"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!
Friday, November 13, 2009
(Why I Am Not a) Gamblin' Man (Mostly)
I have purchased one lottery ticket in my life. It was a loser. I have visited two casinos, one on a "riverboat" in Mississippi and the other on drier land in Louisiana. I put about $1.85 in slot machines, total. I lost it all. In 8th grade I was invited to a "Las Vegas" party, where I played roulette all night long (well, till 9 pm anyway) using poker chips. I bet on 36 time and again. It never came up. I lost, and lost, and lost some more. I eventually needed two loans from the bank. So, I don't gamble (mostly), because I lose (mostly, or maybe always).
Still, there are times when you just have to place that bet...
Yesterday I was in Jan's third and fourth grade class, where students are working on logic and attributes. The focus for the lesson was on Venn diagrams. You know, overlapping circle thingies, like this:
(My college roommate, Bernie, was a double major in science and philosophy. After taking a class on Tibetan Buddhism he once accidentally referred to these things as "Zen diagrams." I'm still tempted to call 'em that sometimes.)
In addition to the Zen, I mean Venn, diagram, we also had a bunch of blocks of various sizes, shapes, and colors, and labels with categories that described the blocks: "triangles," "small," "red," "yellow" and so on. I had chosen two labels at random ("What does that mean, 'at random?'" I'd asked earlier in the day, and the response was "Randomly," which was accurate if not perhaps revealing) and placed one in each circle of the Zen, I mean Venn, diagram. I got to look at the labels. The kids didn't.
The object is for the students to identify the two labels. They name blocks one by one, and I place each piece where it belongs: in the overlapping section of the diagram if it fits both labels; in one of the circles but not the other if it matches just one label; or outside both circles if it matches neither one. For instance, a small green triangle goes inside a circle marked "small," "green," or "triangle." Students use logic and the position of blocks in the diagram to determine what the labels CAN and CANNOT be.
Yesterday, after just three blocks, we had the following situation:
In the left circle, but NOT in the overlap, was a small blue rhombus. (A rhombus, for those not in the know, is not a method of transporting rhoms; it is a four-sided figure in which all sides are equal.)
In the overlap between the circles was a large blue rhombus. And outside both circles looking in was a small blue triangle.
"Talk to each other," I said. "Tell your partner what the labels COULD be and what they COULDN'T be. Then share your ideas with the rest of us."
{WARNING: SPOILERS AHEAD. You may wish to see if you can solve the problem on your own based only on this information. Remember, labels name only colors, shapes, and sizes, and we only choose two labels. Read down when you're ready...}
After the partner conversation, one of the third graders raised her hand. "I think I know what it is," she said. "This circle"--and she pointed to the one on the left--"is the circle for rhombuses. And this circle"--and she pointed to the one on the right--"has to be for big blocks."
"Why couldn't they be for blue blocks?" I asked.
"Because we tried blue with the blue triangle," she said, "and the blue triangle didn't go in either of the circles. So it can't be blue."
I asked a couple more questions like that, inquired if anyone had other ideas, and then turned back to the girl who had spoken first. "How sure are you?" I asked.
"I'm pretty sure," she told me. "Maybe 80% sure. No, 90%." (I like having kids this age express "sureness" in percentages. They seem to like it too.)
I drew a quarter out of my pocket and examined it closely. "90% sure is pretty sure," I said, "but it isn't certain. We only have three blocks so far. It's kind of early to be naming both labels, don't you think? I'm thinking it COULD be something else. I'm thinking it probably IS something else." Pause. "What do you think?"
"Umm." The girl frowned and looked back at the diagram. A classmate next to her whispered something. The girl nodded. "I still think I'm right," she informed me.
I tossed the quarter into the air and caught it nonchalantly. "I have a quarter here that says you're wrong," I said. If the labels were "rhombus" and "large," I explained, the quarter would be hers. (That got everybody's attention.) On the other hand, I added oh-so-casually, if she was wrong she would owe ME a quarter.
"Don't do it!" somebody hissed at my, ah, victim, just as someone else leaned in close to her and said "Go for it!"
"All right," she said, rolling her eyes, "you can have my allowance..."
As it turned out, of course, it wasn't necessary. We went through her reasoning, failed to find any holes (bummer, man), and revealed the labels in the Venn, I mean Zen, diagram. The girl's reasoning had been one hundred percent correct, and she had stuck to her guns despite my best attempt to rattle her. I handed over the quarter as the class cheered and surrounded her to offer their congratulations to the kid who had, if not broken the bank at Monte Carlo, at the very least outwitted the Math Guy.
I'd lost (again). But though my pocket was lighter, I was convinced that the reasoning and confidence the child had demonstrated during the lesson had been worth the very real financial hit to me...
And at any rate, now you know why I am not (generally) a gamblin' man!
Still, there are times when you just have to place that bet...
Yesterday I was in Jan's third and fourth grade class, where students are working on logic and attributes. The focus for the lesson was on Venn diagrams. You know, overlapping circle thingies, like this:
(My college roommate, Bernie, was a double major in science and philosophy. After taking a class on Tibetan Buddhism he once accidentally referred to these things as "Zen diagrams." I'm still tempted to call 'em that sometimes.)
In addition to the Zen, I mean Venn, diagram, we also had a bunch of blocks of various sizes, shapes, and colors, and labels with categories that described the blocks: "triangles," "small," "red," "yellow" and so on. I had chosen two labels at random ("What does that mean, 'at random?'" I'd asked earlier in the day, and the response was "Randomly," which was accurate if not perhaps revealing) and placed one in each circle of the Zen, I mean Venn, diagram. I got to look at the labels. The kids didn't.
The object is for the students to identify the two labels. They name blocks one by one, and I place each piece where it belongs: in the overlapping section of the diagram if it fits both labels; in one of the circles but not the other if it matches just one label; or outside both circles if it matches neither one. For instance, a small green triangle goes inside a circle marked "small," "green," or "triangle." Students use logic and the position of blocks in the diagram to determine what the labels CAN and CANNOT be.
Yesterday, after just three blocks, we had the following situation:
In the left circle, but NOT in the overlap, was a small blue rhombus. (A rhombus, for those not in the know, is not a method of transporting rhoms; it is a four-sided figure in which all sides are equal.)
In the overlap between the circles was a large blue rhombus. And outside both circles looking in was a small blue triangle.
"Talk to each other," I said. "Tell your partner what the labels COULD be and what they COULDN'T be. Then share your ideas with the rest of us."
{WARNING: SPOILERS AHEAD. You may wish to see if you can solve the problem on your own based only on this information. Remember, labels name only colors, shapes, and sizes, and we only choose two labels. Read down when you're ready...}
After the partner conversation, one of the third graders raised her hand. "I think I know what it is," she said. "This circle"--and she pointed to the one on the left--"is the circle for rhombuses. And this circle"--and she pointed to the one on the right--"has to be for big blocks."
"Why couldn't they be for blue blocks?" I asked.
"Because we tried blue with the blue triangle," she said, "and the blue triangle didn't go in either of the circles. So it can't be blue."
I asked a couple more questions like that, inquired if anyone had other ideas, and then turned back to the girl who had spoken first. "How sure are you?" I asked.
"I'm pretty sure," she told me. "Maybe 80% sure. No, 90%." (I like having kids this age express "sureness" in percentages. They seem to like it too.)
I drew a quarter out of my pocket and examined it closely. "90% sure is pretty sure," I said, "but it isn't certain. We only have three blocks so far. It's kind of early to be naming both labels, don't you think? I'm thinking it COULD be something else. I'm thinking it probably IS something else." Pause. "What do you think?"
"Umm." The girl frowned and looked back at the diagram. A classmate next to her whispered something. The girl nodded. "I still think I'm right," she informed me.
I tossed the quarter into the air and caught it nonchalantly. "I have a quarter here that says you're wrong," I said. If the labels were "rhombus" and "large," I explained, the quarter would be hers. (That got everybody's attention.) On the other hand, I added oh-so-casually, if she was wrong she would owe ME a quarter.
"Don't do it!" somebody hissed at my, ah, victim, just as someone else leaned in close to her and said "Go for it!"
"All right," she said, rolling her eyes, "you can have my allowance..."
As it turned out, of course, it wasn't necessary. We went through her reasoning, failed to find any holes (bummer, man), and revealed the labels in the Venn, I mean Zen, diagram. The girl's reasoning had been one hundred percent correct, and she had stuck to her guns despite my best attempt to rattle her. I handed over the quarter as the class cheered and surrounded her to offer their congratulations to the kid who had, if not broken the bank at Monte Carlo, at the very least outwitted the Math Guy.
I'd lost (again). But though my pocket was lighter, I was convinced that the reasoning and confidence the child had demonstrated during the lesson had been worth the very real financial hit to me...
And at any rate, now you know why I am not (generally) a gamblin' man!
Monday, November 9, 2009
If Left and Right Are Opposites, What About Remaining and Wrong?
"Look at the four cards I just gave you," I instructed the first graders I was working with today. We were warming up for some measurement work by reviewing some concepts from last month. "Look at the numbers on the cards. Show me an odd number...good job. Show me an even number...excellent! Which number is the least? Show me a number that's between 5 and 9." And on it went like that, culminating in the following exchange:
Me: "Okay, now find the greatest number. Put that card here in the middle of the table."
Children: [follow directions]
Me: "Now, look at the cards that you still have. Which is the greatest of the numbers that are left?"
One Child [looking back and forth at the three remaining cards, face up on the table]: "Which way is left, again?"
Oh, to have directional words that have just one meaning. That'd be great, right? (Wait...which way is right, again?)
Me: "Okay, now find the greatest number. Put that card here in the middle of the table."
Children: [follow directions]
Me: "Now, look at the cards that you still have. Which is the greatest of the numbers that are left?"
One Child [looking back and forth at the three remaining cards, face up on the table]: "Which way is left, again?"
Oh, to have directional words that have just one meaning. That'd be great, right? (Wait...which way is right, again?)
Sunday, November 1, 2009
Corn, Revisited
I promised to write more about the corn project (see entry of October 13). Picking up the story from there:
Once all the students had the complete and accurate number of kernels, we assembled in the Chapman Room. "Who thinks they have the MOST kernels of anyone in all three classes?" I asked. Several people were pretty sure the honor was theirs, but one young man from Jan's class took the prize: he had 644 kernels on his ear of corn, a full 43 more than the next runner-up.
"Okay, how about the LEAST?" We had a few who coulda been contendahs, but again one student won out--another of Jan's students, down at 289.
All right. We had the greatest and the least values. One way of describing a set of numbers, I explained, is to find the range: the distance between the least and the greatest. (This tells you roughly what kind of a spread you have in the data: are the numbers generally pretty far apart, or are they mostly close together?) As a group, we estimated the difference, then subtracted to find out. "Close together, or far apart?" I asked when we had our result.
"FAR APART," chorused 48 voices.
How right they were. The range was--quite large. Taken together, the two lowest figures were less than the highest. There's plenty of variation among ears of corn, evidently.
Next we turned our attention to the median, or the center value when the numbers were all ordered. We had the students sit in a line--well, technically a curve--arranged from 289 up to 644. When everyone was in order I had them all stand and look around. "Where do you think the median value is?" I asked. "Point to the person who you think had the median amount of corn kernels."
