Week 15: The Third Rail

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Today, in the middle of a multiplication lesson, some words printed on the cover of Billy’s notebook catch his eye. They are taken from a passage in the bible, Philippians 4:8: "Think about all you can praise God for."

The words seem to call out to him. He stops paying attention to me, deciding that now is as good a time as any to practice his reading comprehension. Proceeding staccato like, he reads each word aloud, leading with his fingers.

“Think...uh-bowt….all…you…can…praise…gaw—” He pulls his finger back, recoiling in disgust. The rhythm is broken. 

“What’s wrong?” I ask.

“That’s Christian. That’s for Christians. I’m not Christian.”

A sharp hiss cuts the air. (Did it come from me, or his mom?) I feel my heart palpitating. The Accenture instincts are kicking in: we call this the red zone—the forbiddingly red third rail of client relationships.

His mother is in the kitchen, within earshot, preparing a refreshing couscous salad that has always been the highlight of my afternoon tutoring sessions. I consider what might happen if this dialog were to continue: she might stop feeding me; she might feed me poison; she might bury me up to my neck in the hot desert sand and throw jagged rocks at my face. Instinct can sometime breed irrational fear.

But I want this so bad. I bite my lip; heart racing, I reach for the rail. Instinct can sometimes breed irrational courage.

“Billy, what does ‘x-squared plus y-squared equals one’ mean?”

He furrows his brow in thought. “It’s math language for circle.”

“Right. How else can we say circle? What other languages do we know?” I draw a picture of a circle on a blank sheet of scratch paper and then, next to it, ask him to spell out “circle” in English and "yuvaruj" in Turkish. As a final touch, I add the Chinese character and the mathematical equation for circle. 

“These are all different ways of saying the same thing, just in different languages.” I explain. “It’s the same with the word ‘God’. It doesn’t (or at least it shouldn’t) matter whether you say ‘God’, like in the Christian Bible, or ‘Allah’, like in the Muslim Qu’ran. They are just two different ways of saying the exact same thing.”

“How do you say it in math?” Billy asks.

"That's a good question." I giggle.  

Week 14: Carrying on a Conversation

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An educated populace finds common ground and moves forward; an uneducated populace, paralyzed by divisions, stagnates and dies. Our nation’s founders knew this.

Citizen: What kind of a government have you given us?

Benjamin Franklin: A Republic, madam, if you can hold on to it!

James Howard Kunstler, in his book “The Geography of Nowhere” noted:

"For [Thomas Jefferson], the temple represented the culture that had first conceived of democracy (Greece), and that which had first devised a republican form of government (Rome), and it carried for Jefferson deep associations with his own classical education and his feeling that a democratic republic could only flourish if its citizens were educated.”

My heat transfer professor, Ralph Greif, once said to me:

Don’t worry that you’re not learning how to do or build anything. You’ll have plenty of time for that later. That’s what graduate school is for. That’s the easy stuff. Right now, focus on your literature classes, your social sciences. Relish your humanities—that’s what an education is about. 

Which reminded me of a soul-tingling dialogue between Captain Jean-Luc Picard and Ensign Wesley Crusher in the Star Trek episode Samaritan Snare (TNG 2x17):

Picard: Did you read that book I gave you?

Crusher: Some of it

Picard: That's reassuring.

Crusher: I just don’t have much time.

Picard: There's no greater challenge than the study of philosophy.

Crusher: Well, William James won't be on my Starfleet exam.

Picard: Important things never will be. Anyone can be trained in the mechanics and the piloting of a starship—

Crusher: but Starfleet Academy—

Picard: —takes more. Open your mind to the past—art, history, philosophy—and all this may mean something. 

In the midst of lectures, midterms, finals and projects—it is easy for the goal of education to get lost in the process of education: it is easy to mistake going to school with getting an education.

When we condition ourselves to think that learning only happens in school, we close our minds to so much more.

