This said, casually but not proudly, as one of my tutees tried to figure out 8 times 7 (she'd previously come up with 53).
Three over six, two out of four, 7 over 14, when you see these things I want a giant, golden, glowing one-half symbol to appear instantaneously in your head! I waved my hands in the air, drawing the iconic fraction, then tapped my forehead. I want bells to go off, I want it to be automatic.
It stuns me that it isn't automatic, that all three of my tutees in attendance today spent several minutes struggling with the simplification of 7/14.
All that they don't know presses like water against a levee. Basic multiplication facts. True understanding of what a fraction is, what a whole number is, why you can put a 1 under a whole number to show it in fraction form. Basic division facts. They struggle to divide 72 by two. What's half of 70? I ask. That's easier, but still takes far longer than it should. What's half of 2?
It's been two or three weeks. We've waded through approximately two lessons in a review workbook, first adding and subtracting fractions, now multiplying and dividing them. Multiplying and dividing are easier, but that's only because the algorithm for solving the problems is easier; it belies vast voids of understanding. What does it mean to multiply one fraction by another? What does it mean to divide one fraction into another? (Many adults couldn't tell you that). For that matter, what does it mean to multiply and divide at all? As we deal with fractions, I begin to suspect that their grasp of the essential meaning of these operations is tentative, or non-existent.
I am helping a 7th grader simplify 36 over 56. She knows that two goes into both the numerator and the denominator and can be used to simplify the fraction.
What is half of 36? What's 36 divided by 2?
She thinks for a minute. Nine. I cannot fathom where nine came from. Fifteen.
Okay, I say, trying to help. What's something you really like?
This is not exactly what I had in mind, but she's not easily drawn into the games teachers play, so I run with it.
Okay, so you have, um, 36 boxes of Chinese food, and you are sharing them equally with friend. You have to take half for yourself, give half to her. You're dividing them by two. How many boxes do you each get?
It gets me nowhere. With other kids, I talk about M&M's. I draw slashmarks, circles, rectangles. I hold up six fingers, put down three. How many are left? That's an easy question. So, what's another way to describe what fraction of my fingers I put down? I raise and lower fingers a few times. It's just so clearly half! But he doesn't see it.
I suspect that they don't see numbers. Seven out of fourteen isn't an imaginable quantity, not even in pieces of candy.
I am bringing in Tootsie Rolls tomorrow, although I'm not absolutely sure what we'll do with them. Count them. Sort them, cut them in half, in thirds. I am contemplating making fraction tangrams, so that quarters can be laid over thirds, to show division. I want to do this right. But to do this right feels like breaching the levee, allowing their mathlessness to flood forward. I want them to have the number concepts and to be able to use the algorithms and to know their basic facts by heart, and I want it to happen all at once, and without my really knowing a thing about teaching math... But I don't want to find worksheets and learn about manipulatives and go backwards, farther and farther backwards until we find one piece of solid mathematical ground on which to build. It's not a torrent I had any desire to get caught up in.