Mathlessness
I don't have all my multiplication facts memorized.
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?
Chinese food.
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.
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?
Chinese food.
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.
posted by ms. v. at 6:15 PM
6 Comments:
Today we were told (for the first time) that if a child answers a question on Day 2 using Guess and Check, they are supposed to show 2 other stategies. If not, they will not get any credit for their answer.
Guess and Check has always been another way for children to come up with a solution to a problem. Now the State is saying that the student must also show 2 other ways to solve the problem???? How is this helping the child?? Why was I just told about this with the test only 1 week away???
I would have liked to have seen an example of the scoring so I would have been able to see exactly what was required and show my students how to handle such a problem.
If anyone knows any additional information on this, I would appreciate it.
PS: Did u take the afterschool position?
I don't suppose your tutees will leave hungry, at least. It's awful they've gotten that far without tackling the multiplication tables.
If it were my kid, I'd be horrified. And she's only in fourth grade.
So, this is embarrassing, but I haven't actually memorized all my multiplication facts. For 8x7 I have to say to myself:
7x2=14, 14x2=28, 28x2=56
It all happens within 2 or 3 seconds, but still. I should just sit down one day and memorize them all.
Math teachers have been fighting the silly scoring rules for years. It all falls on deaf ears.
Everyday math, the math program used now in NYC in elementary school does not believe that the memorization of math facts is relevant to success in math.
The elementary and middle school teachers are quite aware that students are passing from grade to grade with limited math skills. We are told to pass these children. Passing has very little to do with the students' skills. It has everything to do with providing an appearance of success.
I understand the dilemma of not wanting to go so far backwards to find sold math understanding on which to build, but if no teachers ever goes back, then increasingly complex math concepts get built on top of non-existent understanding. Sure, students can memorize the alogrithm for multiplying fractions, but will they remember it a month from now if they don't understand what they're doing, never mind what a fraction is? I'm a fourth grade teacher, and I try to use manipulatives and go back to solid math footing as much as time permits because I know that each year it gets harder and harder to build on a shaky foundation of mathlessness. That's one reason I like teaching elementary school, I guess ... it seems easier to fill in gaps before they are insurmountable.
Is teaching math a little bit fun? Or does frustration totally overwhelm fun in your tutoring context?
I also teach 4th grade and I tutor low-performing elementary kids. It's already a struggle to help them develop what we call number sense. The teachers at our school really emphasize the importance of learning their basic facts. So many children just don't get it though for a variety of reasons. I worry so much about the children the move on without that basic understanding.
For my students who do not know their facts, I hold myself out as an example. I did not learn my multiplication facts and I did a great deal of addition on my lap with my fingers. Here's the difference though . . . I knew I had to get the answer. I couldn't just say, "I don't know." I had to figure it out.
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