Week 21: Tell Me Why [Ain’t Nothin’ but a Mistake]

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Tell Me Why [Ain’t Nothin’ but a Mistake]

Billy becomes a mere proxy through which I air my grievances against the world. Is it fair for me to project my own frustrations onto him? Am I asking too much?

“What’s nine times eleven?” I ask.

“Uhh…”

“Use the eleven-trick.”

“Ninety-nine!”

“And nine times twelve?”

Billy starts counting on his fingers, but quickly gives up after realizing that 9 x 12 is too unwieldy a summation to subdue with fingers alone. He can’t remember, and doesn’t understand multiplication well enough to realize that 9 x 12 is just a hop-skip away from 9 x 11. And so he gives up, head collapsing into forearms crossed in quiet resignation on the table.

This has become an all too familiar scene, and it leaves me feeling as trapped and hopeless as he does. We spent weeks learning about what multiplication means. Why doesn’t he get it? Am I pushing him too hard? Is it my fault? Am I a bad teacher?

Or is it him? Is he incapable of learning? Have his parents and their liberal Waldorf approach to education spoiled and coddled Billy into a state of un-teachable complacency?

“Okay, shake it out.” I tell him (“shake it out” is our code word for “empty your mind and start over from the beginning”). He lifts himself from the table, shaking his head and flailing his arms vigorously. “Good. Now, again, what is nine times eleven?”

“Ninety-nine.” he answers.

“Good. Why? Why does nine times eleven equal ninety-nine?”

“Because of the eleven trick.”

“Wrong. The eleven trick helps us remember that nine times eleven equals 99, but it doesn’t tell us why nine times eleven equals ninety-nine.”

“Oh.” He nods in feigned understanding. But I know better.

I ask him to open up his journal and read the entry from a few weeks ago when we first started learning about multiplication: “Multiplication is an addition shortcut. It is an easy way to add the same number over and over again.”

Something about his recitation strikes a nerve...

All my revulsion over literal interpretations of religious texts the world over, and the mental slavery such readings impose; all my contempt for a “people’s movement” that perpetuates the superficial form of marriage over the real tradition of love, simply because “that is how it has always been;” all the weight of the world comes into sharp focus over the head of an unsuspecting 8 year old boy…who promptly bursts into flames and dies a shrieking heat-death.

“So why does 9 x 11 = 99? If you tell me it’s because of the 11-trick, I’m going to throw you out the window.”

Billy giggles a disarming giggle: “No you won’t!”

“Okay come on, I’ll help you.” I write 9 x 11 and its lengthy vertical addition form on a piece of paper. “9 x 11 is the same as 9 + 9 + 9 + 9 + 9 +… 9—how many of them?”

“11 of them!”

“Right. And 9 + 9 + 9 + 9 + 9 +… 9 is equal to 99. That’s why 9 x 11 = 99. 9 x 11 is the same as adding 9, 11 times, which is the same as 99. Now watch what happens when we add one more nine.”

I add an additional 9 to the vertical addition problem. “Now we are adding 9, 12 times. Do we really need to add 9, 12 times?”

“Yes..?” he offers meekly, studying my face for clues.

“No!" I snarl, slamming my fist down on the table. "We know that 9 x 11 = 99, so we can just skip all the way to the end here, and add just one more 9 to get 108. 9 x 12 = 108. This is why multiplication is called an addition shortcut. It’s okay to memorize your times tables, but it’s not okay to forget why. If you don’t know why 9 x 12 = 108, you don’t really know anything.”

I write the word “why” in his notebook, furiously tracing dark blue circles around it with my pen. “If you don’t know why,” I repeat, jabbing the paper repeatedly with the tip of my pen, “you don’t know anything!”

Week 18: Iron Chef

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In Week 6, I explained to Billy that quantity:numeral/place value as time:hour/minute as place:city/street/house. I taught Billy the relationship between quantity and place value by drawing analogies to other abstract concepts and the social conventions that we use to describe them. 

My hidden agenda in all this, if not obvious by now, is to develop in Billy a keen sense of form versus function, so as to "emancipate" him from the "chains of mental slavery", as Bob Marley would say. 

But at the same time, I am wary of getting too abstract, of teaching him everything except what is pertinent.  While I can see a distinction between the concept of quantity and the social convention of numbering, I doubt that Billy, at the tender young age of 8, is able to appreciate or even care about such a nuance. It's simply not relevant to someone who can barely read numbers as it is. 

So today, rather than wax philosophical about the biochemical orgy that ignites in my brain when thinking about number systems based on anything other than the quantity described by the Hindu-Arabic numeral 10—I teach Billy to cook.

How to Make a Number

Step 1) Find the Recipe: To make the number 512, you first need to figure out how many of each ingredient to add. Each place value represents one ingredient, and the number in that place value tells you how much of that ingredient you need. So in the number 512, you need 2 ones, 1 ten, and 5 hundreds.

Step 2) Prepare the Ingredients: The preparation process requires two tools which you have already mastered the use of: addition and multiplication. Use multiplication first to measure out enough of each ingredient: (5 x 100), (1 x 10), and (2 x 1).

Step 3) Combine the Ingredients: Once you have measured out enough of each ingredient, use addition to combine them all together: (5 x 100) + (1 x 10) + (2 x 1).

Step 4) Reduce the Ingredients: Reduce all the ingredients until you end up with a single number: (500) + (10) + (2) = 512.

Step 5) Garnish and Serve: Check to make sure everything reduced correctly, and serve!

Pretty damn concrete, right?

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 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.