Fingers waved toward the middle of the line. Most people in the middle of the line pointed to themselves. To find out the real answer, we started at the outside of the line and had students sit down two by two: 644 matched with 289, 601 matched with 293, and so on. Like a very slow row of falling dominoes, or perhaps like spectators doing the wave at a baseball stadium, they sat down, or fell down, depending on their level of coordination and their penchant for dramatics. Little by little, the number of children standing diminished. The 500s disappeared altogether, so did the 300s. The upper 400s took their seats. People began revising their predictions.
Before long, we were down to two students. One had amassed a total of 408 kernels. The other had--412. There was an even number of people. The answer, someone realized, was to split the difference, and that's exactly what we did. The median was 410. If you wanted to choose one number to stand for all the numbers in the group, you could do a lot worse than choose 410.
(The picture below shows the Final Two. Everyone else has been eliminated from contention as the Merry Median of the Corn Kernels. Thanks to Jan for the photo.)
One more project remained. You've heard of the Living Flag? Well, this was to be a Living Histogram. (A histogram has nothing to do with allergies--it's a bar graph in which the bars stand for a range of numbers rather than a single figure or response.) We had the students divide themselves into groups, according to the number of kernels: up to 299 over here, 300-349 over there, 350 to 400 in that corner. Then we called the groups over one by one and had group members sit in a line, creating eight lines of varying lengths in all. "What do you notice?" I asked, and they noticed quite a lot. The longest line was in the middle, they explained, the shortest lines on the outside. It was like stairs, someone said; it was like a roller coaster, said someone else. They were quite right, too. It was about the normal-est curve I'd encountered in the last few months--the nice bell shape you read about.
(Here are the lines. You might recognize the two almost-median-winners, smack dab in the center of the longest line there in the middle of the photo. See how neatly all these things work out?)
So, a nice way to spend a misty, mathy morning. The kids enjoyed getting their minds around the concept of range and median--and did it very well, I might add. They were surprised to see how big the range actually was, and they very much liked using their own bodies to locate the median. And while some of the players were beginning to get a bit restless toward the end, they kept their sense of curiosity about the graph and loved the idea of constructing it themselves. We'll continue to explore range and median--and who knows, we may get back out to the Chapman Room with a different set of data someday!
Once all the students had the complete and accurate number of kernels, we assembled in the Chapman Room. "Who thinks they have the MOST kernels of anyone in all three classes?" I asked. Several people were pretty sure the honor was theirs, but one young man from Jan's class took the prize: he had 644 kernels on his ear of corn, a full 43 more than the next runner-up.
"Okay, how about the LEAST?" We had a few who coulda been contendahs, but again one student won out--another of Jan's students, down at 289.
All right. We had the greatest and the least values. One way of describing a set of numbers, I explained, is to find the range: the distance between the least and the greatest. (This tells you roughly what kind of a spread you have in the data: are the numbers generally pretty far apart, or are they mostly close together?) As a group, we estimated the difference, then subtracted to find out. "Close together, or far apart?" I asked when we had our result.
"FAR APART," chorused 48 voices.
How right they were. The range was--quite large. Taken together, the two lowest figures were less than the highest. There's plenty of variation among ears of corn, evidently.
Next we turned our attention to the median, or the center value when the numbers were all ordered. We had the students sit in a line--well, technically a curve--arranged from 289 up to 644. When everyone was in order I had them all stand and look around. "Where do you think the median value is?" I asked. "Point to the person who you think had the median amount of corn kernels."
Fingers waved toward the middle of the line. Most people in the middle of the line pointed to themselves. To find out the real answer, we started at the outside of the line and had students sit down two by two: 644 matched with 289, 601 matched with 293, and so on. Like a very slow row of falling dominoes, or perhaps like spectators doing the wave at a baseball stadium, they sat down, or fell down, depending on their level of coordination and their penchant for dramatics. Little by little, the number of children standing diminished. The 500s disappeared altogether, so did the 300s. The upper 400s took their seats. People began revising their predictions.
Before long, we were down to two students. One had amassed a total of 408 kernels. The other had--412. There was an even number of people. The answer, someone realized, was to split the difference, and that's exactly what we did. The median was 410. If you wanted to choose one number to stand for all the numbers in the group, you could do a lot worse than choose 410.
(The picture below shows the Final Two. Everyone else has been eliminated from contention as the Merry Median of the Corn Kernels. Thanks to Jan for the photo.)
One more project remained. You've heard of the Living Flag? Well, this was to be a Living Histogram. (A histogram has nothing to do with allergies--it's a bar graph in which the bars stand for a range of numbers rather than a single figure or response.) We had the students divide themselves into groups, according to the number of kernels: up to 299 over here, 300-349 over there, 350 to 400 in that corner. Then we called the groups over one by one and had group members sit in a line, creating eight lines of varying lengths in all. "What do you notice?" I asked, and they noticed quite a lot. The longest line was in the middle, they explained, the shortest lines on the outside. It was like stairs, someone said; it was like a roller coaster, said someone else. They were quite right, too. It was about the normal-est curve I'd encountered in the last few months--the nice bell shape you read about.
(Here are the lines. You might recognize the two almost-median-winners, smack dab in the center of the longest line there in the middle of the photo. See how neatly all these things work out?)
So, a nice way to spend a misty, mathy morning. The kids enjoyed getting their minds around the concept of range and median--and did it very well, I might add. They were surprised to see how big the range actually was, and they very much liked using their own bodies to locate the median. And while some of the players were beginning to get a bit restless toward the end, they kept their sense of curiosity about the graph and loved the idea of constructing it themselves. We'll continue to explore range and median--and who knows, we may get back out to the Chapman Room with a different set of data someday!
Saturday, October 31, 2009
Mathtubs
Give me a mathtub each morning,
Give me a mathtub at noo-oo-oon,
Give me a mathtub each evening,
But give me a mathtub soon.
One of the little perks of my job is distributing the mathtubs each Friday. {See the link below for more info about these tubs--they're boxes filled with math games, math materials, suggestions for math-related projects, and picture/storybooks with math connections; kids take them home for a few days at some point during the year.}
I walk into a classroom, around about the time pizza is delivered, carrying one or two tubs, and deliver the tubs to the children (CHOSEN AT RANDOM--NOTHING UP MY SLEEVES) who will get them for the next few days.
Children vary, of course, in how they express their excitement over getting the tub (from a small, self-satisfied smile up to an enthusiastic fist-pump and a chanted "Oh yeah, oh yeah, oh yeah"), but about 97% are very pleased to have their turn. It's quite gratifying.
The other children, meanwhile, are full of helpful comments such as "Is it my turn yet?" "When is it my turn?" "She's so LUCKY," and so on. For the moment, at least, the arrival of the Math Guy and the Math Materials outranks everything--even pizza. No small feat.
I thought I had invented the mathtub idea, or at the very least, um, repurposed it from a similar idea I'd read about somewhere in which teachers sent books home in backpacks for kids and families to enjoy. A couple of years into the mathtub project, though I was cleaning out some old papers and discovered to my surprise that in 1997, while at a conference in Rochester (NY), I had actually attended a workshop in which the presenter was describing how teachers could package up some math materials for use at home. I still take credit for the name "mathtubs"--I think that teacher used shopping bags or something similar--but as for the concept, well, I should know by now that there are few truly original ideas in education. Hey, it works, and the kids enjoy it, and that's what counts--right?
You can read more about mathtubs here, in an article I published in a teacher magazine a few years back:
http://www.highlightsteachers.com/archives/articles/the_mathtubs_are_coming_by_stephen_currie.html
Give me a mathtub at noo-oo-oon,
Give me a mathtub each evening,
But give me a mathtub soon.
One of the little perks of my job is distributing the mathtubs each Friday. {See the link below for more info about these tubs--they're boxes filled with math games, math materials, suggestions for math-related projects, and picture/storybooks with math connections; kids take them home for a few days at some point during the year.}
I walk into a classroom, around about the time pizza is delivered, carrying one or two tubs, and deliver the tubs to the children (CHOSEN AT RANDOM--NOTHING UP MY SLEEVES) who will get them for the next few days.
Children vary, of course, in how they express their excitement over getting the tub (from a small, self-satisfied smile up to an enthusiastic fist-pump and a chanted "Oh yeah, oh yeah, oh yeah"), but about 97% are very pleased to have their turn. It's quite gratifying.
The other children, meanwhile, are full of helpful comments such as "Is it my turn yet?" "When is it my turn?" "She's so LUCKY," and so on. For the moment, at least, the arrival of the Math Guy and the Math Materials outranks everything--even pizza. No small feat.
I thought I had invented the mathtub idea, or at the very least, um, repurposed it from a similar idea I'd read about somewhere in which teachers sent books home in backpacks for kids and families to enjoy. A couple of years into the mathtub project, though I was cleaning out some old papers and discovered to my surprise that in 1997, while at a conference in Rochester (NY), I had actually attended a workshop in which the presenter was describing how teachers could package up some math materials for use at home. I still take credit for the name "mathtubs"--I think that teacher used shopping bags or something similar--but as for the concept, well, I should know by now that there are few truly original ideas in education. Hey, it works, and the kids enjoy it, and that's what counts--right?
You can read more about mathtubs here, in an article I published in a teacher magazine a few years back:
http://www.highlightsteachers.com/archives/articles/the_mathtubs_are_coming_by_stephen_currie.html
Friday, October 30, 2009
People v. Tables
"I am going to have a party," read the question given to a number of our 1-2 students the other day. "I want to invite ___ people." (The blank is standard: everybody gets a different number, which a) cuts down on the Problem of Roving Eyes and b) allows us to give somewhat harder numbers to kids who are ready for a challenge while keeping the same problem frame for everyone.)
"I have ____ tables where my guests can sit," the problem continues. "Each table has room for _____ people. Do I have enough tables, or do I need to get more?"
Different kids had different ways of attacking the problem, as usual. Some sketched the tables, drew chairs around them, and counted by ones. Others dispensed with the chairs and simply wrote the number at each table, then counted by that number if they knew how. A couple didn't bother with a sketch at all. One or two made groups with checkers or other materials--7 groups of 6 checkers, for instance, to represent 7 tables with 6 people at each--and then checked the number of people to see if they'd gone over or not. The strategies were generally quite accurate, if not consistently efficient: the next step will be to move kids away from the pictures and toward more abstract skip-counting and other strategies.
At any rate, children needed to show or describe their work and then answer the question (which, if you recall, had something to do with whether there were enough tables or whether we needed to get more). Several children didn't recall--they needed a reminder to do this part--but eventually we had the answers we sought.
"I have enough tables," one child wrote confidently and accurately. (Actually, she wrote "enuff," but let that pass...)
"You have enough tables. Am I rite?" wrote another child, perhaps a little less confident than the first. (Yup, I told him, you're rite. Um, right.)
"You need to get more tables," wrote a third responder, "because seven tables is going to be a smaller number of people. You need 8 tables." She included a careful sketch with the correct number of heads jowl by jowl at each table: an arrow then pointed to the last table, with the helpful label "extra."
