"Okay," says the teacher, relentlessly cheerful as teachers of youngish children are expected to be, "does everyone understand this chapter?"
"YES!" bellow the assembled students.
"Have you all finished the sample problems?" the teacher continues.
"YES!" the kids chorus again.
"Are there any questions about the homew--"
"NO!"
You probably can guess the punch line. But if not, check it out (just copy and paste):
http://www.babyblues.com/archive/index.php?formname=getstrip&GoToDay=11/29/2009
Saturday, December 19, 2009
Saturday, December 12, 2009
The Week with Less Pizza
As you may know, the 3-4 students have been keeping track of pizza sales thus far this year. Yes, we have records stretching back as far as, let me see, September 10 or so!
For quite some time, as you'll see on the graph pictured below (in two parts), the total pizza order was a rather dull oscillation between 144 slices (18 whole pizza pies) and 152 slices (19 pies). Week after week, 144 or 152, 152 or 144. You could set your watch by it. It was like, I don't know, jazz music or Blue's Clues or driving on Interstate 65 in northern Indiana or something. As the graph shows, the median (the middle value when the data points are ordered) stayed within a very constrained band of numbers, and the range (the difference between the lowest and highest values) remained absolutely, boringly, even mindnumbingly consistent.
[Note that the number of slices actually ordered by lower school students doesn't match the number of slices we actually buy. Why is that, I wonder? Hmmm...]
Then, all of a sudden one Friday, the number of slices ordered took a nosedive. Fell off a cliff, or at least rolled down a slope, as the graph makes clear. Woke us all up, I tell you that. Boom, all the way down from the 150 region to...104. 104! Think of it! The median didn't change (why it didn't was food for thought for some of the students), but the range changed, oh boy did it ever.
Why would things be so different this week? I asked the gathered third and fourth grade children (after swearing to secrecy Ellen's class, which had handled the order and therefore knew the answer). What possibilities do you think there are?
They came up with four:
A) There were a LOT of kids out with swine flu.
B) Some of the classes were on a field trip.
C) The pizza place ran out of pizza partway through.
And
D) Not very many people were hungry for pizza that day.
I wonb't tbell ybou thbe rbeal ansbwer. But wbith anby lbuck, yobu cban gbuess.
For quite some time, as you'll see on the graph pictured below (in two parts), the total pizza order was a rather dull oscillation between 144 slices (18 whole pizza pies) and 152 slices (19 pies). Week after week, 144 or 152, 152 or 144. You could set your watch by it. It was like, I don't know, jazz music or Blue's Clues or driving on Interstate 65 in northern Indiana or something. As the graph shows, the median (the middle value when the data points are ordered) stayed within a very constrained band of numbers, and the range (the difference between the lowest and highest values) remained absolutely, boringly, even mindnumbingly consistent.
[Note that the number of slices actually ordered by lower school students doesn't match the number of slices we actually buy. Why is that, I wonder? Hmmm...]
Then, all of a sudden one Friday, the number of slices ordered took a nosedive. Fell off a cliff, or at least rolled down a slope, as the graph makes clear. Woke us all up, I tell you that. Boom, all the way down from the 150 region to...104. 104! Think of it! The median didn't change (why it didn't was food for thought for some of the students), but the range changed, oh boy did it ever.
Why would things be so different this week? I asked the gathered third and fourth grade children (after swearing to secrecy Ellen's class, which had handled the order and therefore knew the answer). What possibilities do you think there are?
They came up with four:
A) There were a LOT of kids out with swine flu.
B) Some of the classes were on a field trip.
C) The pizza place ran out of pizza partway through.
And
D) Not very many people were hungry for pizza that day.
I wonb't tbell ybou thbe rbeal ansbwer. But wbith anby lbuck, yobu cban gbuess.
Wednesday, December 9, 2009
What to Wear in Winter
As a high school student, I came up with a foolproof (and very mathematical) way to determine what winter clothes I needed.
I attended a PK-12 school, and the estimate depended on the behavior of two very different student groups: kindergarteners and sixth/seventh graders.
The K students were sent in (by parents) with masses of winter protection--coats, boots, hats, mittens, earmuffs, scarves, alpenstocks, beeveils, etc--and sent out (by teachers, into the elements) the same way.
