Week 19: Black Magic

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Arthur C. Clarke's Third Law of Prediction states that "any sufficiently advanced technology is indistinguishable from magic." If we can assume that all technology has its basis in electromechanical algorithms, then Clarke's Third Law can be restated as "any sufficiently advanced or complex algorithm is indistinguishable from magic." 

To illustrate, allow me to read your mind:

1. Pick a number from 1 - 10.
2. Multiply that number by 9.
3. Subtract 5 from your answer.
4. Assign a letter to the number that you get. (A=1,B=2,C=3,etc)
5. Think of a country in Europe that starts with that letter. 
6. Take the second letter in that country's name, and think of an animal that starts with it.
7. Are there really elephants in Denmark? << Highlight this magic row of text.

My 3rd grade teacher mystified my entire class with that "magic" trick. Here are some other useful spells (taken from Pat Gilliland's Book of Black Arts) for all you young witches and warlocks out there practicing your times tables!  

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 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 4: Know Your Place

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The Mathematics Content Standards for California Public Schools, adopted by the Board of Education in 1997, exemplifies my ideal of what government's role in society should be. The standards define "what" a student must look like at each grade level, but leaves it up to the teachers, the parents, the Sylvan Learning Centers, the ACI Institutes, and the indie math tutors (locally grown and organic, like me!) of the world to figure out "how" to actually go about creating those students.

One of the larger categories within the standards is something called "number sense". I interpret number sense as being able to understand numbers abstractly: what numbers mean, what they symbolize, and how they are related. A primary requirement for number sense, (1.0), states:  

Students understand the relationship between numbers, quantities, and place value in whole numbers up to 1,000.

The sub-requirements are:

Count, read, and write whole numbers to 1,000 and identify the place value for each digit. (1.1)

Use words, models, and expanded forms (e.g. 45 = 4 tens + 5) to represent numbers to 1,000. (1.2)

I've been struggling with how to teach Billy the concept of place value. He can identify the place value of each digit in whole numbers up to 1,000, per (1.1) but I know he doesn't really "get" it, because he's still struggling with (1.2). It's great that he knows how to identify the ones, tens, and hundreds place of a number, but it doesn't do him any good if he can't use this knowledge to get a "sense" of what the number actually means, abstractly.

Today, I tried introducing a game involving different colored poker chips. I would give him a number and ask him to create that number using a combination of various green (1), blue (10), and red (100) chips. He started getting very fidgety after about 1 minute of gameplay. I was losing him, and we were both getting frustrated.

I decided to give up on the game when I realized that, while it was a functioning game, it wasn't functioning to teach Billy what he needed to learn. The poker chip game was teaching him to express abstract quantities through color, when it should have been teaching him to express them through number. Hippies would argue that both are equally valid forms of expression, but I say hippies be damned. The reason we have standards is to ensure that, at some basic level, we can all speak a common language.

I think next week, I'll replace the poker chips with monopoly money. I figure, if the kid is gonna gamble, it's better he learn to gamble on something that the government is almost guaranteed to subsidize, to prop up, to perpetuate as a false promise--an opiate of the masses.

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."