"I have to sell one more chair," wrote still another girl. A somewhat convoluted way of saying that she not only had enough tables--she had an extra seat. I'm not entirely clear whether the sale would be an auction for the right to attend the party, or simply an attempt to convert an unwanted and unnecessary item into cold hard cash. Either way, this is a girl who knows the value of a buck.
And perhaps my favorite: the boy who discovered that he had space for 52 when he only needed to seat 49. After showing his method, he concluded: "You need more people."
"I have ____ tables where my guests can sit," the problem continues. "Each table has room for _____ people. Do I have enough tables, or do I need to get more?"
Different kids had different ways of attacking the problem, as usual. Some sketched the tables, drew chairs around them, and counted by ones. Others dispensed with the chairs and simply wrote the number at each table, then counted by that number if they knew how. A couple didn't bother with a sketch at all. One or two made groups with checkers or other materials--7 groups of 6 checkers, for instance, to represent 7 tables with 6 people at each--and then checked the number of people to see if they'd gone over or not. The strategies were generally quite accurate, if not consistently efficient: the next step will be to move kids away from the pictures and toward more abstract skip-counting and other strategies.
At any rate, children needed to show or describe their work and then answer the question (which, if you recall, had something to do with whether there were enough tables or whether we needed to get more). Several children didn't recall--they needed a reminder to do this part--but eventually we had the answers we sought.
"I have enough tables," one child wrote confidently and accurately. (Actually, she wrote "enuff," but let that pass...)
"You have enough tables. Am I rite?" wrote another child, perhaps a little less confident than the first. (Yup, I told him, you're rite. Um, right.)
"You need to get more tables," wrote a third responder, "because seven tables is going to be a smaller number of people. You need 8 tables." She included a careful sketch with the correct number of heads jowl by jowl at each table: an arrow then pointed to the last table, with the helpful label "extra."
"I have to sell one more chair," wrote still another girl. A somewhat convoluted way of saying that she not only had enough tables--she had an extra seat. I'm not entirely clear whether the sale would be an auction for the right to attend the party, or simply an attempt to convert an unwanted and unnecessary item into cold hard cash. Either way, this is a girl who knows the value of a buck.
And perhaps my favorite: the boy who discovered that he had space for 52 when he only needed to seat 49. After showing his method, he concluded: "You need more people."
Monday, October 26, 2009
Virtual Manipulatives
We sometimes haul out the laptops during lower school math times and have kids work with Utah State University's National Library of Virtual Manipulatives website. We've made good use of this site for projects with both the 3-4 and the 1-2 classes, but my personal favorite is the subtraction.
See, you get these rods and cubes, just like base blocks only they're on the screen and exist only in pixel form, so they don't fall off the table and get lost and they can't be used as hockey sticks and pucks, or as drumsticks or grenade launchers or whatever else creative minds have in store for them.
--Oh, and then when you model regrouping (which I prefer not to call "borrowing," as I've said before, because you don't ever give it back--I prefer to use the phrase "stealing") you actually grab one of the virtual tens rods and bring it over to the ones column and let go and watch as it separates itself into ten little ones cubes.
Then you hear the kids saying WHOA! and COOL! and NEATO TORPEDO! (well, not that one, maybe) and the like.
Then you get to separate hundreds into tens the same way and thousands into hundreds and the whole thing is utterly charming and truly awesome and the best thing next to...
[Down, boy.]
[The picture below isn't actually from the virtual manipulatives website--it's from a powerpoint presentation I made dramatizing the process. What you see here is the ones stealing a ten, in the dead of the night of course, dragging it back to Ones Street, and breaking it into ten little ones cubes so there'll be enough ones to carry out the subtraction.]
Anyhow, children sometimes ask how they can get to the site at home. Unfortunately, the address isn't straightforward. If you google "virtual manipulatives," it's the first site that comes up (as of today, anyway).
The whole site's URL is http://nlvm.usu.edu/en/nav/vLibrary.html. If you're interested, take a spin around the site with your child(ren). It may not be the equivalent of a medieval European cathedral, but as the Michelin guide would put it, it's quite definitely worth a visit.
See, you get these rods and cubes, just like base blocks only they're on the screen and exist only in pixel form, so they don't fall off the table and get lost and they can't be used as hockey sticks and pucks, or as drumsticks or grenade launchers or whatever else creative minds have in store for them.
--Oh, and then when you model regrouping (which I prefer not to call "borrowing," as I've said before, because you don't ever give it back--I prefer to use the phrase "stealing") you actually grab one of the virtual tens rods and bring it over to the ones column and let go and watch as it separates itself into ten little ones cubes.
Then you hear the kids saying WHOA! and COOL! and NEATO TORPEDO! (well, not that one, maybe) and the like.
Then you get to separate hundreds into tens the same way and thousands into hundreds and the whole thing is utterly charming and truly awesome and the best thing next to...
[Down, boy.]
[The picture below isn't actually from the virtual manipulatives website--it's from a powerpoint presentation I made dramatizing the process. What you see here is the ones stealing a ten, in the dead of the night of course, dragging it back to Ones Street, and breaking it into ten little ones cubes so there'll be enough ones to carry out the subtraction.]
Anyhow, children sometimes ask how they can get to the site at home. Unfortunately, the address isn't straightforward. If you google "virtual manipulatives," it's the first site that comes up (as of today, anyway).
The whole site's URL is http://nlvm.usu.edu/en/nav/vLibrary.html. If you're interested, take a spin around the site with your child(ren). It may not be the equivalent of a medieval European cathedral, but as the Michelin guide would put it, it's quite definitely worth a visit.
Tuesday, October 13, 2009
Corn
How many kernels on an ear of corn? we asked the third and fourth graders the other day. They've been studying the Mayan people, who called themselves "People of the Corn," so it was a worthwhile question.
We started by having students find approximations; as you should know by now if you've been reading this blog, us Math Guys consider this a very important step. We asked students to choose a round number (a number that is a multiple of 10); the point, after all, wasn't to guess the exact number, but to use a number that makes some sense and is relatively easy to work with. You can always revise your estimate later, we assured them.
What is the estimate based on? Well, we gave them each an ear of dried corn to eyeball. Some did some quick-n-dirty calculations, fourth graders in particular. (Yes, we asked them to justify their reasoning. Some of them HATE this, but it's oh-so-good for them.)
"About 20 in each row," wrote one student. "Maybe 10 rows. 10 x 20 = 200. I estimate 200 kernels in all."
"I think there are 20 rows and 30 in each row," reported someone else, "but that might not be enough so I added a few more. I say 640."
"I think 260," wrote a third grader, who would have been happy to leave it at that, but who added, under duress from a teacher, "because it looks right. And because it's a good number." We might call this strategy "Pick-a-large-number, any-large-number, and-assign-it-great-virtue-so-critics-will-be-cowed."
The next step: Count the kernels! The classroom teachers had prepared egg cartons with ten cuplets (better them than me). Kids used their fingernails to push the kernels off the cob (great fun). Then they distributed the kernels 5 or 10 at a time into the cups, making groups of 50 or 100. Record the number, dump out the kernels, lather, rinse, repeat.
At some point along the way several students noticed that their estimates weren't looking as accurate as they had back before counting had begun. This was especially true for those whose initial strategy had been "Pick-a-large-number, any-large-number &c," but other more careful estimators ran into this difficulty too. No problem! we said. Just revise your estimate, record it--oh, and explain why you wanted to change your original prediction. (My favorite: "Because I passed my first estimate a long time ago.") You will no doubt be shocked to learn that the second set of estimates were considerably closer than the first.
Eventually, all corn kernels were off the cobs and had traveled through the eggcups and into plastic bowls or paper bags (except for a few strays which had found their way onto the floor), and everyone had an exact answer. Some were surprised to see how many there were. Others found the results unsurprising in the extreme, or claimed they did: "I knew it," crowed one boy whose answer was not, perhaps, as close as he thought.
As students finished, they compared their totals with friends and thought about questions such as Why aren't all the totals the same?, What could you do to get a better estimate next time? ("Nothing," said the young man quoted above), and About how many kernels do you think there might be in the whole class?
So, three-digit numbers, ordering, estimating, grouping by tens, fives, 50s, and 100s, and explaining reasoning. Plus, a fun project (there's something truly satisfying about flicking those kernels off the cob, and something even more satisfying about running your fingers through a nice big tub full of everyone's kernels), and one that relates to science and social studies. A worthwhile math period indeed. Next up: data analysis with these results. On Thursday we'll be in the Chapman Room calculating the median and range of the data and forming a Living Histogram. Pictures to follow, assuming my camera behaves itself...
We started by having students find approximations; as you should know by now if you've been reading this blog, us Math Guys consider this a very important step. We asked students to choose a round number (a number that is a multiple of 10); the point, after all, wasn't to guess the exact number, but to use a number that makes some sense and is relatively easy to work with. You can always revise your estimate later, we assured them.
What is the estimate based on? Well, we gave them each an ear of dried corn to eyeball. Some did some quick-n-dirty calculations, fourth graders in particular. (Yes, we asked them to justify their reasoning. Some of them HATE this, but it's oh-so-good for them.)
"About 20 in each row," wrote one student. "Maybe 10 rows. 10 x 20 = 200. I estimate 200 kernels in all."
"I think there are 20 rows and 30 in each row," reported someone else, "but that might not be enough so I added a few more. I say 640."
"I think 260," wrote a third grader, who would have been happy to leave it at that, but who added, under duress from a teacher, "because it looks right. And because it's a good number." We might call this strategy "Pick-a-large-number, any-large-number, and-assign-it-great-virtue-so-critics-will-be-cowed."
The next step: Count the kernels! The classroom teachers had prepared egg cartons with ten cuplets (better them than me). Kids used their fingernails to push the kernels off the cob (great fun). Then they distributed the kernels 5 or 10 at a time into the cups, making groups of 50 or 100. Record the number, dump out the kernels, lather, rinse, repeat.
At some point along the way several students noticed that their estimates weren't looking as accurate as they had back before counting had begun. This was especially true for those whose initial strategy had been "Pick-a-large-number, any-large-number &c," but other more careful estimators ran into this difficulty too. No problem! we said. Just revise your estimate, record it--oh, and explain why you wanted to change your original prediction. (My favorite: "Because I passed my first estimate a long time ago.") You will no doubt be shocked to learn that the second set of estimates were considerably closer than the first.
Eventually, all corn kernels were off the cobs and had traveled through the eggcups and into plastic bowls or paper bags (except for a few strays which had found their way onto the floor), and everyone had an exact answer. Some were surprised to see how many there were. Others found the results unsurprising in the extreme, or claimed they did: "I knew it," crowed one boy whose answer was not, perhaps, as close as he thought.
As students finished, they compared their totals with friends and thought about questions such as Why aren't all the totals the same?, What could you do to get a better estimate next time? ("Nothing," said the young man quoted above), and About how many kernels do you think there might be in the whole class?