The 6th/7th graders, regardless of what they were sent in wearing or sent out wearing, very quickly removed as much outer clothing as possible. There was something truly cool about wearing short sleeves as the mercury dipped down to the single digits Fahrenheit. (Also something truly frostbitten about it, but when you're in middle school you don't care.)
My formula was simple: to determine what level of clothing I needed, I found the halfway point between the overdressed kindergarteners and the underdressed middle school students. That was what I put on before leaving school.
Worked every time. And to judge by what I see out on our playground this winter, the formula continues to work today!
I attended a PK-12 school, and the estimate depended on the behavior of two very different student groups: kindergarteners and sixth/seventh graders.
The K students were sent in (by parents) with masses of winter protection--coats, boots, hats, mittens, earmuffs, scarves, alpenstocks, beeveils, etc--and sent out (by teachers, into the elements) the same way.
The 6th/7th graders, regardless of what they were sent in wearing or sent out wearing, very quickly removed as much outer clothing as possible. There was something truly cool about wearing short sleeves as the mercury dipped down to the single digits Fahrenheit. (Also something truly frostbitten about it, but when you're in middle school you don't care.)
My formula was simple: to determine what level of clothing I needed, I found the halfway point between the overdressed kindergarteners and the underdressed middle school students. That was what I put on before leaving school.
Worked every time. And to judge by what I see out on our playground this winter, the formula continues to work today!
Tuesday, December 1, 2009
On Family Size
It's important to connect numbers with real-life situations. Which is why I had 4th graders tell "stories" about the multiplication expression 4 x 6 as a warmup for a lesson this week. By "stories," I hasten to say, I don't mean great literary efforts, with foreshadowing and metaphor and plot twists and poetic license and all those great things. No, I mean simple situations like these:
"There were 4 glasses and each glass had 6 ice cubes in it."
"There were 4 people and each one ate 6 hot dogs."
"I saw 4 flowers. Each flower had 6 petals."
You'll note that in each case the 4 [the first number in the expression] represents the number of groups, and the 6 [the second number in the expression] represents the number in each group. Of course, 4 x 6 is equal to 6 x 4, which all the children I was working with that day knew perfectly well; but it's useful to think of the first and second numbers each playing a slightly different role in the expression.
And we were progressing swimmingly until one boy said, "There were 4 families and each family had 6..." Then his voice trailed off, and he thought, and then he said, "I mean, there were SIX families, and each family had 4 people in it."
Real-life situations indeed. No prizes for guessing how many people there were in his family!
"There were 4 glasses and each glass had 6 ice cubes in it."
"There were 4 people and each one ate 6 hot dogs."
"I saw 4 flowers. Each flower had 6 petals."
You'll note that in each case the 4 [the first number in the expression] represents the number of groups, and the 6 [the second number in the expression] represents the number in each group. Of course, 4 x 6 is equal to 6 x 4, which all the children I was working with that day knew perfectly well; but it's useful to think of the first and second numbers each playing a slightly different role in the expression.
And we were progressing swimmingly until one boy said, "There were 4 families and each family had 6..." Then his voice trailed off, and he thought, and then he said, "I mean, there were SIX families, and each family had 4 people in it."
Real-life situations indeed. No prizes for guessing how many people there were in his family!
Sunday, November 22, 2009
"You Must Be Smart at This"
I've been reading a fascinating book called "How We Decide," by a guy named Jonah Lehrer. The book contains many odd and interesting (and useful) tidbits of information relating to psychology, probability, and more. I'd mark it up with lots of underlining and margin notes, only I won't, because it's a library book.
One of the more intriguing stories in the book details an experiment done by a researcher named Carol Dweck. I've read about this study before, but not in such detail as it appears in this book. Here's what Dweck did:
1. She gathered a bunch of fifth graders and had researchers give them some simple nonverbal puzzles .
2. Then she had her researchers offer the children a one-sentence statement of praise--EITHER "You must be smart at this" OR "You must have worked really hard."
3. Then the researchers offered the kids a choice of two followup puzzles.
--Option A: A "harder" puzzle, "but you'll learn a lot just from trying it," or
--Option B: A puzzle that's "about as easy as the one you just tried."
