David had his students look closely at some addition problems this morning. He set up 5-6 large sheets of paper on the tables in the room, each with a problem such as 16,732 + 5,895. Each of the problems came pre-solved, with every step of the solution carefully worked out and on display.
However, each solution used a different algorithm.* In each case, the students had to figure out how the solving strategy worked and respond to the strategy. Was it easy to understand? Was it clear what was going on? Had they tried it before? Would they consider trying it in the future? Would it work better with some kinds of numbers than others? They moved around the room, analyzing each algorithm and writing briefly about it as well.
*Of course, you remember what an algorithm is--it's an alligator that wins So You Think You Can Dance, right? Right? I mean, think about it, Al's got rhythm? Rhythm? Al? Al-go-rithm? -- Oh, all right. An algorithm is a series of steps which, if performed accurately and sequentially, will always lead to a correct answer.
The kids were very familiar with some of the algorithms on display, less so with others. The "traditional" algorithm was in there: you know, the one with carrying, which we actually call regrouping or trading because it's a more descriptive term thankyouverymuch. ("Seven plus five is twelve, carry the one" should actually be "Seven plus five is twelve, that's a ten and two more; the ten goes upstairs...") There were a couple of other models in which numbers were written in expanded form (that is, 2,368 becomes 2000 + 300 + 60 + 8); there was one which used a series of partial sums...all of them were useful in at least some circumstances. "Think about them," David instructed the students, "compare them, see which ones make sense to you."
A few of the responses, written and oral:
"This one's interesting. It uses expanded form, but then it puts the numbers back together in a very mathy way." (This student and I are going to write a song, perhaps: "A Very Mathy Way." Listen for it at the Grammys.)
*"I recommend this one if you're in calculator school. If not, try different ways."
*"This part over here--that's just like a sloppy copy when you're writing a story. You don't have to look at it too closely."
*"Easy and quick if you fully understand the base ten system." A nice companion for "An example of using the base ten system in an advanced way." Different algorithms, and (I think) different responders as well.
*"[The person who solved it this way] is a strong mathematician that made a hard question easy."
*"Our group thinks it's too messy. There's too many lines and numbers." (But do you understand what's going on? I inquired.) "Oh, we understand it. We just think it's messy."
*"EASY" (Upper case letters and lack of punctuation in the original)
*"This one doesn't make very much sense. But it does make a little bit of sense."
*"I am going to try it because it looks cool and fun."
Wednesday, September 28, 2011
Thursday, September 15, 2011
First Week of School
As you very likely know, PDS opened last Wednesday. I must say it's been very nice to be back with the kids again. There's something quite wonderful about children waving enthusiastically at you from halfway across the room and yelling out"Hi, Math Guy!" Makes a man feel worthwhile!
This blog is also back from vacation, so perhaps it will have some children yelling, "Hi, PDSMathGuy blog!" from across the room at it...Or texting "Hi, PDSMathGuy blog!" Or emailing it. Or perhaps not.
In any case, a few little stories, vignettes from the first few days of school to get you smiling (I hope):
***I am presenting a lesson to first graders about odd and even numbers. Part of the followup includes a grid of numbers: 1, 2, 3, and so on. Kids are supposed to determine which numbers are even and which are odd, then color odd numbers blue and even numbers red. One child digs in the colored pencil box and pulls out a blue one, which--alone among all the pencils in the box, it seems--has been sharpened on both ends. "It's a good thing I'm supposed to color the odd numbers blue," she tells me. "This is a blue pencil. And it's very odd."
***A second grade class. A child is paying extremely close attention to her teacher, who is showing the children how to play a game. You can see the child's eyes locked onto the board, her body still, her jaw determined. When she gets up and heads for her seat, there is a Weary Expression on her face. Dropping down on her chair like a rag doll, she shakes her head and looks up at me. "Hard...work," she says emphatically. "Hard...work!"
***Another second grader. The question is to come up with two numbers you can add to make 27. I'm asking him to try the problem in his head, without manipulatives and without writing numbers on a sheet of paper. "Okay, 15," he says proudly. Well, that's a fine start, I tell him, but I'm asking for two numbers and an addition expression, so just saying 15... He nods. "Well, 15," he says, "plus whatever number you put with 15 to make 27."