So, three-digit numbers, ordering, estimating, grouping by tens, fives, 50s, and 100s, and explaining reasoning. Plus, a fun project (there's something truly satisfying about flicking those kernels off the cob, and something even more satisfying about running your fingers through a nice big tub full of everyone's kernels), and one that relates to science and social studies. A worthwhile math period indeed. Next up: data analysis with these results. On Thursday we'll be in the Chapman Room calculating the median and range of the data and forming a Living Histogram. Pictures to follow, assuming my camera behaves itself...
Friday, October 9, 2009
p-a-t-t-e-r-n-s
What do you call it
When things repeat?
We call it...
A pattern.
Head, shoulders, knees, and feet,
Head, shoulders, knees, and feet,
That
Is a pattern.
A, B, C, A, B, C, A, B, C, A, B, C
That
Is a pattern,
1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3
That
Is a pattern.
What do you call it
When things repeat?
We call it...
A pattern.
Wednesday, September 30, 2009
Pizza!
As many of you know, the third and fourth grade classes order pizza each Friday. Children throughout the lower school put in their order; runners from the 3-4s pick up the orders and the money, determine the number of pizzas to buy, and hand-deliver it when it arrives.
Pizza is a major undertaking. There are times when we teachers wonder whether it is all worthwhile, especially when we discover that one class is short $15 or that a dozen or so children neglected to sign up until the pizza, you know, arrived... BUT we continue to do it because pizza a) tastes good, b) is convenient for parents, and c) IS A GREAT TOOL FOR PRACTICING MATH SKILLS. Of the three, c) is by far the most important in my book, though your mileage may vary.
How does pizza relate to mathematics? Glad you asked. Let us count the ways...
1. Pizza order takers get good practice in counting money and determining if it matches the number of slices ordered (hint: it does only about half the time).
2. Students get practice in giving and making change.
3. Students round the number of slices ordered per class to the nearest multiple of 10 to make estimation easier.
4. Kids practice addition skills by calculating the total number of slices ordered.
5. We look at number patterns. Hmmm: when a class orders 14 slices at $1.50 per slice, we get $21. Interestingly enough, 14 plus half-of-14 equals 21--the same number, only in regular numbers rather than in money. Now why would that be?
6. Especially later in the year, we use pizza as a real-life example of multiplication--if there are 8 slices per pizza, how many slices in 5 pizzas? In 7 pizzas? In 13 pizzas?
7. Kids calculate the profit for each week's worth of pizza: if we charge a dollar-fifty per slice after buying it for [sorry, trade secrets removed--suffice it to say "less"] per slice, how much money is left over? What operation can we use to calculate it?
And there are many other ways we mathicize pizza, especially this year, but I've been typing all day and my fingers are about to fall off. So you'll have to wait for another post. Sorry! In the meantime, how about some pictures? ...Yes, yes, the very thing!
Some of the proceeds, up close and personal:
This young man is clearly enjoying himself.
Think Scrooge McDuck.
Doublechecking that the amount of money from one of the 1-2 classes actually matches the number of slices ordered:
One of the "Grand Totalers," making bundles of ten for easier counting:
And at last, the fruits of our labors--or seven eighths of them at least (did I mention that pizza and FRACTIONS go well together? No? Consider it mentioned...):
Pizza is a major undertaking. There are times when we teachers wonder whether it is all worthwhile, especially when we discover that one class is short $15 or that a dozen or so children neglected to sign up until the pizza, you know, arrived... BUT we continue to do it because pizza a) tastes good, b) is convenient for parents, and c) IS A GREAT TOOL FOR PRACTICING MATH SKILLS. Of the three, c) is by far the most important in my book, though your mileage may vary.
How does pizza relate to mathematics? Glad you asked. Let us count the ways...
1. Pizza order takers get good practice in counting money and determining if it matches the number of slices ordered (hint: it does only about half the time).
2. Students get practice in giving and making change.
3. Students round the number of slices ordered per class to the nearest multiple of 10 to make estimation easier.
4. Kids practice addition skills by calculating the total number of slices ordered.
5. We look at number patterns. Hmmm: when a class orders 14 slices at $1.50 per slice, we get $21. Interestingly enough, 14 plus half-of-14 equals 21--the same number, only in regular numbers rather than in money. Now why would that be?
6. Especially later in the year, we use pizza as a real-life example of multiplication--if there are 8 slices per pizza, how many slices in 5 pizzas? In 7 pizzas? In 13 pizzas?
7. Kids calculate the profit for each week's worth of pizza: if we charge a dollar-fifty per slice after buying it for [sorry, trade secrets removed--suffice it to say "less"] per slice, how much money is left over? What operation can we use to calculate it?
And there are many other ways we mathicize pizza, especially this year, but I've been typing all day and my fingers are about to fall off. So you'll have to wait for another post. Sorry! In the meantime, how about some pictures? ...Yes, yes, the very thing!
Some of the proceeds, up close and personal:
This young man is clearly enjoying himself.
Think Scrooge McDuck.
Doublechecking that the amount of money from one of the 1-2 classes actually matches the number of slices ordered:
One of the "Grand Totalers," making bundles of ten for easier counting:
And at last, the fruits of our labors--or seven eighths of them at least (did I mention that pizza and FRACTIONS go well together? No? Consider it mentioned...):
Getting to School
If you're like most of us, you've always wanted to know how kindergarteners get to school. Lucky you! Now's your chance to find out--because the kindergarten recently put together a graph showing that information. Voila--!
(You can click on the picture to enlarge it.)
The kids enjoy the drawing, of course (some of them enjoy it quite a lot--I'm very fond of the multicolored, creatively shaped cars at the top of the cars column, along with the 3-wheeled truck on the far right). But it's also a good learning experience for these newly minted K students.
There are the reading-the-graph questions, of course:
*Which way of getting to school was the most common?
*Which was second most common?
*How many children came to school by truck?
*Which way of getting to school was used by 7 children?
And the interpreting-the-graph questions too:
*Suppose we asked the same question tomorrow and made a graph about that. Would the graph look exactly the same? Mostly the same? Not at all the same? Why?
*What if we made a graph showing how people got home? What do you think that would look like? Why?
*Why do you think no one got here on a skateboard? A surfboard? A motorboat?
*Do you think the school needs a bigger parking lot? Why?
But mostly, there's the notion that you can take information and show it in a way that makes it available to everybody who comes along. You can tell, just by looking, that more people in your class come to school by bus than come by van, and that a LOT more come by car than come by truck. You can locate your own name (or your own truck) on the graph and show a friend how you got to school that day. You can find out how a friend arrived. The information will be there today--and tomorrow--and the next day--and it will remain available forever, or at least as long as the teachers choose to hang it in the hall. Knowledge is power, we like to say; and graphs, I believe, are a really good example of that adage.
(You can click on the picture to enlarge it.)
The kids enjoy the drawing, of course (some of them enjoy it quite a lot--I'm very fond of the multicolored, creatively shaped cars at the top of the cars column, along with the 3-wheeled truck on the far right). But it's also a good learning experience for these newly minted K students.
There are the reading-the-graph questions, of course:
*Which way of getting to school was the most common?
*Which was second most common?
*How many children came to school by truck?
*Which way of getting to school was used by 7 children?
And the interpreting-the-graph questions too:
*Suppose we asked the same question tomorrow and made a graph about that. Would the graph look exactly the same? Mostly the same? Not at all the same? Why?
*What if we made a graph showing how people got home? What do you think that would look like? Why?
*Why do you think no one got here on a skateboard? A surfboard? A motorboat?
*Do you think the school needs a bigger parking lot? Why?
But mostly, there's the notion that you can take information and show it in a way that makes it available to everybody who comes along. You can tell, just by looking, that more people in your class come to school by bus than come by van, and that a LOT more come by car than come by truck. You can locate your own name (or your own truck) on the graph and show a friend how you got to school that day. You can find out how a friend arrived. The information will be there today--and tomorrow--and the next day--and it will remain available forever, or at least as long as the teachers choose to hang it in the hall. Knowledge is power, we like to say; and graphs, I believe, are a really good example of that adage.
Tuesday, September 22, 2009
n (or maybe n+1) Flies on the Wall
The 3-4 classes generally begin the year with work on number sense, number patterns, place value, and number in general. This year we're starting off with some projects involving functions and some simple algebraic ideas.
Here's a fly-on-the-wall view of an introductory lesson (shh; don't let them know you're in the room):
Teacher: Suppose we choose a number from 1 to 100. We'll call that number n. We often use the letter n in math to stand for any number. Someone pick a number for n--
Student: 38!
Good enough. So if n is 38, what's n + 10? 38 + 10, right? Which is--
Students: 48.
That's right. Okay, let's make a table and try it using some other numbers for n:
n n + 10 Result
--- ------- ----
38 38 + 10 48
17 17 + 10 27
90 90 + 10 100
45 45 + 10 55
Looks good. Okay, what patterns do you see? How does n change when you add 10?
Students: The ones digit stays the same.
Yeah? Always, or only most of the time?
Students, a bit hesitantly, because you always have to watch out for trick questions: Always...always so far, anyway.
That's right. Can you think of a number n where the ones digit would change after you add 10?
Students: several suggestions, all of them withdrawn upon further reflection.
...Why doesn't it change?
Student: The number 10 has 0 in the ones column, so it doesn't change the ones.
Another student: Oh, and when you add ten on the hundreds board you just go down to the next row, so if you're in the threes column you stay in the threes column...[We use the hundred board a lot; one is pictured here.]
What happens with the tens? The tens go up? Good; by how much? By one? Always, or only sometimes?
Students, less hesitantly than before: Always.
How do you know? So, okay, let's put the rule into words: When you add 10 to a number n, the ones digit stays the same but the tens digit goes up by one.
Nice job! Okay, let's try it again, only this time we'll look at what happens when you add 11 to n.
n n + 11 Result
--- ------- ----
12 12 + 11 23
28 28 + 11 39
77 77 + 11 88
Student, bursting to be the first: I know, I know! I know the rule! It's the tens digit goes up and the ones digit goes up too!
Student, bursting to be the second: Yeah! It's the tens digit goes up and the ones digit goes up too!
By how much? Let's say it as a rule.
Students, cautiously: It goes up by one in the tens column and one in the ones column.
Always, or just sometimes?
n-2 or n-3 students, where n is the total population of the class: Always.
Two or three students: Sometimes.
Why sometimes?
2 or 3 students: Because what happens when the number is in the nines? Say you add 11 to a number like 59...
2 or 3 more students: Ohhh!
Let's extend the table--
59 59 + 11 70
69 69 + 11 80
n/2 students: It goes up two in the tens!
The other n/2 students: And it goes down to 0 in the ones.
Okay, let;s make the rule. Help me out here:
[And so we develop the rule: When you add 11 to a number n, the tens digit usually goes up one and the ones digit goes up one as well, EXCEPT that when the ones digit is 9, the tens digit goes up by 2 and the ones digit goes back to 0. We talk about why this might be the case--and then out go the students to work on developing rules for n+1, or n+19, or n-2, or perhaps even n x 5...]
Okay, class is over for the day. You can come down from the wall now! Aren't you glad none of the kids brought flyswatters today??
[Edited to add: I should note that this lesson is adapted from a set of activities in a new book by math education guru Marilyn Burns. In 2008, I spoke at a national conference of math teachers. I was disappointed to discover that I was scheduled at the same time as Marilyn, which was disappointing for two reasons...first, I didn't get to hear her, and second, hardly anybody was left to come to my workshop...]