The results? 90% of the "worked really hard" group opted for choice A. Less than 50% of the "must be smart" group did.
Dweck wasn't done. She gave the kids a REALLY hard puzzle. The "must be smart" group worked at it for a little while and got discouraged and frustrated. They gave up pretty quickly, on the whole. The "worked really hard" group--well, they worked really hard. "This is my favorite test," many of them claimed, even some of those who never actually solved it.
And when Dweck told the kids that they could see the work of students who'd done better than them or the work of kids who'd done worse, the "must be smart" kids typically chose to see the work of kids who'd done worse. The "worked really hard" kids, in contrast, tended to look at the work of kids who'd done better than they had. The "must be smart" group, Lehrer sums up, "chose to bolster their self-esteem" by looking at the work of students who hadn't done as well--who weren't as smart. The "worked really hard" group "wanted to understand their mistakes, to learn from their errors, to figure out how to do better."
The reasons for the split, to Dweck, were clear. "When we praise children for their intelligence," she writes, "we tell them that this is the name of the game: Look smart, don't risk making mistakes." The "smart" kids acted in ways that avoided putting their supposed level of intelligence to the test. In particular, they did their best to put themselves in situations where they'd be unlikely to make mistakes. "Mistakes," reports Lehrer, "were seen as signs of failure; perhaps [the children] really weren't smart after all." The "worked really hard" group, on the other hand, acted in ways that tended to reinforce the notion that they really WERE hard workers. The results were telling: they showed more curiosity, enjoyed themselves more, and in the end LEARNED more. Which is, after all, the point of school.
This has implications for all subjects, but perhaps especially for math. People tend to believe that math is something that you either CAN do or you CAN'T: you're "smart" at math or you're not. "I was never any good at math," parents (and teachers!) sometimes tell me. "I just don't have the knack for it....It's like other people have a math brain and I don't." I don't usually hear those kinds of things about social studies or even about reading.
For the record, there are lots of good reasons to reject the notion that some people have a "math brain" and others don't. But EVEN IF IT WERE TRUE, it isn't something I'd ever want to hear, because it simply isn't helpful. Dweck's research strongly suggests that if we changed the question "Which kids are smart when it comes to math?" to "Which kids work hard when it comes to math?", we'd all be better off--that kids who find math a little alarming might develop a more resourceful and positive attitude toward it; that kids who are already quick with numbers but accustomed to coasting might find themselves motivated to delve a little deeper and think a little harder; that kids of all ability and interest levels might be inclined to take more risks, show more persistence, and in the end, like the students in Dweck's study, learn more.
So. Two conclusions (for now, anyway).
One: when we teacher types say, "Mistakes are a natural part of learning," we really MEAN it.
And two: Yes, we know your kids are smart. Of course they're smart; they've got good genes, they've grown up in wonderful homes, they're verbal, they're curious, and they're as bright and funny as all-get-out. But do us (and yourselves, and your children) a favor:
Don't tell them.
One of the more intriguing stories in the book details an experiment done by a researcher named Carol Dweck. I've read about this study before, but not in such detail as it appears in this book. Here's what Dweck did:
1. She gathered a bunch of fifth graders and had researchers give them some simple nonverbal puzzles .
2. Then she had her researchers offer the children a one-sentence statement of praise--EITHER "You must be smart at this" OR "You must have worked really hard."
3. Then the researchers offered the kids a choice of two followup puzzles.
--Option A: A "harder" puzzle, "but you'll learn a lot just from trying it," or
--Option B: A puzzle that's "about as easy as the one you just tried."
The results? 90% of the "worked really hard" group opted for choice A. Less than 50% of the "must be smart" group did.
Dweck wasn't done. She gave the kids a REALLY hard puzzle. The "must be smart" group worked at it for a little while and got discouraged and frustrated. They gave up pretty quickly, on the whole. The "worked really hard" group--well, they worked really hard. "This is my favorite test," many of them claimed, even some of those who never actually solved it.
