Math Thoughts, Uncollected:
-There was a change in vocabulary, maybe sometime in the late nineties. When you and I studied math in first and second grade, there was a procedure called "borrowing." I would "borrow" from the tens column to make sure I had enough ones to complete my subtraction problem. But there was an issue, here: I wasn't really borrowing. I was never going to return the ten. Now, the same procedure is called "regrouping." You're taking a ten and "grouping" it anew, as ten ones. There are actual toys--called, pretentiously, "manipulatives"--to illustrate this operation. (Teachers sometimes have an addiction to pretentious language. A teacher is an "educator." A test is an "assessment." A book is "a text." This is to be avoided--but I'd argue that the switch from "borrowing" to "regrouping" really is a good one.)
-One very simple question I loved, from a mentor: Ask a child what happens when I have three thousand units, and I remove one.
Some children--and I would have been one of these children--will perform the tedious algorithm, as a knee-jerk reaction. (I'll regroup my thousand as ten hundreds, and I'll regroup one of those ten hundreds as ten tens, and so on.) Other kids will understand that I'm just going to have one fewer unit: I can tell you I'll have two thousands, and nine hundreds, and ninety tens, and nine units, without performing any real work. Then there are students who will not only reach for the unnecessary algorithm, but who will then also fail to perform the algorithm correctly. You get a great deal of information from one small question. (The mentor who invented this question was a dreamy, older, heterosexual man, and he often wore a De La Vega shirt with the inscription: "Become Your Dream." It had a squiggle of a small fish dreaming of becoming a larger fish.)
-A Marilyn Burns masterstroke was to have children explain their sums, all the time.
So, if she asked, "What is the sum of eight and four?" she wouldn't stop once the right answer was produced. She would ask: "How did you get that?" Then she would illustrate each child's procedure on the board. One child might count on her fingers. Another might split two off to make ten, then add on the other two. A third, wacky child, might do a doubles fact (eight plus eight) and then subtract four. There's subtext here. Burns wants kids avoiding their fingers as much as possible. She wants kids "making tens" and using doubles facts all the time. Because these tricks are the most efficient, and because they encourage further mastery, further risk-taking. Burns also encourages skip-counting as much as possible--because it's simply faster than counting by ones, and because it gets you ready for multiplication.
-I remember struggling to impart something as simple as "evens" and "odds," when I began teaching. But the answer is right there. Teachers now have a tool called "unifix cubes," which are just cubes that interlock. You take two and snap them together. You say: "See? My cube has a buddy. They are even, they are a pair." But if you try to make pairs with nine cubes, you see that one will not have a buddy; one will be an "odd man out." So you know that nine is an odd number. So much more satisfying than saying--for example--"You just should know it. You should memorize it!"
-Sometimes, fury would govern my soul. I remember colleagues saying, "I make a point of never complimenting a student's outfit, because it invites thoughts of materialism, and it suggests that some outfits are prettier than others." I can never, never sign on for this absurd and tiresome thinking. It's a cruel world out there. One way we all get by is by noticing things we like in a colleague's--or neighbor's--outfit. This makes the world more pleasant. I will always, always compliment a person's outfit when I see fit to do so.
Teachers would wring hands if students began discussing their summers, because this might result in the production of challenging, life-altering data: A student might discover that her classmate went on a fabulous trip, and thus learn that there is such a thing as socioeconomic injustice in the world. I found this conspiracy-of-silence attitude so condescending and infuriating: You think children don't realize there is socioeconomic injustice in the world? I'm generally in favor of laying bare all--all!--of the facts. For example: You could say, "Yes, certain people will have more money than you, but those people will have dazzling problems of their own. Drug addiction, paralyzing guilt, learned helplessness. We all suffer in our own spectacular and unique ways." (No, I never said that--but why not?)
And another thing that enraged me: A ban on war games. The policing of a child's imaginative life. I could never hop on board for that. Do you think these things don't come up in a math lesson? They do. That's the thing about elementary-school education: Everything comes up, in all contexts, all the time. A math lesson is also an early introduction to dating, a survey of the natural world, a bit of philosophical discourse, and a flirtation with new language skills. This is how children experience the world. Several plates spin at once, regardless of the subject you're discussing. This is what makes a classroom so much fun.
