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?

Week 8: An Old Friend

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To me, video games are visual/audio sensory theaters that engage our brains in ways very difficult to replicate in real life. Take my old green friend, the Number Muncher, for example. 

Watching him traverse the vast electronic grid on his evolutionarily questionable little green feet, chasing down and devouring numerical expressions to the sounds of 8-bit electronic gusto, while being chased in turn by a roving band of nameless numerical cannibalistic nightmares known collectively as the troggles; anyone with an ounce of humanity (or PETA membership) is left with no other choice: you must do the math! Replicating such a dire threat in real life is impossible. Math doesn’t kill people—people kill people. 

Compared to Number Munchers, flashcard games have the visual/audio appeal of a battery-powered shower radio.  The visual stimulus is sub-par: Arabic numerals in dull blue ink scrawled out hastily on square white index cards. The only sound to latch onto is the sound of your own, tremulous, 8-year old voice reciting the addition problem followed by what you hope and pray (you’ve made the mistake of crossing your fingers before) is the correct sum. An electronic stopwatch on your tutor’s fancy smartphone offers a modicum of technical delight, but nothing that compares to the integrated world of Number Munchers.    
 

It is with this understanding that I brought my old friend back from the edge of oblivion. I installed an MS-DOS emulator, loaded up a freeware version of the old classic downloaded off the internet, and let Billy loose! The game is a great way to keep Billy’s five senses engaged in math, and the best part is that it doubles as a highly effective 8-bit carrot of oppression.  

As needed.