***Yet another second grader. "What are some things you know about math?" I ask. "Well, ONE thing I know is that math is fun!" he responds immediately. Then, thinking a moment and remembering who he's talking to, he adds, "And I bet you would agree with that!" (P.S. He's right. I would!)
This blog is also back from vacation, so perhaps it will have some children yelling, "Hi, PDSMathGuy blog!" from across the room at it...Or texting "Hi, PDSMathGuy blog!" Or emailing it. Or perhaps not.
In any case, a few little stories, vignettes from the first few days of school to get you smiling (I hope):
***I am presenting a lesson to first graders about odd and even numbers. Part of the followup includes a grid of numbers: 1, 2, 3, and so on. Kids are supposed to determine which numbers are even and which are odd, then color odd numbers blue and even numbers red. One child digs in the colored pencil box and pulls out a blue one, which--alone among all the pencils in the box, it seems--has been sharpened on both ends. "It's a good thing I'm supposed to color the odd numbers blue," she tells me. "This is a blue pencil. And it's very odd."
***A second grade class. A child is paying extremely close attention to her teacher, who is showing the children how to play a game. You can see the child's eyes locked onto the board, her body still, her jaw determined. When she gets up and heads for her seat, there is a Weary Expression on her face. Dropping down on her chair like a rag doll, she shakes her head and looks up at me. "Hard...work," she says emphatically. "Hard...work!"
***Another second grader. The question is to come up with two numbers you can add to make 27. I'm asking him to try the problem in his head, without manipulatives and without writing numbers on a sheet of paper. "Okay, 15," he says proudly. Well, that's a fine start, I tell him, but I'm asking for two numbers and an addition expression, so just saying 15... He nods. "Well, 15," he says, "plus whatever number you put with 15 to make 27."
***Yet another second grader. "What are some things you know about math?" I ask. "Well, ONE thing I know is that math is fun!" he responds immediately. Then, thinking a moment and remembering who he's talking to, he adds, "And I bet you would agree with that!" (P.S. He's right. I would!)
Tuesday, December 7, 2010
In the Fullness of Time
It was a He-Man action figure, I think, that started the trouble on that spring day early in my teaching career. If you're old enough to remember they heyday of He-Man and his, um, business associates (the name Skeletor ring a bell?), congratulations. (And you do need to have some age on you, as these figures were popular in my kindergarten classroom around the same time as the A-Team and just before Cabbage Patch Kids (bless their ***** little hearts) and Transformers.) If you're not old enough to remember the excitement, if you grew up outside the cultural influence of the USA, or if your memory is like a sieve, never mind. It was a toy, that's all you need to know; a coveted toy, a toy with a definite cool factor, the kind of toy that conferred automatic social standing on its owner, even in kindergarten.
Anyhow, there was a stray He-Man figure, a slightly dusty one that someone had found wedged halfway under a shelf near the block corner. He immediately claimed it as his own property, based on two incontrovertible legal principles (incontrovertible among five-year-olds, at least): first, he'd found it; and second, just in case that didn't carry any weight with others, he was almost certain he'd lost that very figure some time earlier, and lost it in a part of the room that was surprisingly near the shelf in question. At first no one challenged the incontrovertibility of these claims, and he was happily showing it off to all his friends, and growing no doubt in social standing all the while...when up came another child, who promptly rocked the boat by claiming this very He-Man as his own property.
It soon became apparent that this was going to be one of those situations: a he-said, he-said situation, long before Anita Hill and Clarence Thomas came along to popularize that expression, or one almost exactly like it. A glance at the toy in question was of no help in resolving ownership. One He-Man toy looked very like another, after all, extruded as they were from the same plastic molding machine and colored by the same shade of dyes, and other than the dust there were no particular marks that distinguished this particular figure as belonging to either of the boys. No parent had scrawled his or her child's initials on the figure's lower back; no child had affixed a slice of black tape to He-Man's mighty foot for easy recognition; no serial numbers appeared on He-Man's warrious helmet.