Here's a fly-on-the-wall view of an introductory lesson (shh; don't let them know you're in the room):
Teacher: Suppose we choose a number from 1 to 100. We'll call that number n. We often use the letter n in math to stand for any number. Someone pick a number for n--
Student: 38!
Good enough. So if n is 38, what's n + 10? 38 + 10, right? Which is--
Students: 48.
That's right. Okay, let's make a table and try it using some other numbers for n:
n n + 10 Result
--- ------- ----
38 38 + 10 48
17 17 + 10 27
90 90 + 10 100
45 45 + 10 55
Looks good. Okay, what patterns do you see? How does n change when you add 10?
Students: The ones digit stays the same.
Yeah? Always, or only most of the time?
Students, a bit hesitantly, because you always have to watch out for trick questions: Always...always so far, anyway.
That's right. Can you think of a number n where the ones digit would change after you add 10?
Students: several suggestions, all of them withdrawn upon further reflection.
...Why doesn't it change?
Student: The number 10 has 0 in the ones column, so it doesn't change the ones.
Another student: Oh, and when you add ten on the hundreds board you just go down to the next row, so if you're in the threes column you stay in the threes column...[We use the hundred board a lot; one is pictured here.]
What happens with the tens? The tens go up? Good; by how much? By one? Always, or only sometimes?
Students, less hesitantly than before: Always.
How do you know? So, okay, let's put the rule into words: When you add 10 to a number n, the ones digit stays the same but the tens digit goes up by one.
Nice job! Okay, let's try it again, only this time we'll look at what happens when you add 11 to n.
n n + 11 Result
--- ------- ----
12 12 + 11 23
28 28 + 11 39
77 77 + 11 88
Student, bursting to be the first: I know, I know! I know the rule! It's the tens digit goes up and the ones digit goes up too!
Student, bursting to be the second: Yeah! It's the tens digit goes up and the ones digit goes up too!
By how much? Let's say it as a rule.
Students, cautiously: It goes up by one in the tens column and one in the ones column.
Always, or just sometimes?
n-2 or n-3 students, where n is the total population of the class: Always.
Two or three students: Sometimes.
Why sometimes?
2 or 3 students: Because what happens when the number is in the nines? Say you add 11 to a number like 59...
2 or 3 more students: Ohhh!
Let's extend the table--
59 59 + 11 70
69 69 + 11 80
n/2 students: It goes up two in the tens!
The other n/2 students: And it goes down to 0 in the ones.
Okay, let;s make the rule. Help me out here:
[And so we develop the rule: When you add 11 to a number n, the tens digit usually goes up one and the ones digit goes up one as well, EXCEPT that when the ones digit is 9, the tens digit goes up by 2 and the ones digit goes back to 0. We talk about why this might be the case--and then out go the students to work on developing rules for n+1, or n+19, or n-2, or perhaps even n x 5...]
Okay, class is over for the day. You can come down from the wall now! Aren't you glad none of the kids brought flyswatters today??
[Edited to add: I should note that this lesson is adapted from a set of activities in a new book by math education guru Marilyn Burns. In 2008, I spoke at a national conference of math teachers. I was disappointed to discover that I was scheduled at the same time as Marilyn, which was disappointing for two reasons...first, I didn't get to hear her, and second, hardly anybody was left to come to my workshop...
Friday, September 18, 2009
Learning from the DVD...Player
Teachers of today can choose from a wide array of technologies to spice up their lessons and increase students' understanding. There's Powerpoint, of course, and calculators, smart boards and video cameras, wikis and spellcheckers, voice-to-text programs and DVDs, Excel spreadsheets and Activote systems, GPSes and, um, electric pencil sharpeners; the list goes on.
Most of these educational technologies get plenty of respect within the educational world. (Well, maybe not the pencil sharpeners.) Whole conferences are organized around these technologies and how they can help teachers do a better job of preparing students for the 21st century [Q: At what point will we start saying "preparing students for the 22nd century"?]. BUT there is one technology that is sadly overlooked. It is the Rodney Dangerfield of the educational technology world. I refer, of course, to the lowly DVD player. Not the DVD; the player.
"How did you know so quickly that 8 + 8 was 16?" I asked a first grade girl earlier this week. (If this question sounds familiar, it's probably because you read the previous entry in this blog.)
"Well," she said, "we have this DVD player at home and it has arrows. And if you want to speed through the movie it says 2, 4, 8, 16, 32, and then it goes back to 2 again. And I know that 2+2 is 4 and that 4+4 is 8, so 8+8 must be 16, and I guess that 16+16 would be 32. But then the pattern stops because it goes back to 2 and 32+32 is...something, but it isn't 2."
What can I say? Clearly, we should as a nation reduce our spending on old-boring-and-ineffective technologies such as computers, projectors, smart boards, and digital cameras, and load up classrooms instead with DVD players. Who's with me?
--Actually, this is a really good example of a child not only noticing but using math in everyday life. No one taught her that 8 + 8 was 16. She was struck by a sequence of numbers that appeared in her environment, and spent time and energy deciphering the pattern--learning, and evidently mastering, the fact that 8+8=16 along the way. This is the kind of thinking we always want to see in our students. As our report form puts it, one of our goals for children is that they "recognize and construct mathematics in daily life." It's lovely to see such a clear example.
Most of these educational technologies get plenty of respect within the educational world. (Well, maybe not the pencil sharpeners.) Whole conferences are organized around these technologies and how they can help teachers do a better job of preparing students for the 21st century [Q: At what point will we start saying "preparing students for the 22nd century"?]. BUT there is one technology that is sadly overlooked. It is the Rodney Dangerfield of the educational technology world. I refer, of course, to the lowly DVD player. Not the DVD; the player.
"How did you know so quickly that 8 + 8 was 16?" I asked a first grade girl earlier this week. (If this question sounds familiar, it's probably because you read the previous entry in this blog.)
"Well," she said, "we have this DVD player at home and it has arrows. And if you want to speed through the movie it says 2, 4, 8, 16, 32, and then it goes back to 2 again. And I know that 2+2 is 4 and that 4+4 is 8, so 8+8 must be 16, and I guess that 16+16 would be 32. But then the pattern stops because it goes back to 2 and 32+32 is...something, but it isn't 2."
What can I say? Clearly, we should as a nation reduce our spending on old-boring-and-ineffective technologies such as computers, projectors, smart boards, and digital cameras, and load up classrooms instead with DVD players. Who's with me?
--Actually, this is a really good example of a child not only noticing but using math in everyday life. No one taught her that 8 + 8 was 16. She was struck by a sequence of numbers that appeared in her environment, and spent time and energy deciphering the pattern--learning, and evidently mastering, the fact that 8+8=16 along the way. This is the kind of thinking we always want to see in our students. As our report form puts it, one of our goals for children is that they "recognize and construct mathematics in daily life." It's lovely to see such a clear example.
Monday, September 14, 2009
That's All She (w)Rote
You have 8 cubes, I say.
The child, a first grader, nods happily. He's just counted them, accurately, and showed me how you could split them up so that we each had the same number (4 apiece, if you were curious), and answered several other questions about them as well.
What if you had 8 cubes and I had 8 cubes too? I ask. How many would we have in all?
This isn't necessarily an easy question for six-year-olds, and they vary in their approaches--also in the speed with which they answer. 28, says the boy, just as automatically as you please. There's no lack of confidence here. (Not a lot of accuracy, either, but hey, it's still early in the year.)
28? I ask, just to make sure.
28, he says. No. I mean, um, 34. Yeah, 34.
34, I repeat, resisting the temptation to ask, Regis-style, whether this is his final answer. And how did you know?
Oh, I didn't know, he says with a grin. I guessed.
Okay, I say, and go on to do a few more activities with him. I wrap up with a nice open-ended question: What else do you know about math that you'd like to tell me?
Well, he says eagerly, one thing I know is that 8 plus 8 is 16...
SMACK! goes my hand (metaphorically at least) against the side of my head.
This little anecdote nicely illustrates the difference between knowing a fact and KNOWING it. This boy knew that 8+8 was 16, but he didn't KNOW it--that is, while he could repeat it, he couldn't use that information in a real-life context. His verbal knowledge isn't yet supported by his conceptual understanding.
There's nothing wrong with learning some kinds of things by rote. Indeed, sometimes it's necessary. It's just that you have to be careful with kids and not automatically assume they KNOW everything they know....if you know (KNOW?) what I mean!
Sunday, September 13, 2009
Spiiiiiiiiders
So there I am in the Pre-K, "just visiting" as they say on the Monopoly board, and the children are doing watercolors, and one child brings over her picture to show me.
"I painted a tabantula," she explains, her eyes as wide as a four-year-old's can get. Wider, if possible.
"A tabantula, huh?" I say. "Sounds mighty scary."
"It has eight legs," she says, and proceeds to count them, which she does very well (us Math Guys notice these kinds of things), and lo and behold, guess what, there ARE eight.
"Well, that's a good thing," I say, "because taban, I mean, taRANtulas are spiders, and spiders are supposed to have eight legs. Good for you for knowing that. I guess you're an expert on spiders."
She ignores this comment as the typical babbling of the Adult and points instead at a swirl of red paint. "That's the tabantula's head," she explains. "Do you want to know what that red is for?"
"Tell me," I say.
She leans in very close, stretches up, finds my ear, and stage whispers "IT'S BLOOD."
Can't wait for Halloween!
"I painted a tabantula," she explains, her eyes as wide as a four-year-old's can get. Wider, if possible.
"A tabantula, huh?" I say. "Sounds mighty scary."
"It has eight legs," she says, and proceeds to count them, which she does very well (us Math Guys notice these kinds of things), and lo and behold, guess what, there ARE eight.
"Well, that's a good thing," I say, "because taban, I mean, taRANtulas are spiders, and spiders are supposed to have eight legs. Good for you for knowing that. I guess you're an expert on spiders."
She ignores this comment as the typical babbling of the Adult and points instead at a swirl of red paint. "That's the tabantula's head," she explains. "Do you want to know what that red is for?"
"Tell me," I say.
She leans in very close, stretches up, finds my ear, and stage whispers "IT'S BLOOD."
Can't wait for Halloween!
Wednesday, September 9, 2009
The whole nine yards; or, Dressed to the nines
If I'm timing this one correctly (and I'm probably not as the margin for error is not exactly huge), this post will be time-stamped
09 [month]
09 [day]
09 [year]
09 [hour]
09 [minute]
or
09/09/09 at 09:09
The nines have it!
[Edited to add: Ooh! Missed by ONE MINUTE. Oh well...I tried.]
09 [month]
09 [day]
09 [year]
09 [hour]
09 [minute]
or
09/09/09 at 09:09
The nines have it!
[Edited to add: Ooh! Missed by ONE MINUTE. Oh well...I tried.]
Tuesday, August 4, 2009
SummerMath, Part 4: On the Road Again
If you're like most Americans, you and your children spend a lot of time on the road. There are plenty of ways for kids to pass the time on car trips, whether of 5 minutes or 500 miles. Several of them I'm quite sure you know:
Screaming "She's on MYYYY side!!!! Mom, make her get off of MYYYY side!"
