mind, you’ve captured skills that are easy for you to transfer to a variety of applications. In other words, you’ll have acquired the equivalent of general language-learning skills. You may be a physics student, for example, but you could use your knowledge of abstract math to quickly grasp how some of that math could apply to very different biological, financial, or even psychological processes.

This is part of why mathematicians like to teach math in an abstract way, without necessarily zooming in on applications. They want you to see the essence of the ideas, which they feel makes it easier to transfer the ideas to a variety of topics.14 It’s as if they don’t want you to learn how to say a specific Albanian or Lithuanian or Icelandic phrase meaning I run but rather to understand the more general idea that there is a category of words called verbs, which you conjugate.

The challenge is that it’s often easier to pick up on a mathematical idea if it is applied directly to a concrete problem—even though that can make it more difficult to transfer the mathematical idea to new areas later. Unsurprisingly, there ends up being a constant tussle between concrete and abstract approaches to learning mathematics. Mathematicians try to hold the high ground by stepping back to make sure that abstract approaches are central to the learning process. In contrast, engineering, business, and many other professions all naturally gravitate toward math that focuses on their specific areas to help build student engagement and avoid the complaint of “When am I ever going to use this?” Concretely applied math also gets around the issue that many “real-world” word problems in mathematics textbooks are simply thinly disguised exercises. In the end, both concrete and abstract approaches have their advantages and disadvantages.

Transfer is beneficial in that it often makes learning easier for students as they advance in their studies of a discipline. As Professor Jason Dechant of the University of Pittsburgh says, “I always tell my students that they will study less as they progress through their nursing programs, and they don’t believe me. They’re actually doing more and more each semester; they just get better at bringing it all together.” 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.15 You may think you’re learning in between checking your phone messages, but in reality, your brain is not focusing long enough to form the solid neural chunks that are central to transferring ideas from one area to another.

TRANSFERRING IDEAS WORKS!

“I took fishing techniques from the Great Lakes and tried using them down in the Florida Keys this past year. Completely different fish, different bait, and a technique that had never been used but it worked great. People thought I was crazy and it was funny to show them that it actually caught fish.” —Patrick Scoggin, senior, history

SUMMING IT UP

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’s important to grasp the chunked essence of 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.

PAUSE AND RECALL

Close the book and look away. What were the main ideas of this chapter? Can you picture some of these ideas with symbols in your mind’s eye?

ENHANCE YOUR LEARNING

  1. Write an equation poem—several unfolding lines that provide a sense of what lies beneath a standard equation.

  2. Write a paragraph that describes how some concepts you are studying could be visualized in a play. How do you think the actors in your play might realistically feel and react to one another?

  3. Take a mathematical concept you have learned and look at a concrete example of how that concept is applied. Then step back and see if you can sense the abstract chunk of an idea underlying the application. Can you think of a completely different way that concept might be used?

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