"It's mine," said one of the boys, lower lip trembling, "I know it is. I remember."
"It's mine," said the other boy, a catch in his voice. "I recognize it."
Both looked pleadingly at the adult, a.k.a me. A Solomonic situation, I remember thinking, and briefly toyed with the notion of threatening to cut the figure in half. But no, I ducked, and simply suggested that they try to work out a fair way to solve the problem. That, after all, was high on the list of values at the school, the ability to talk about problems and come to an agreement that would be, um, agreeable to both parties. (I note here that few of my former students seem to have gone into politics.) "Take a few minutes," I said, "talk about it, see what you come up with. When you have a plan that you think will work, let me know."
Off they went. And a few minutes later they were both back. The quivering lower lip was still, the tremor in the voice had ceased. "We worked it out," they told me in unison. And they had. "First I get it for ten years..." said the boy who had found it below the shelf.
"And then I get it for ten years," added the other boy.
It was a fair deal, on its surface, but I couldn't help feeling there was a flaw in their plan somewhere or other...
Anyhow, there was a stray He-Man figure, a slightly dusty one that someone had found wedged halfway under a shelf near the block corner. He immediately claimed it as his own property, based on two incontrovertible legal principles (incontrovertible among five-year-olds, at least): first, he'd found it; and second, just in case that didn't carry any weight with others, he was almost certain he'd lost that very figure some time earlier, and lost it in a part of the room that was surprisingly near the shelf in question. At first no one challenged the incontrovertibility of these claims, and he was happily showing it off to all his friends, and growing no doubt in social standing all the while...when up came another child, who promptly rocked the boat by claiming this very He-Man as his own property.
It soon became apparent that this was going to be one of those situations: a he-said, he-said situation, long before Anita Hill and Clarence Thomas came along to popularize that expression, or one almost exactly like it. A glance at the toy in question was of no help in resolving ownership. One He-Man toy looked very like another, after all, extruded as they were from the same plastic molding machine and colored by the same shade of dyes, and other than the dust there were no particular marks that distinguished this particular figure as belonging to either of the boys. No parent had scrawled his or her child's initials on the figure's lower back; no child had affixed a slice of black tape to He-Man's mighty foot for easy recognition; no serial numbers appeared on He-Man's warrious helmet.
"It's mine," said one of the boys, lower lip trembling, "I know it is. I remember."
"It's mine," said the other boy, a catch in his voice. "I recognize it."
Both looked pleadingly at the adult, a.k.a me. A Solomonic situation, I remember thinking, and briefly toyed with the notion of threatening to cut the figure in half. But no, I ducked, and simply suggested that they try to work out a fair way to solve the problem. That, after all, was high on the list of values at the school, the ability to talk about problems and come to an agreement that would be, um, agreeable to both parties. (I note here that few of my former students seem to have gone into politics.) "Take a few minutes," I said, "talk about it, see what you come up with. When you have a plan that you think will work, let me know."
Off they went. And a few minutes later they were both back. The quivering lower lip was still, the tremor in the voice had ceased. "We worked it out," they told me in unison. And they had. "First I get it for ten years..." said the boy who had found it below the shelf.
"And then I get it for ten years," added the other boy.
It was a fair deal, on its surface, but I couldn't help feeling there was a flaw in their plan somewhere or other...
Thursday, October 28, 2010
In Which the Math Guy Is Reminded (Yet Again) of the Importance of Not Making Assumptions
The second graders were measuring. They'd cut out replicas of their feet (exact size, natch) and were busily determining how many of these footprints (feetprints?) it took to equal the length of a shelf, the width of the room, and other various and sundry distances. Then they were converting the number of feetprints (footprints?) to inches and recording it all on a chart.
I plunked myself down next to a child who was recording the number of feetsprint she had needed to cover the distance across a table. She'd written a 7, which sounded reasonable--seven second-grade-sized footsprint looked about right--but what was this next to it? A zero? Seventy? Surely she was putting 70 in the wrong place of the chart. Or she'd mismeasured. Or--
Wait a minute.