For me, once I realized that it was education for the sake of education, and not education for the sake of a job; once I realized to look beyond the specialized knowledge of an engineering education to see the universal truths driving all interaction—I was able to recognize my true, limitless potential.

Week 13: Your World. Delivered.

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In the process of education, we immerse ourselves in different fields of study, laying down a sample size of data in our brain large enough to illuminate a coherent pattern. The more we learn, the more this pattern is refined, reinforced, and strengthened. This pattern then lifts us up, allowing us to see the world not as a forebodingly complex jumble of dissonance, but as infinite variations on a singular theme: the world becomes one. 

Through education, we are empowered to infer beyond the world of our five senses, beyond our perception of reality. Through education—we find faith.

In Week 11, when Billy tried to sound out the word “mathematics”, he was transforming a single, ominous and impossibly complex “sound” into a logical sequence of more familiar, easy-to-spell syllables. He took a complex word, framed it as a collection of syllables, and used this new perspective to decompose the word into smaller, more manageable pieces.

This is the exact same approach we take to add any multi-digit sum, such as 88 + 66. The down and dirty approach would be to start at 88 and then count a collection of 66 sticks, rocks, fingers, toes or some witches brew of all the above in order to arrive at the sum. But nobody does this—not anymore; modern mathematics allows us to sense and manipulate numbers more efficiently than that. In school, we all learned how to take a multi-digit addition problem, frame it as a collection of place values, and then use this new perspective to decompose it into a series of easy-to-calculate single-digit sums. The algorithm is simple: start at the ones place, carry over if necessary, move on to the next place value, and repeat.

Taking a complex problem, orienting it around a familiar frame of reference, and then decomposing it into more manageable components: this is problem solving! Through education, we expose ourselves to so many different frames of reference and familiarize ourselves with such a diversity of viewpoints that no problem is too complex.

Education: the entire world—decomposed at your fingertips.

Week 12: No Training Wheels

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In Week 11, I taught Billy how to “see”. I gave him a new tool called “sounding it out” that would help him break down big scary words into simple sequences of easy-to-spell syllables. But what if Billy never learned to “see”?

Billy could hypothetically grow up never learning how to “see” syllables. To learn how to spell a new word, he would copy it onto a piece of paper, paying close attention to the sequence of each letter, and then repeat this process over and over again until the word finally commits itself to his aural, visual, and muscle memory—it becomes instinct.

If that hypothetical sounds way too hypothetical—you must not be Chinese.

Consider that there is no phonetic alphabet in the Chinese language. When students of Chinese learn to write, they are committing thousands of individual characters to memory, associating words with patterns that often contain no phonetic cues whatsoever. They have no tool other than brute force memorization to help them through this process.

Memorizing three to four thousand different characters in order to write: sounds like hell, doesn't it? I thought so too until I recalled that one, really cool Princeton study that people plastered all over their AIM profiles and mass-forwarded in e-mails several years ago. Remember? It went something like this:

A Priecnton stduy reaelved that flnuet Eglnsih reaedrs don't denped on phonetic cues but rahter learn to reogncize shpaes and pattrens in context wehn reidang. 

If you showed that to a first year "English as a foreign language" student, his head would probably explode! But assuming you have native English fluency, that previous passage should have been pretty damn cool! It means that after a while, we don't really "read" anymore; that is,(take a deep breath) we're not really grouping individual letters into syllables and then connecting them to form a complete sound that our brain then recognizes as a word with meaning (exhale). Like the Chinese, we just look at pictures.

Now the benefit of having a written language based on phonetic characters is that if you can’t remember how to draw the picture, you can always “sound it out”. For new initiates to the language, this tool functions like a pair of training wheels, easing them into fluency. After repeated exposure over many many years through books, newspapers, magazines, and TV, they learn to draw most pictures by heart, and “sounding it out” is no longer necessary.

(How many pictures, you ask? Estimates put the figure for basic English literacy at about 3 to 4 thousand words. Interestingly enough, this is pretty much the same number of characters a literate Chinese speaker must commit to memory.)