-There was a change in vocabulary, maybe sometime in the late nineties. When you and I studied math in first and second grade, there was a procedure called "borrowing." I would "borrow" from the tens column to make sure I had enough ones to complete my subtraction problem. But there was an issue, here: I wasn't really borrowing. I was never going to return the ten. Now, the same procedure is called "regrouping." You're taking a ten and "grouping" it anew, as ten ones. There are actual toys--called, pretentiously, "manipulatives"--to illustrate this operation. (Teachers sometimes have an addiction to pretentious language. A teacher is an "educator." A test is an "assessment." A book is "a text." This is to be avoided--but I'd argue that the switch from "borrowing" to "regrouping" really is a good one.)
-One very simple question I loved, from a mentor: Ask a child what happens when I have three thousand units, and I remove one.
Some children--and I would have been one of these children--will perform the tedious algorithm, as a knee-jerk reaction. (I'll regroup my thousand as ten hundreds, and I'll regroup one of those ten hundreds as ten tens, and so on.) Other kids will understand that I'm just going to have one fewer unit: I can tell you I'll have two thousands, and nine hundreds, and ninety tens, and nine units, without performing any real work. Then there are students who will not only reach for the unnecessary algorithm, but who will then also fail to perform the algorithm correctly. You get a great deal of information from one small question. (The mentor who invented this question was a dreamy, older, heterosexual man, and he often wore a De La Vega shirt with the inscription: "Become Your Dream." It had a squiggle of a small fish dreaming of becoming a larger fish.)
-A Marilyn Burns masterstroke was to have children explain their sums, all the time.
So, if she asked, "What is the sum of eight and four?" she wouldn't stop once the right answer was produced. She would ask: "How did you get that?" Then she would illustrate each child's procedure on the board. One child might count on her fingers. Another might split two off to make ten, then add on the other two. A third, wacky child, might do a doubles fact (eight plus eight) and then subtract four. There's subtext here. Burns wants kids avoiding their fingers as much as possible. She wants kids "making tens" and using doubles facts all the time. Because these tricks are the most efficient, and because they encourage further mastery, further risk-taking. Burns also encourages skip-counting as much as possible--because it's simply faster than counting by ones, and because it gets you ready for multiplication.
-I remember struggling to impart something as simple as "evens" and "odds," when I began teaching. But the answer is right there. Teachers now have a tool called "unifix cubes," which are just cubes that interlock. You take two and snap them together. You say: "See? My cube has a buddy. They are even, they are a pair." But if you try to make pairs with nine cubes, you see that one will not have a buddy; one will be an "odd man out." So you know that nine is an odd number. So much more satisfying than saying--for example--"You just should know it. You should memorize it!"
-Sometimes, fury would govern my soul. I remember colleagues saying, "I make a point of never complimenting a student's outfit, because it invites thoughts of materialism, and it suggests that some outfits are prettier than others." I can never, never sign on for this absurd and tiresome thinking. It's a cruel world out there. One way we all get by is by noticing things we like in a colleague's--or neighbor's--outfit. This makes the world more pleasant. I will always, always compliment a person's outfit when I see fit to do so.
Teachers would wring hands if students began discussing their summers, because this might result in the production of challenging, life-altering data: A student might discover that her classmate went on a fabulous trip, and thus learn that there is such a thing as socioeconomic injustice in the world. I found this conspiracy-of-silence attitude so condescending and infuriating: You think children don't realize there is socioeconomic injustice in the world? I'm generally in favor of laying bare all--all!--of the facts. For example: You could say, "Yes, certain people will have more money than you, but those people will have dazzling problems of their own. Drug addiction, paralyzing guilt, learned helplessness. We all suffer in our own spectacular and unique ways." (No, I never said that--but why not?)
And another thing that enraged me: A ban on war games. The policing of a child's imaginative life. I could never hop on board for that. Do you think these things don't come up in a math lesson? They do. That's the thing about elementary-school education: Everything comes up, in all contexts, all the time. A math lesson is also an early introduction to dating, a survey of the natural world, a bit of philosophical discourse, and a flirtation with new language skills. This is how children experience the world. Several plates spin at once, regardless of the subject you're discussing. This is what makes a classroom so much fun.
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