Asking as often as possible, "Are we there yet?"
Threatening to throw up.
Singing vaguely risque songs like "The boys and girls are kissing in the D-R-A-K, D-R-A-K, D-R-A-K, Dark!"
Singing "George Washington Bridge." OH MY EARS QUICK RINSE THEM OUT WITH RUBBING ALCOHOL (George Washington Bridge, for those of you not in the know, repeats the lyrics "George Washington Bridge" again and again and again to an essentially unmelodic melody. My sister pulled this one on a 5-hour car ride between Chicago and La Crosse, Wisconsin, many years ago, and when she was finally told she could NO LONGER sing "George Washington Bridge" she said "Okay, then I will sing 'Adobe Bricks.' 'Adobe Bricks,' as it turns out, is 10 times worse. We eventually threw her out of the car)
But those days are over, for here is a list of WONDERFUL MATH ACTIVITIES that you can do in the car. (Assuming that your children can actually see out the windows.)
1. LICENSE PLATES. In New York State, most (not all) plates are of the form AAA 1111--three letters, then four numbers. Have kids hunt for license plates withe certain characteristics. Who can find one that has four even digits? Four odd digits? Four digits that are ascending (like 4689)? Descending (say, 8521)? Two digits that are the same? Three digits that are the same? If you're stuck at a stoplight, ask your child to add the four digits on the license plate of the vehicle in front of you--which two are best to add first? What shortcuts can your child find? (To add 6364, kids might start with 6+6=12, a "doubles fact," or with the "ten-friends" fact that 6+4=10.) Estimate by looking if the sum will be greater than or less than 20. Then check. Whoops, green light--better move on...
2. STOPLIGHTS. How many stoplights do you think there will be between here and the mall/camp/Grandma's house? Let's keep track. Will they mostly be green when we get to them, or red? You count the red ones, I'll count the green ones. Do you think it'll be about the same on the way back, or will it be different? Let's find out. For a route you drive frequently, choose a couple of lights and keep track of whether they're red or green over a period of 8-10 days. These are exercises in counting; they also ask kids to gather, use, and interpret data. More than half of the lights are green? Why do you think that might be?...That light where we cross Route 55 is almost always red when we get there--how come?
3. VROOM, VROOM. On a 4-lane highway, have kids count the cars you pass and the cars that pass you. Make this an exercise in counting forwards and backwards: start with a score of 10, add one for every car you pass, subtract one for every car that passes you. Or, start with 50 or even 100. Try not to give in to your children's pleadings to do whatever it takes to avoid being passed by that in-your-face Oldsmobile or to overtake that weirdly painted appliance truck. Does it matter who's driving?...why yes, yes, it might. (And can we correlate that with speeding tickets received? Why yes, yes, we can...)
4. NUMBERS ON SIGNS. There are lots of these running around the roads: speed limits, mileage markers, route numbers, distances to upcoming cities. We call Route 376 "route three-seventy-six," but what's its "proper" name? (Three hundred seventy-six. Yes, I know it isn't *really* a number, because it doesn't indicate three hundred seventy-six of anything...) Who can find a two-digit number on a road sign? A three-digit number? An odd number? The sign tells how many miles to Montreal and how many to Buffalo. Which is further away from us right now? How do you know?
5. MENTAL ARITHMETIC. We're at milepost 27--look, there's the sign. What milepost will we pass in five miles? Ten miles? (Careful--are we driving towards milepost 0 or away from it?) How many more miles till milepost 40? (See above.) The sign says Albany is 65 miles away. Our speed is, guess what, 65 miles per hour. About how long till we're in Albany? (Only use easy numbers for this kind of question!) The car can be a good place to go over basic facts as well: "7 + 2." "9!" "Tell me two ways to make 10." "Um--"
6. MAPS and DIRECTIONS. Have children direct you to a location they've visited many times before. Ask them to tell you where to go straight and where to turn, and whether you should go left or right when you turn. Obviously, don't break any traffic laws--you are still the captain of the ship! As for maps: Print out a map showing your route to a (relatively) nearby place. Go over the map with your child before you leave. Have your child hold the map and try to track your position along the route as you drive. Try using a road map for longer distances: Find a long thick blue line with a shield and the number 84. That's the road we're on now. We're heading west...which way is west? Can you find a city called Middletown? Excellent--we're just a little bit west of that right now. What's the next town you see? Some third and fourth graders can become quite good at navigating. Just be sure that the "road" they're having you follow isn't just a marmalade stain on the map (this happened once to Paddington Brown and his family, I believe).
As always, these are ideas, nothing more; you can come up with others yourself. Be sure not to push too hard. REPEAT: BE SURE NOT TO PUSH TOO HARD. If you find you're suddenly more invested in these activities than the kids are, cut the games short and do something else: talk, sing, tell jokes, look at scenery. But if people start yelling about siblings being on theirrrrr side, it helps to have games like find-a-license-plate-with-4-odd-digits in your back pocket; and if you can find the right combination of math activities for the car ride on any given day, you will NEVER EVER EVER have to put up with anybody singing George Washington Bridge, and that will make any hardship worthwhile.
Screaming "She's on MYYYY side!!!! Mom, make her get off of MYYYY side!"
Asking as often as possible, "Are we there yet?"
Threatening to throw up.
Singing vaguely risque songs like "The boys and girls are kissing in the D-R-A-K, D-R-A-K, D-R-A-K, Dark!"
Singing "George Washington Bridge." OH MY EARS QUICK RINSE THEM OUT WITH RUBBING ALCOHOL (George Washington Bridge, for those of you not in the know, repeats the lyrics "George Washington Bridge" again and again and again to an essentially unmelodic melody. My sister pulled this one on a 5-hour car ride between Chicago and La Crosse, Wisconsin, many years ago, and when she was finally told she could NO LONGER sing "George Washington Bridge" she said "Okay, then I will sing 'Adobe Bricks.' 'Adobe Bricks,' as it turns out, is 10 times worse. We eventually threw her out of the car)
But those days are over, for here is a list of WONDERFUL MATH ACTIVITIES that you can do in the car. (Assuming that your children can actually see out the windows.)
1. LICENSE PLATES. In New York State, most (not all) plates are of the form AAA 1111--three letters, then four numbers. Have kids hunt for license plates withe certain characteristics. Who can find one that has four even digits? Four odd digits? Four digits that are ascending (like 4689)? Descending (say, 8521)? Two digits that are the same? Three digits that are the same? If you're stuck at a stoplight, ask your child to add the four digits on the license plate of the vehicle in front of you--which two are best to add first? What shortcuts can your child find? (To add 6364, kids might start with 6+6=12, a "doubles fact," or with the "ten-friends" fact that 6+4=10.) Estimate by looking if the sum will be greater than or less than 20. Then check. Whoops, green light--better move on...
2. STOPLIGHTS. How many stoplights do you think there will be between here and the mall/camp/Grandma's house? Let's keep track. Will they mostly be green when we get to them, or red? You count the red ones, I'll count the green ones. Do you think it'll be about the same on the way back, or will it be different? Let's find out. For a route you drive frequently, choose a couple of lights and keep track of whether they're red or green over a period of 8-10 days. These are exercises in counting; they also ask kids to gather, use, and interpret data. More than half of the lights are green? Why do you think that might be?...That light where we cross Route 55 is almost always red when we get there--how come?
3. VROOM, VROOM. On a 4-lane highway, have kids count the cars you pass and the cars that pass you. Make this an exercise in counting forwards and backwards: start with a score of 10, add one for every car you pass, subtract one for every car that passes you. Or, start with 50 or even 100. Try not to give in to your children's pleadings to do whatever it takes to avoid being passed by that in-your-face Oldsmobile or to overtake that weirdly painted appliance truck. Does it matter who's driving?...why yes, yes, it might. (And can we correlate that with speeding tickets received? Why yes, yes, we can...)
4. NUMBERS ON SIGNS. There are lots of these running around the roads: speed limits, mileage markers, route numbers, distances to upcoming cities. We call Route 376 "route three-seventy-six," but what's its "proper" name? (Three hundred seventy-six. Yes, I know it isn't *really* a number, because it doesn't indicate three hundred seventy-six of anything...) Who can find a two-digit number on a road sign? A three-digit number? An odd number? The sign tells how many miles to Montreal and how many to Buffalo. Which is further away from us right now? How do you know?
5. MENTAL ARITHMETIC. We're at milepost 27--look, there's the sign. What milepost will we pass in five miles? Ten miles? (Careful--are we driving towards milepost 0 or away from it?) How many more miles till milepost 40? (See above.) The sign says Albany is 65 miles away. Our speed is, guess what, 65 miles per hour. About how long till we're in Albany? (Only use easy numbers for this kind of question!) The car can be a good place to go over basic facts as well: "7 + 2." "9!" "Tell me two ways to make 10." "Um--"
6. MAPS and DIRECTIONS. Have children direct you to a location they've visited many times before. Ask them to tell you where to go straight and where to turn, and whether you should go left or right when you turn. Obviously, don't break any traffic laws--you are still the captain of the ship! As for maps: Print out a map showing your route to a (relatively) nearby place. Go over the map with your child before you leave. Have your child hold the map and try to track your position along the route as you drive. Try using a road map for longer distances: Find a long thick blue line with a shield and the number 84. That's the road we're on now. We're heading west...which way is west? Can you find a city called Middletown? Excellent--we're just a little bit west of that right now. What's the next town you see? Some third and fourth graders can become quite good at navigating. Just be sure that the "road" they're having you follow isn't just a marmalade stain on the map (this happened once to Paddington Brown and his family, I believe).
As always, these are ideas, nothing more; you can come up with others yourself. Be sure not to push too hard. REPEAT: BE SURE NOT TO PUSH TOO HARD. If you find you're suddenly more invested in these activities than the kids are, cut the games short and do something else: talk, sing, tell jokes, look at scenery. But if people start yelling about siblings being on theirrrrr side, it helps to have games like find-a-license-plate-with-4-odd-digits in your back pocket; and if you can find the right combination of math activities for the car ride on any given day, you will NEVER EVER EVER have to put up with anybody singing George Washington Bridge, and that will make any hardship worthwhile.
My Heart's in the Heartland
A cool map for you geography geeks out there:
http://strangemaps.wordpress.com/2009/07/27/402-homeland-is-where-the-heartland-is/#comments
Note that some of the sizes are a bit off. Also, something seems to have happened to Dutchess County. STILL!
http://strangemaps.wordpress.com/2009/07/27/402-homeland-is-where-the-heartland-is/#comments
Note that some of the sizes are a bit off. Also, something seems to have happened to Dutchess County. STILL!
More Rabbit News
--> NEWS FLASH <--
THE MATH GUY'S RABBIT CAN MULTIPLY!!!!
(Technically, this is NOT the Math Guy's rabbit. It belongs to his daughter, who purchased it from a pet store a couple of years ago and then hid it in the garage for three days while she worked up the nerve to tell us what she'd done. However, now that she's away at school, guess who does the bulk of caretaking for Miss Rabbit? --That's right.)
