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.