My friend Steven, who taught English in Shenzhen, China, mentioned something interesting once about how his less advanced English students deal with new vocabulary words. Apparently, the idea of a standardized system of phonetic cues is so foreign to Chinese students that they won't even try to pronounce new words.

Take the word CAT. Show it to any American preschool student and they will go: "Cat. Kuh- Ah - Tuh. Cat."

Flash to China: when Steven shows the word CAT to his students, they will stare at him blankly or fidget in their seats until he gives up in frustration and screams "MIAO! MIAO! MIAO! IT'S A $%&^*# CAT!" His students then go home and patiently write the word CAT fifty times, repeating it out loud each time, until it is finally committed to memory.

Chinese students never learn how to "see" syllables; "sounding it out" is not a tool in their toolbox. When it comes to writing, the Chinese don't believe in training wheels—it's do or die.

Week 11: What Do You See?

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Billy keeps a journal of the cool and exciting things we learn each week. Today, I asked him to write down the following insight: Multiplication is an easy way to add the same number over and over again. It is an addition shortcut.

“How do you spell muh-ti-pi-kay-shun?” he asks.

“Muhl-tih-plih-kay-shun,” I enunciate slowly. “Sound it out.”

A few weeks ago, had I asked him to “sound it out”, this is what he would have done: write down the first two or three letters correctly, stare blankly at his paper, start chewing on his pencil, and then begin throwing out one random letter after another rapid-fire hoping I get so frustrated that I just spell the word out for him—and I usually do.
 
A few weeks ago, after many failed attempts at getting Billy to “sound it out”, I began to fear he might be dyslexic. According to Wikipedia, signs of dyslexia include difficulty counting syllables in words, called phonological awareness; and difficulty segmenting words into individual sounds, called phonemic awareness. His inability to spell polysyllabic words seemed consistent with these symptoms, but then again—it could have just been that I was a bad teacher.
 
So a few weeks ago, I started to pay attention, to open my eyes to the shapes and sounds of the words I saw around me. New clarity began bubbling to the surface from the depths of my unconscious mind. I began to see all handwritten, printed and pixilated words as simple sequences of easy-to-spell three- four- and five-letter sounds; for the first time in a long time, I was seeing syllables again! I had been given this tool in early childhood to help commit the spelling of new and formidable words to memory. But as the words grew more familiar, the tool lost its ubiquity, until finally it was forgotten—discarded like a pair of old training wheels.
 
I realized that Billy wasn’t having difficulty segmenting words into syllables because he was dyslexic; he was having difficulty segmenting words into syllables because he had no concept of syllable to begin with! He had not been taught to “see syllables” as I had; as such, he experienced an unfamiliar polysyllabic word as a single, ominous and impossibly complex “sound” that defied all attempts at decomposition. He would try to “sound it out” in its entirety, and fail, not because he was stupid, but because he couldn’t “see”.
 
It was truly a humbling experience to realize that my frustration at Billy’s inability to spell was rooted in my own ignorance of his situation. Not everybody can “see” syllables; it is not a trait inherent to our biology, but a skill to be mastered. I presumed that Billy could see, when in fact he was blind, and then got angry when he kept walking into walls.

A few weeks ago, I opened my eyes. And now Billy can see. 

Week 10: Easy Come, Easy Go

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Today, I ask Billy to set up a two column table in his notebook, labeling one column “Addition (+)” and the other “Multiplication (x)”. I want to return to the idea of math as a language, to teach him that “times” is just another “word” we use in mathematics to describe a special type of addition problem.

I write the expression 2 x 3 in the multiplication column, and ask him if he knows what it means.

“Two times three.” he responds.

“What does two times three equal?”

“Six!” he proclaims smugly. Time to shut him down.

“Why?”

“Uhhh,” he fumbles, “I just remember it from school.”

I write 2 + 2 + 2 in the adjacent column, and ask him if he knows what it means.

“Two plus two plus two.”