Anyway. The other day I was communing with the rabbit and asked it a question, offhandedly:
"Rabbit!" I said. "What is 3 x 5?"
And darned if she didn't woffle her nose 15 times.
"Remarkable!" I said. "And, Rabbit, what is 2 x 4?"
I counted eight woffles.
"Such a mathematical rabbit!" I crowed, and took her through several more examples: 6 x 2, 4 x 4, and 2 x 10. Woffle, woffle, woffle, and she was right on the money EVERY TIME.
My family seems suspicious of her awesome abilities. They seem to believe that I am cheating or perhaps misinterpreting her responses, but I swear she woffles the correct number of times. Anyhow, who cares what they say. I'm considering taking her on the talk show circuit--your thoughts?
THE MATH GUY'S RABBIT CAN MULTIPLY!!!!
(Technically, this is NOT the Math Guy's rabbit. It belongs to his daughter, who purchased it from a pet store a couple of years ago and then hid it in the garage for three days while she worked up the nerve to tell us what she'd done. However, now that she's away at school, guess who does the bulk of caretaking for Miss Rabbit? --That's right.)
Anyway. The other day I was communing with the rabbit and asked it a question, offhandedly:
"Rabbit!" I said. "What is 3 x 5?"
And darned if she didn't woffle her nose 15 times.
"Remarkable!" I said. "And, Rabbit, what is 2 x 4?"
I counted eight woffles.
"Such a mathematical rabbit!" I crowed, and took her through several more examples: 6 x 2, 4 x 4, and 2 x 10. Woffle, woffle, woffle, and she was right on the money EVERY TIME.
My family seems suspicious of her awesome abilities. They seem to believe that I am cheating or perhaps misinterpreting her responses, but I swear she woffles the correct number of times. Anyhow, who cares what they say. I'm considering taking her on the talk show circuit--your thoughts?
Tuesday, July 21, 2009
Of Rabbits and Math Guys
Several years ago, early in my incarnation as Math Guy, I walked into Sue's third and fourth grade classroom ready to present a lesson. I was surprised to see that a bunch of rabbits had replaced the children that day.
The class had been reading a novel about rabbits or rabbitlike creatures, Sue explained, and several children had come up with the idea of dressing like rabbits one day, and the idea had met with approval from basically everybody.
Some had done just the basics--a few face-paint whiskers, a kush ball for a tail. Others had added a carefully-stapled set of ears made from construction paper. A few had gone whole hog (whole bunny?) and dressed all in white or brown or black with socks and slippers and even gloves. They looked...different. They looked...creative.
"Greetings, rabbits," I said, and asked them to take their seats so we could begin the math instruction for the day. For rabbits, they did reasonably well sitting still, and they did an even better job of listening (must've been the big ears).
My planned lesson was on what kids often like to call "timesing." We began by reviewing some basic multiplication facts and then moved on to multiplication strategies and the link between multiplication and addition, and just before I sent them to the tables to do some independent work, it suddenly occurred to me that I was--
--that's right--
--teaching rabbits to multiply.
Bada-bing!
True story, too.
The class had been reading a novel about rabbits or rabbitlike creatures, Sue explained, and several children had come up with the idea of dressing like rabbits one day, and the idea had met with approval from basically everybody.
Some had done just the basics--a few face-paint whiskers, a kush ball for a tail. Others had added a carefully-stapled set of ears made from construction paper. A few had gone whole hog (whole bunny?) and dressed all in white or brown or black with socks and slippers and even gloves. They looked...different. They looked...creative.
"Greetings, rabbits," I said, and asked them to take their seats so we could begin the math instruction for the day. For rabbits, they did reasonably well sitting still, and they did an even better job of listening (must've been the big ears).
My planned lesson was on what kids often like to call "timesing." We began by reviewing some basic multiplication facts and then moved on to multiplication strategies and the link between multiplication and addition, and just before I sent them to the tables to do some independent work, it suddenly occurred to me that I was--
--that's right--
--teaching rabbits to multiply.
Bada-bing!
True story, too.
Thursday, July 16, 2009
SummerMath, Part 3: The Bikepath, the Ballpark, and Beyond
My family went to the ballgame the other day, attracted by among other things a "Henry Hudson Bobblehead" giveaway (see picture). My son is quite eager to show off his Hudson Valley roots with this, um, iconic image when he heads west for his next college semester, and as for the rest of us, well, how could we pass up such a quality and historic freebie??
(Ol' Henry)
Anyhow, the game put me in my mind of Sports and Math. I spent most of my childhood free time engaged in one of five activities:
1) eating
2) reading the classics, mainly the Hardy Boys books
3) writing short stories with meandering plots and lots of unnecessary characters
4) playing board games and card games (see SummerMath Parts 1 and 2)
and
5) playing, watching, reading about, or thinking about baseball.
Baseball and math are closely linked, and in fact I learned quite a lot about math from my interest in baseball. My 1972 Sports Illustrated baseball board game (see 4 and 5 above) helped inform me about probability. I can remember the power I felt when I realized that I could use what (little) I knew about ratios to compare teams' won-lost records in my head--was it better to be 38-37 or 37-36, and how could I prove it? And I developed some facility with division, if not comprehension of WHY it worked, by virtue of calculating my batting average every day back when I was ten or so. (My batting average was very good. I counted it as a hit, of course, if someone muffed a ball I'd put in play. Or if the umpire mistakenly called me out when I was CLEARLY safe at first--don't laugh, it happened all the time. Or if I hit a line drive or a deep fly ball that somebody managed to corral, but which clearly SHOULD'VE been a hit--why should I be penalized just because my opponents had good hands? That was in addition to the occasional, you know, REAL hits I got. As I said, my batting average was very good.)
In any case, there are lots of ways to combine math with sports, for those of you whose children like to watch baseball, play soccer, ride bikes, or mess around with balls and such in the back yard after dinner. Here are some ideas of questions you can ask and projects you can do:
*Counting and estimating. "I wonder how many pitches the pitcher will throw this inning. Do you think it'll be more than 15 or less than 15?" "Take ten shots on goal from right here. Let's see how many go in...Now let's move you back a few feet. How many do you think will go into the net now?" "Good job! We just did 6 throws back and forth in a row without dropping a single one. Think we can beat that record? Let's keep track."
*Adding and subtracting, multiplying and dividing. "The scoreboard says the Renegades are winning 7 to 2. How many runs are they winning by?" "That's your third basket. Each basket is worth 2 points. How many points do you have so far?" I'll just add that I have taught many primary graders who could count rapidly by twos, fives, and tens when they came to my class, and a few who could rattle off threes, fours, and nines; but the only one I ever had who could count fluently by sevens was the one who lived and died with the NY Giants. Sevens...football...hmm!
*Measuring. "You sure hit that one a long way! I wonder how far it went.." You can measure with "nonstandard units," such as steps or rake lengths, which tends to be a little more meaningful for younger children, or with standard units--feet, yards, meters. "14 rake lengths--that's a lot. Whoa, that one went even further! Would you say 15, or 20, or even more?" How long does it take to run around the yard or the perimeter of the park? Time your child; let your child time you. Write it down. Try it again another day. Look at the map of one of the local bike paths. "It's 10 and a half miles long! How far do you think we'll get before I'll be ready to turn around?...I see another mileage marker up ahead--4 miles and still going!"
*Graphing. These take a little more time and energy, but they're great for kids who really love sports, especially team spectator sports. Work with your child to make a bar graph showing his or her favorite team's wins and losses.
(A sample bar graph)
Update it daily; use the internet or the newspaper to get the scores.
Or, make a line graph showing the number of runs your team scores on a daily basis. Look how the line moves around. What has the trend been? More runs over time, or fewer or about the same? How could you show the number of runs they gave up each day on the same graph?
(A sample line graph)
Can you make a graph showing how many times you go swimming/bicycling/hiking this month? We'll write the words down here; put up a blue sticker for the water whenever we swim, a red sticker for the color of your bike to show each time you go for a ride, a green sticker for the color of the leaves to stand for a hike.
(A sample picture graph)
Which has the most so far? The fewest? How many more bike rides have you taken than hikes?
Of course, I don't mean to reduce sports and physical activity to numbers. Nor is the point for kids to quantify their outside play. Be sure that timing and measuring are just for fun, a nice way of bringing a little math into children's lives, not an opportunity for frustration and embarrassment because they can't seem to beat their old record; be sure that a graph is a cute little add-on, not another chore that has to be done or the sole reason for taking a bike ride or going out for a hike. Sports are their own reward. Though, now that I think about, the ability to hit .658 (and calculate it properly!) might be its own reward, too...
(Ol' Henry)Anyhow, the game put me in my mind of Sports and Math. I spent most of my childhood free time engaged in one of five activities:
1) eating
2) reading the classics, mainly the Hardy Boys books
3) writing short stories with meandering plots and lots of unnecessary characters
4) playing board games and card games (see SummerMath Parts 1 and 2)
and
5) playing, watching, reading about, or thinking about baseball.
Baseball and math are closely linked, and in fact I learned quite a lot about math from my interest in baseball. My 1972 Sports Illustrated baseball board game (see 4 and 5 above) helped inform me about probability. I can remember the power I felt when I realized that I could use what (little) I knew about ratios to compare teams' won-lost records in my head--was it better to be 38-37 or 37-36, and how could I prove it? And I developed some facility with division, if not comprehension of WHY it worked, by virtue of calculating my batting average every day back when I was ten or so. (My batting average was very good. I counted it as a hit, of course, if someone muffed a ball I'd put in play. Or if the umpire mistakenly called me out when I was CLEARLY safe at first--don't laugh, it happened all the time. Or if I hit a line drive or a deep fly ball that somebody managed to corral, but which clearly SHOULD'VE been a hit--why should I be penalized just because my opponents had good hands? That was in addition to the occasional, you know, REAL hits I got. As I said, my batting average was very good.)
In any case, there are lots of ways to combine math with sports, for those of you whose children like to watch baseball, play soccer, ride bikes, or mess around with balls and such in the back yard after dinner. Here are some ideas of questions you can ask and projects you can do:
*Counting and estimating. "I wonder how many pitches the pitcher will throw this inning. Do you think it'll be more than 15 or less than 15?" "Take ten shots on goal from right here. Let's see how many go in...Now let's move you back a few feet. How many do you think will go into the net now?" "Good job! We just did 6 throws back and forth in a row without dropping a single one. Think we can beat that record? Let's keep track."
*Adding and subtracting, multiplying and dividing. "The scoreboard says the Renegades are winning 7 to 2. How many runs are they winning by?" "That's your third basket. Each basket is worth 2 points. How many points do you have so far?" I'll just add that I have taught many primary graders who could count rapidly by twos, fives, and tens when they came to my class, and a few who could rattle off threes, fours, and nines; but the only one I ever had who could count fluently by sevens was the one who lived and died with the NY Giants. Sevens...football...hmm!