It wasn't just a zero. It was a bubble letter--you know, the puffy letters that kids love to make, especially when time is of the essence. The ones that slow kids' work pace down to a crawl. The ones that drive me faintly crazy. The ones that--
Hold on.
Now she was decorating the thing. Shading in part of the inside ring, drawing something unrecognizable in the middle. Decorating--during math time! Bubble letters--during math time! I mean, gee whillikers!
I opened my mouth to say something gentle, yet pointed. Okay, something not-so-gentle yet pointed. Something about saving the artistry for art and getting back to math, and by-the-way was 70 really a reasonable answer, and if you'd been paying closer attention to the math rather than to the art you'd know...But then I didn't. "Tell me about what you're drawing," I said instead, pointing. Just in case my assumption was wrong and there was method to her madness.
"Oh, that's a quarter," she explained, barely looking up.
"The coin?" I asked. "The thing that's worth twenty-five cents?" I peered closer. Okay, now that she'd mentioned it I could see that the bubble-letter zero did indeed resemble a quarter. Fine and dandy, but that didn't explain why she drawn a coin as part of this measurement project. I opened my mouth again...but instead of the pointed comment I'd intended, I found myself with a different response, again a response that didn't automatically assume that she'd messed up.
"Why a quarter?" I asked.
"Well," she said, "when I measured the table I found it was seven and a quarter of my footsprints." She tapped the seven on the chart, then the quarter beside it. "So I wrote seven, and then I drew a quarter. That's why."
And that's why I'm glad I asked!
I plunked myself down next to a child who was recording the number of feetsprint she had needed to cover the distance across a table. She'd written a 7, which sounded reasonable--seven second-grade-sized footsprint looked about right--but what was this next to it? A zero? Seventy? Surely she was putting 70 in the wrong place of the chart. Or she'd mismeasured. Or--
Wait a minute.
It wasn't just a zero. It was a bubble letter--you know, the puffy letters that kids love to make, especially when time is of the essence. The ones that slow kids' work pace down to a crawl. The ones that drive me faintly crazy. The ones that--
Hold on.
Now she was decorating the thing. Shading in part of the inside ring, drawing something unrecognizable in the middle. Decorating--during math time! Bubble letters--during math time! I mean, gee whillikers!
I opened my mouth to say something gentle, yet pointed. Okay, something not-so-gentle yet pointed. Something about saving the artistry for art and getting back to math, and by-the-way was 70 really a reasonable answer, and if you'd been paying closer attention to the math rather than to the art you'd know...But then I didn't. "Tell me about what you're drawing," I said instead, pointing. Just in case my assumption was wrong and there was method to her madness.
"Oh, that's a quarter," she explained, barely looking up.
"The coin?" I asked. "The thing that's worth twenty-five cents?" I peered closer. Okay, now that she'd mentioned it I could see that the bubble-letter zero did indeed resemble a quarter. Fine and dandy, but that didn't explain why she drawn a coin as part of this measurement project. I opened my mouth again...but instead of the pointed comment I'd intended, I found myself with a different response, again a response that didn't automatically assume that she'd messed up.
"Why a quarter?" I asked.
"Well," she said, "when I measured the table I found it was seven and a quarter of my footsprints." She tapped the seven on the chart, then the quarter beside it. "So I wrote seven, and then I drew a quarter. That's why."
And that's why I'm glad I asked!
Sunday, October 10, 2010
at10tion!
Yup, it's the day we've all been waiting for--at least, the day us Math Guys have all been waiting for: Ten Ten Ten, or 10/10/10. (As many of you Eager Readers no doubt know, 10 holds a special place in every math guy's heart. 2 1/2, 8, 92, 753.6, even 3.14159...., they all have their points (some of them even have decimal points (sorry)), but none of them can hold their own next to Ten.) However you write it, it's as decimal a day as it gets (well, okay, 10/10/1010, a thousand years ago this afternoon, was maybe a skoonch better, and there's something quite appealingly, I don't know, clean, about 10/10/10 back in 10 CE, not that anyone knew it WAS 10 CE at the time).