And when Dweck told the kids that they could see the work of students who'd done better than them or the work of kids who'd done worse, the "must be smart" kids typically chose to see the work of kids who'd done worse. The "worked really hard" kids, in contrast, tended to look at the work of kids who'd done better than they had. The "must be smart" group, Lehrer sums up, "chose to bolster their self-esteem" by looking at the work of students who hadn't done as well--who weren't as smart. The "worked really hard" group "wanted to understand their mistakes, to learn from their errors, to figure out how to do better."
The reasons for the split, to Dweck, were clear. "When we praise children for their intelligence," she writes, "we tell them that this is the name of the game: Look smart, don't risk making mistakes." The "smart" kids acted in ways that avoided putting their supposed level of intelligence to the test. In particular, they did their best to put themselves in situations where they'd be unlikely to make mistakes. "Mistakes," reports Lehrer, "were seen as signs of failure; perhaps [the children] really weren't smart after all." The "worked really hard" group, on the other hand, acted in ways that tended to reinforce the notion that they really WERE hard workers. The results were telling: they showed more curiosity, enjoyed themselves more, and in the end LEARNED more. Which is, after all, the point of school.
This has implications for all subjects, but perhaps especially for math. People tend to believe that math is something that you either CAN do or you CAN'T: you're "smart" at math or you're not. "I was never any good at math," parents (and teachers!) sometimes tell me. "I just don't have the knack for it....It's like other people have a math brain and I don't." I don't usually hear those kinds of things about social studies or even about reading.
For the record, there are lots of good reasons to reject the notion that some people have a "math brain" and others don't. But EVEN IF IT WERE TRUE, it isn't something I'd ever want to hear, because it simply isn't helpful. Dweck's research strongly suggests that if we changed the question "Which kids are smart when it comes to math?" to "Which kids work hard when it comes to math?", we'd all be better off--that kids who find math a little alarming might develop a more resourceful and positive attitude toward it; that kids who are already quick with numbers but accustomed to coasting might find themselves motivated to delve a little deeper and think a little harder; that kids of all ability and interest levels might be inclined to take more risks, show more persistence, and in the end, like the students in Dweck's study, learn more.
So. Two conclusions (for now, anyway).
One: when we teacher types say, "Mistakes are a natural part of learning," we really MEAN it.
And two: Yes, we know your kids are smart. Of course they're smart; they've got good genes, they've grown up in wonderful homes, they're verbal, they're curious, and they're as bright and funny as all-get-out. But do us (and yourselves, and your children) a favor:
Don't tell them.
Wednesday, November 18, 2009
How to Measure: An Illustrated Manual
The Definitive Treatise, by PDS First Graders.
1. “There can’t be gaps when you measure. You have to push the measuring things together, like this.”
2. “And you can’t make the rods all zigzag. You have to put them together in a straight line, like this. Don’t get off track.”
3. “The first thing is you have to estimate how many rods will fit.”
4. “You should look at it carefully. Then you can use your fingers to help you estimate.”
5. “You start by putting the first rod down so its end is right at the edge of the thing you’re measuring.”
6. “Then put more of them along the side, like this.”
7. “Go until you get to the other end of what you’re measuring! Then count how many rods you used.”
And now you know how to measure!
1. “There can’t be gaps when you measure. You have to push the measuring things together, like this.”
2. “And you can’t make the rods all zigzag. You have to put them together in a straight line, like this. Don’t get off track.”
3. “The first thing is you have to estimate how many rods will fit.”
4. “You should look at it carefully. Then you can use your fingers to help you estimate.”
5. “You start by putting the first rod down so its end is right at the edge of the thing you’re measuring.”
6. “Then put more of them along the side, like this.”
7. “Go until you get to the other end of what you’re measuring! Then count how many rods you used.”
And now you know how to measure!
Monday, November 16, 2009
A Dot Can Be...
This morning, Bill and Robbie's kindergarten read Donald Crews's picture book Ten Black Dots. It's a lovely book showing imaginative ways of transforming static black dots into familiar objects...
...as in the children's responses pictured below.
One dot can make a Cyclops...
...or a squirrel hole...
...or a window in a house...
As for 2 dots, they make great eyes...
Stop by the kindergarten to see the whole series!
...as in the children's responses pictured below.
One dot can make a Cyclops...
...or a squirrel hole...
...or a window in a house...
As for 2 dots, they make great eyes...
Stop by the kindergarten to see the whole series!
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