“And what does two plus two plus two equal?”

“Six.”

“Why?”

“Because.” he shows me two raised fingers, raises two more, and then another two, and then counts them out loud: “One, two, three, four, five, six—six!”

“Good. How many twos are you adding together?”

“Three.”

I point him back to the original 2 x 3 expression, and explain to him that 2 x 3 is the same as 2 + 2 + 2. I explain that, in math, we often need to add the same number over and over and over again, and so we created this new word called “times” to make things easier. Saying “two times three” is much easier, and much more concise, than saying “add three twos together”. We use it so much, in fact, that it’s easier to remember that 2 x 3 = 6, rather than have to calculate 2 + 2 + 2 each time.

To drive home this point, I ask him to write the problem “2 x 10 = ” in addition form, and to solve for the answer. He scribbles a long and unwieldy vertical addition problem in his notebook, and after a long and laborious series of calculations, arrives at 22—one two too many. I correct him, to his dismay, and then ask him if he wants to do another one.

His eyes bulge outwards in silent rage as he shakes his head vigorously no. I make him do another one anyway. 2 x 15. He bites down on his pencil in frustration.

With some additional prodding, he finally arrives at an answer, correct this time around. I ask him how he feels.

“I hate math.” he answers quietly, careful to avoid making any eye contact.

“I’m sorry it had to happen this way, but don’t worry, you’ll never have to do that again.” I explain that I have a special present for him that will ensure that something like this will never, ever, happen again.

“A calculator?” he asks.

“No. Even better.” I smirk. Rummaging through my bag, I pull out the holy grail of multiplication—the times table.

It was a hard lesson, but hey—nothing good ever comes easy.

Week 9: On Tradition Versus Nostalgia

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He's ready. 

Last week, Billy finished all 100 of the single-digit addition flashcards in under 8 minutes! I was so proud of him! He's discovered the 9 trick on his own. The 9 trick, like any algorithm, doesn't translate into prose very well, but it goes something like this:

9 trick--Any number plus 9 is equal to that number minus 1 plus 10.

Billy still relies on his fingers for 7+6, 7+8, and 6+8, but hey, so do I. And what?

So I think he's ready. I've decided to hold off on drilling single-digit subtraction problems, in favor of teaching him multiplication first. My reasons for this deviation from plan are twofold:

First, I see this as a chance to build Billy's mathematical ego, which has suffered from a state of perpetual bruising over the last year. I want him to feel confident in class. I want him to feel that sense of achievement and ego-building self-satisfaction that will stimulate the pleasure centers of his brain, creating a positive feedback loop that makes math more exciting for social reasons (read: educated elite), as opposed to purely economic reasons (read: I need to learn math to get a job).

Second, multiplication naturally follows addition, since both are simply variations on a single theme: summation.  2 x 5 is merely a superficial and arbitrary representation, a mask that conceals a deeper meaning. Peeling away that mask reveals the expression 2 + 2 + 2 + 2 + 2, a lengthy but simple summation.  

2 x 5 = 2 + 2 + 2 + 2 + 2 = 10

This long, unwieldy mathematical equation is the truth behind 2 x 5, a truth that is easily forgotten once we commit 2 x 5 = 10 to ritual memory.

But none of that is really important, unless you're trying to teach someone what multiplication is, conceptually. It is easy to teach children to memorize 2 x 5 = 10, using your authority as a teacher to force them to accept this at face value. It is harder to teach children what 2 x 5 really means.

It is easy to hide behind the shield of authoritarianism, because then you never have to explain why--only how. 

Authoritarianism works by creating a legion of machines that work for the sake of work, oblivious to the goal, oblivious to what it all means. Should the man behind the machine be lost, the machine will, out of ignorance, out of nostalgia, continue to function, even as it has no function. Ignorant of purpose, slave to ritual and form, its existence--meaningless.

They say that to teach is to know. I say that to teach function, practice form, and be able to explain the distinction--that is when you truly know.

Are we still talking about math here?