I had a distant relative who I met back in 1978 or so. She showed me her passport and called my attention to her birthdate. I know, I know, ladies of a certain age are not supposed to reveal their ages, but she was so pleased with the day she was born that she couldn't resist. It was, of course, 10/10/10--1910, that is. I promised not to do the subtraction necessary to calculate her age at the time. Here's to you, Cousin Lily.
So have a happy Ten Ten Ten. Ten cheers for this day, and long may it wave.
I had a distant relative who I met back in 1978 or so. She showed me her passport and called my attention to her birthdate. I know, I know, ladies of a certain age are not supposed to reveal their ages, but she was so pleased with the day she was born that she couldn't resist. It was, of course, 10/10/10--1910, that is. I promised not to do the subtraction necessary to calculate her age at the time. Here's to you, Cousin Lily.
So have a happy Ten Ten Ten. Ten cheers for this day, and long may it wave.
Saturday, October 9, 2010
Or We Could Just Call 'em Number Cube(s)
First grade, a lesson on estimation.
"I'm going to show you a cup with dice inside," I explained. "I won't show it for very long. So, you won't be able to count how many there actually are. Instead, I want you to decide how many it COULD be and how many it COULDN'T be. Got it?"*
*The purpose of this activity is to get kids to establish a zone of possible answers. Very often kids think of estimation as a sport in which the goal is to guess exactly the right answer. It isn't. This project asks kids to identify numbers that they think are reasonable. The zone can be quite large (when working independently later on, one partnership ruled out 1, 2, 3, 4, and 5, and decided that anywhere from 6 up to 40 was perfectly reasonable) or quite small (another pair, in direct contrast, established a zone that went all the way from 10 to 11). Either way, you get a sense of kids' ability to think about large quantities and a sense of the confidence (or overconfidence) they bring to the table when it comes to mathematical thinking.
"Okay," I said when everyone had taken a look. "Would you say the cup is full?"
NO, they chorused. Some said it was mostly empty, others about half full, but all agreed that it absolutely was not full or even close.
"Could you see all the dice at the same time?" NO, again.
I had a number line of sorts on the board. I touched the number 1. "Could there be just one object in the cup?" I asked. NO. "How about 2?" NO. "Three?" NO, NO, NO. "Okay," I challenged, because after all explaining your thinking is an important part of mathematics, "you sound awfully sure. How can you be so sure?"
A boy raised his hand. "I could see if it was three," he said, holding up three fingers. "I couldn't tell how many there were, so I knew it wasn't more than three." He could tell just by looking, because he knew what three looked like. A couple of other children followed by explaining their own reasoning in remarkably similar terms. Great minds and all that.
One child remained with her hand up. "Yes?" I asked.
She smiled. "I knew it couldn't be just 1," she said, "because you said you had dice in the cup, and if there was just one then you would have to say you had a die in the cup."
True, too!
"I'm going to show you a cup with dice inside," I explained. "I won't show it for very long. So, you won't be able to count how many there actually are. Instead, I want you to decide how many it COULD be and how many it COULDN'T be. Got it?"*
*The purpose of this activity is to get kids to establish a zone of possible answers. Very often kids think of estimation as a sport in which the goal is to guess exactly the right answer. It isn't. This project asks kids to identify numbers that they think are reasonable. The zone can be quite large (when working independently later on, one partnership ruled out 1, 2, 3, 4, and 5, and decided that anywhere from 6 up to 40 was perfectly reasonable) or quite small (another pair, in direct contrast, established a zone that went all the way from 10 to 11). Either way, you get a sense of kids' ability to think about large quantities and a sense of the confidence (or overconfidence) they bring to the table when it comes to mathematical thinking.
"Okay," I said when everyone had taken a look. "Would you say the cup is full?"
NO, they chorused. Some said it was mostly empty, others about half full, but all agreed that it absolutely was not full or even close.
"Could you see all the dice at the same time?" NO, again.
I had a number line of sorts on the board. I touched the number 1. "Could there be just one object in the cup?" I asked. NO. "How about 2?" NO. "Three?" NO, NO, NO. "Okay," I challenged, because after all explaining your thinking is an important part of mathematics, "you sound awfully sure. How can you be so sure?"