Week 3: The Five Senses

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I'm not trying to teach Billy addition anymore. He gets it. I know that he can answer 5 + 1 by methodically working through the problem. He can put up 5 fingers, raise 1 more finger, and then count them all to arrive at 6. He goes from A (the problem) to B (the process) to C (the solution). This is great! Now the next step is to create shortcuts in his brain that cut out that middleman (B) and take him directly from (A) to (C). Memorization is the name of the game!

 

We always begin and end the day with a run through the deck of flashcards. The rules of the game require that he answer each problem in a very specific way. For example, if I show him a 5 + 1 flashcard, I expect him to say "five plus one is six," and not just "six". This rule requires him to practice holding a solution in memory as he goes back to recite the original problem, connecting A to C, if you will. He hates this rule.

Today, after getting tripped up on a particularly nasty flashcard because of this rule, Billy asked me the question that all teachers, parents, and figures of authority around the world dread: why?

"Why do I have to say the problem every time? Why can't I just tell you the answer?" 

I knew he was testing me. Billy wasn't seeking personal edification. Rather, like Socrates before him, his question was intended to poke holes in my rigid methodology, force me to concede my own ignorance, to undermine my authority and humiliate me. 

I wasn't falling for it. I kept my cool. I tilted my gaze upwards to focus on a nondescript spot on the ceiling, pausing long enough to reinforce my dominance in this relationship. I then reverted my gaze, looked him straight in the eyes, and proceeded to answer his question with another question of my own design (touche, Billy!).

"Do you know what the five senses are?"

He shakes his head no. I proceed to teach him the five senses: eyes see, nose smells, hands feel, ears hear, tongue tastes. I explain to him that the five senses are what our brains use to remember and learn new things, like math.

"Our brains needs to see math, touch math, hear math, smell math, and taste math (okay maybe not the last two) in order to learn. So when I show you a flashcard, your eyes "see" the problem and your brain remembers it. The more you see the flashcard, the more your brain remembers it.  When I make you write, 50 times, each math problem you've answered incorrectly, your hands "feel" the problem, your eyes "see" it, and your brain remembers it. Does that make sense?"

Billy shakes his head yes.

"So the reason I make you say the problem every time, the reason you can't just say the answer, is because I want your ears to "hear" the math. The more times you say the problem and answer together, the more your ears "hear", and the more your brain will remember, until one day, you won't need your fingers anymore! You'll just remember!"

Week 2: Connecting the Dots

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Billy's 3rd grade class is moving on to multiplication; meanwhile, Billy is still counting basic sums on his fingers. This has obviously been a nightmare for him, and it makes him feel stupid in class when others can answer basic multiplication questions but he can not. Multiplication requires a facility with basic, single-digit sums that Billy currently lacks, so until he is comfortable with addition, multiplication will always be out of reach. 

To help him catch up, I have created some flashcards that we use to repeatedly drill everything from 0 + 1 =1 to 10 + 10 = 20. I am hoping that, as we progress day-by-day, week-by-week, Billy will start to recognize helpful patterns that will aid him in committing these sums to memory.

He's already noticed some patterns involving sums with the number 1, and sums with the number 0. When we first started last week, he would recite the problem, hold up some fingers, and wiggle them awkwardly before coming up with the answer to a simple problem like 1 + 5. Today, I showed  him 1 + 5, and with fingers at bay, he responded almost instantly with "SIX!".

"How did you get that so fast?"

"I dunno."

"What's one plus six?"

"Seven."

"What's one plus nine?"

"Ten."

"What's one plus four?"

"Five."

"What's one plus any number?"

"Uhh......I just count one more number..?"

"Right! Exactly!" 

It was a proud moment for me. Even if he couldn't articulate the pattern, I knew that he was getting a sense for it. It manifests itself as a connection between two neurons uniting two separate regions of the brain guarding what seem to be two seemingly disparate ideas; dots collide, giving off random sparks that, every so often, grow to illuminate the world in a whole new way.

Once he knew it implicitly, we worked on creating an explicit, more formal definition. We call it the "Rule of One", and it states that any number plus one is equal to the next number.

Week 1: Mathematical Expression

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"This is SO HARD!" he exclaims as he throws his head back, a tortured look on his face. We had been working through some basic addition flashcards for the last 45 minutes, and by some I mean about 10. 5 minutes of awkward finger counting per problem: it was painful for us both.

"Of course it's hard. You're learning a new language. Learning a new language is always hard." I explain. 

He scratches his head. "Huh?"

"Math. Think of it like English, or Chinese, or Turkish. It's just another way of speaking to other people."

He looks at me like I'm crazy. I tear out a fresh sheet of notebook paper and set up a horizontal grid. I sketch a rough circle in the leftmost box, point to it, and ask him what it is.

"A circle." he responds. 

"Right. So in English, we say circle. I write 'circle' in the second box. "How do you say this in Turkish?"

"Uhh...yu-ah-rruj."

"Okay..so in Turkish it's yooo-ahhh-rruj." Billy giggles as I struggle with the word. I add my phonetic interpretation to the third box, and then begin writing the Chinese character for circle in the fourth box.

"What is that?" Billy asks, leaning forward in his chair for a closer look.

"This is Chinese for circle. This character is pronounced: yuan." 

"Wow cool!" Billy picks up his pencil and tries to copy the character.

"Yep. Now watch this." I write the mathematical equation for a circle in the last box. "Do you know what this says?"

"X-two plus y-two equals 1!" he proclaims. His attempt is valiant. I laugh out loud.

"It says circle. This is how we say 'circle' in math--in the language of math. You read it like this: x-squared plus y-squared equals 1."

"Did Einstein create this? My dad says Einstein was the smartest guy in the world."

"Well, he didn't really create this. But if you showed this to him, he would know that it says circle. He spoke math."

"Can I speak math?" 

"Yes, you can speak some math, but you still have a lot more words to learn."