A boy raised his hand. "I could see if it was three," he said, holding up three fingers. "I couldn't tell how many there were, so I knew it wasn't more than three." He could tell just by looking, because he knew what three looked like. A couple of other children followed by explaining their own reasoning in remarkably similar terms. Great minds and all that.
One child remained with her hand up. "Yes?" I asked.
She smiled. "I knew it couldn't be just 1," she said, "because you said you had dice in the cup, and if there was just one then you would have to say you had a die in the cup."
True, too!
Saturday, September 18, 2010
You're Wrong HA HA HA
Ellen asked her 4th graders to write down what helped them in solving math problems, and what did precisely the opposite. This is a great assignment, as it requires kids to think about their own study habits and student skills: metacognition at its finest.
It was also fun to read the replies. Most students said that a quiet room helped. (More students, it strikes me, than there are students who actually help KEEP the room quiet, but it's the thought that counts.) Many said that having manipulatives to work with was helpful. And a few were very specific: What helps, wrote one student, is "people keeping their feet to themselves."
More interesting, however, were the things that children said did NOT help. Here they were usually quite detailed and focused in their complaints.
It makes it hard, said one child, when "people [are] invading my personal space." (Presumably that includes invading it with feet; see above.)
"Being told that your idea is terrible," suggested a classmate.
"When people are unnice," commented a third. (I agree. I much prefer it when people are unmean. Not to mention unloud.)
Another student wrote: "To hear any bragging or any kind of DISTRACTION." [caps in the original]
Another hated hearing "I wish you'd catch up with me so we can work together, since I'm so much farther than you." (A subtle slam, couched in nice enough words but with a very unnice message.)
"To say 'I'm right and you're wrong HA HA HA," wrote another student. (Bad enough to be wrong, worse for someone else to be right, but I agree, the HA HA HA really puts it over the edge into no-jury-would-ever-convict territory.)
And this one, which sounds like a couple of lines from a 50s song, maybe sung by one of those girl groups I can never remember the name of:
"I don't like to be pressured
I don't wanna be bugged."
(If you'd like to take a stab at completing the lyric, please note that "pressured" rhymes nicely with "Eschered"--presumably the act of being turned into an impossible figure or a slightly bizarre tessellation--and "bugged" with "hugged," "chugged," "plugged," "tugged," and "mugged." Among others. Good luck!)
It was also fun to read the replies. Most students said that a quiet room helped. (More students, it strikes me, than there are students who actually help KEEP the room quiet, but it's the thought that counts.) Many said that having manipulatives to work with was helpful. And a few were very specific: What helps, wrote one student, is "people keeping their feet to themselves."
More interesting, however, were the things that children said did NOT help. Here they were usually quite detailed and focused in their complaints.
It makes it hard, said one child, when "people [are] invading my personal space." (Presumably that includes invading it with feet; see above.)
"Being told that your idea is terrible," suggested a classmate.
"When people are unnice," commented a third. (I agree. I much prefer it when people are unmean. Not to mention unloud.)
Another student wrote: "To hear any bragging or any kind of DISTRACTION." [caps in the original]
Another hated hearing "I wish you'd catch up with me so we can work together, since I'm so much farther than you." (A subtle slam, couched in nice enough words but with a very unnice message.)
"To say 'I'm right and you're wrong HA HA HA," wrote another student. (Bad enough to be wrong, worse for someone else to be right, but I agree, the HA HA HA really puts it over the edge into no-jury-would-ever-convict territory.)
And this one, which sounds like a couple of lines from a 50s song, maybe sung by one of those girl groups I can never remember the name of:
"I don't like to be pressured
I don't wanna be bugged."
(If you'd like to take a stab at completing the lyric, please note that "pressured" rhymes nicely with "Eschered"--presumably the act of being turned into an impossible figure or a slightly bizarre tessellation--and "bugged" with "hugged," "chugged," "plugged," "tugged," and "mugged." Among others. Good luck!)
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