*Measuring. "You sure hit that one a long way! I wonder how far it went.." You can measure with "nonstandard units," such as steps or rake lengths, which tends to be a little more meaningful for younger children, or with standard units--feet, yards, meters. "14 rake lengths--that's a lot. Whoa, that one went even further! Would you say 15, or 20, or even more?" How long does it take to run around the yard or the perimeter of the park? Time your child; let your child time you. Write it down. Try it again another day. Look at the map of one of the local bike paths. "It's 10 and a half miles long! How far do you think we'll get before I'll be ready to turn around?...I see another mileage marker up ahead--4 miles and still going!"
*Graphing. These take a little more time and energy, but they're great for kids who really love sports, especially team spectator sports. Work with your child to make a bar graph showing his or her favorite team's wins and losses.
(A sample bar graph)Update it daily; use the internet or the newspaper to get the scores.
Or, make a line graph showing the number of runs your team scores on a daily basis. Look how the line moves around. What has the trend been? More runs over time, or fewer or about the same? How could you show the number of runs they gave up each day on the same graph?
(A sample line graph)Can you make a graph showing how many times you go swimming/bicycling/hiking this month? We'll write the words down here; put up a blue sticker for the water whenever we swim, a red sticker for the color of your bike to show each time you go for a ride, a green sticker for the color of the leaves to stand for a hike.
(A sample picture graph)
Which has the most so far? The fewest? How many more bike rides have you taken than hikes?
Of course, I don't mean to reduce sports and physical activity to numbers. Nor is the point for kids to quantify their outside play. Be sure that timing and measuring are just for fun, a nice way of bringing a little math into children's lives, not an opportunity for frustration and embarrassment because they can't seem to beat their old record; be sure that a graph is a cute little add-on, not another chore that has to be done or the sole reason for taking a bike ride or going out for a hike. Sports are their own reward. Though, now that I think about, the ability to hit .658 (and calculate it properly!) might be its own reward, too...
Tuesday, July 14, 2009
A Games-in-Education Site
This one is intended mainly for teachers who want to make better use of games of all kinds in their classrooms (and I'll be passing on the link to lower school teachers, of course), but parents should be able to benefit from it too. I recommend it!
http://www.gamesforeducators.com
Happy July 14, by the way. In, let's see, 12 years this will be a special date of its own: 7/14/21. No prizes for guessing the pattern, but you might try it out on your third grader.
http://www.gamesforeducators.com
Happy July 14, by the way. In, let's see, 12 years this will be a special date of its own: 7/14/21. No prizes for guessing the pattern, but you might try it out on your third grader.
Tuesday, July 7, 2009
SummerMath, Part 2: Card Games
I wrote a few days ago about board games. Now it's the card games' turn. What can I say? They whined and groaned until I HAD to include them...
Card games are if anything even more math-y than board games. In fact, cards themselves are pretty solidly mathematical. Consider: There are 4 suits with 13 cards in each; there are 4 seasons in the year, with 13 weeks (give or take a day here and there) in each. Coincidence? Nah!
Many of you know the game Crazy Eights, a version of which is marketed under the trade name UNO. This is a great game for getting kids to think about attributes--the different categories that cards fit into. On a seven of diamonds, for instance, you can play any diamond or any seven. Young children often scan their hands and then say with disappointment "I don't have any cards that will work." "You don't have any diamonds?" I'll ask. "No," they'll say. "And no sevens either?" "No--oh, wait!" The ability to keep two attributes (such as suit AND rank) in mind at the same time is extremely useful in math. In geometry, for instance, kids will need to know that a square is a kind of a rhombus and a kind of a quadrilateral (and on and on); in numbers, kids should recognize that 44, say, is divisible by both 2 and 11 (not to mention 4 and 22). So a hand of Crazy Eights before dinner is a nice painless way to encourage mathematical thinking--and knock off a few of those prerequisites for geometry, division, and more.
Okay, okay, the game War is exceedingly dull and involves no strategy whatever. I get that (boy, do I ever). But your 4-7-year-old is busy practicing concepts of greater than and less than while playing, which ALMOST makes up for the boredom issue. Ask questions as the game goes on, too. (And it DOES go on...okay, enough carping.) "Your 9 beats my 2...by a little, or by a lot?" "I'm going to put my card down first--oh, a 3. Do you think I will probably win with a 3? Let's check your prediction."
There are any number of variations on rummy. These games are especially good for third grade and up. Basically, players try to get groups of three (or more) cards that are all the same rank (as in three queens) or same suit and in a run (as in 4, 5, 6, 7 of spades). You pick up and discard various cards in an attempt to make these groups. We're talking strategic thinking and probability in addition to attributes and sequencing. Scoring requires adding the values of cards, too.
Then there are the approximately one zillion forms of solitaire. Many of these games deal with attributes, or with addition, or with sequencing; all of them are good for strategic thinking. The game Spit was extremely popular as a snacktime/rainyday activity for third and fourth graders last year; despite its unsavory name it helps develop sequencing skills, both backwards and forwards, and encourages kids to know what's one less or one more than a given number automatically. Concentration isn't much of a math game, but you can make it one by playing only with cards A-9 and having the object be to draw 2 cards that have a sum of 10. (Instead of matching two 8s, say, you match an 8 and a 2.) The same principle applies to Go Fish, another not-very-mathy-game, which becomes "Tens Go Fish" when you ask for a card that goes with one of your own to make 10.
A little more purely mathematical, but still fun: For younger kids you can try Close to 10 or Close to 20. For Close to 10, deal out 3 cards after removing the face cards from the deck. Focus only on the rank (ace = 1). Choose two cards with a sum that is as close to 10 as possible. How close are you? That's your score. Record it. Play 5 rounds. High score loses. You can play this cooperatively or competitively, which each player having a different set of cards. For Close to 20, use five cards and choose three, or try some other variation. This game is great for estimation, for practicing addition strategies, and again for strategic thinking.
Then there are various betting games. "Can we play Cash Cab poker?" one of Ellen's fourth graders used to ask me almost every day last year, and though the answer was usually "Not today," kids ages 7 and up very much enjoy the mixture of skill and luck in --> HIGH STAKES <-- card games. I don't advise using actual money, but counters work just fine. Here's a basic template, which permits a whole mess of variations:
*Remove the face cards (and sometimes the tens). Remind players that ace counts as 1.
*Deal each player a card face up. High card bets (or folds). Other players follow (or fold). (I generally don't do raises, but you can if you like.)
*Next, deal a second card face down. High card showing bets again; others follow.
*Finally, deal a third card face down. High card showing bets again; others follow.
Who gets the dough? Here are some possible ways to do it.
*Multiplication practice. Choose two of your three cards. Find the product (what you get when you multiply them). Greatest product wins the pot. Alternatively, play high/low in which players who have low cards still can win. Before revealing their cards, players announce whether they're going for high or going for low. Those who announce they're going for high reveal their products; highest product gets half the pot. Those who announce they're going for low do the same; lowest product gets the other half the pot. Sneaky, huh?
*Greatest 3-digit number. Or greatest 2-digit number chosen from the 3 cards. Or high/low. Which way should you order 4, 7, and 2 if you're going for high? Which way for low? Which gives you a better chance of winning?
*Greatest sum. Make a 2-digit number and a 1-digit number (so if your cards are 4, 7, 2 you can do 47 and 2, or 24 and 7, or...). Add them, mentally or with paper and pencil. Greatest sum wins; or do high/low...
*Make it 5 cards instead of 3. Your goal is to have the 5 cards that add to a total nearer 25 than anyone else. This one's especially interesting because what looks like a "good" hand early on may prove to be a "bad" hand as those nines and tens don't stop coming
Or other variations that you and your children come up with.
As before, these games should be considered an opportunity for some fun rather than a chore. They're games, after all. Be aware of when your child starts to squirm, or when the brain begins to turn off, or when the beautiful day outside is becoming more appealing than the king of hearts. But if you don't overdo it and play your cards right (hardy-har-har), these games can be great ways to help your child have fun--and practice a little math in the bargain.
Card games are if anything even more math-y than board games. In fact, cards themselves are pretty solidly mathematical. Consider: There are 4 suits with 13 cards in each; there are 4 seasons in the year, with 13 weeks (give or take a day here and there) in each. Coincidence? Nah!
Many of you know the game Crazy Eights, a version of which is marketed under the trade name UNO. This is a great game for getting kids to think about attributes--the different categories that cards fit into. On a seven of diamonds, for instance, you can play any diamond or any seven. Young children often scan their hands and then say with disappointment "I don't have any cards that will work." "You don't have any diamonds?" I'll ask. "No," they'll say. "And no sevens either?" "No--oh, wait!" The ability to keep two attributes (such as suit AND rank) in mind at the same time is extremely useful in math. In geometry, for instance, kids will need to know that a square is a kind of a rhombus and a kind of a quadrilateral (and on and on); in numbers, kids should recognize that 44, say, is divisible by both 2 and 11 (not to mention 4 and 22). So a hand of Crazy Eights before dinner is a nice painless way to encourage mathematical thinking--and knock off a few of those prerequisites for geometry, division, and more.
Okay, okay, the game War is exceedingly dull and involves no strategy whatever. I get that (boy, do I ever). But your 4-7-year-old is busy practicing concepts of greater than and less than while playing, which ALMOST makes up for the boredom issue. Ask questions as the game goes on, too. (And it DOES go on...okay, enough carping.) "Your 9 beats my 2...by a little, or by a lot?" "I'm going to put my card down first--oh, a 3. Do you think I will probably win with a 3? Let's check your prediction."
There are any number of variations on rummy. These games are especially good for third grade and up. Basically, players try to get groups of three (or more) cards that are all the same rank (as in three queens) or same suit and in a run (as in 4, 5, 6, 7 of spades). You pick up and discard various cards in an attempt to make these groups. We're talking strategic thinking and probability in addition to attributes and sequencing. Scoring requires adding the values of cards, too.
Then there are the approximately one zillion forms of solitaire. Many of these games deal with attributes, or with addition, or with sequencing; all of them are good for strategic thinking. The game Spit was extremely popular as a snacktime/rainyday activity for third and fourth graders last year; despite its unsavory name it helps develop sequencing skills, both backwards and forwards, and encourages kids to know what's one less or one more than a given number automatically. Concentration isn't much of a math game, but you can make it one by playing only with cards A-9 and having the object be to draw 2 cards that have a sum of 10. (Instead of matching two 8s, say, you match an 8 and a 2.) The same principle applies to Go Fish, another not-very-mathy-game, which becomes "Tens Go Fish" when you ask for a card that goes with one of your own to make 10.
A little more purely mathematical, but still fun: For younger kids you can try Close to 10 or Close to 20. For Close to 10, deal out 3 cards after removing the face cards from the deck. Focus only on the rank (ace = 1). Choose two cards with a sum that is as close to 10 as possible. How close are you? That's your score. Record it. Play 5 rounds. High score loses. You can play this cooperatively or competitively, which each player having a different set of cards. For Close to 20, use five cards and choose three, or try some other variation. This game is great for estimation, for practicing addition strategies, and again for strategic thinking.
Then there are various betting games. "Can we play Cash Cab poker?" one of Ellen's fourth graders used to ask me almost every day last year, and though the answer was usually "Not today," kids ages 7 and up very much enjoy the mixture of skill and luck in --> HIGH STAKES <-- card games. I don't advise using actual money, but counters work just fine. Here's a basic template, which permits a whole mess of variations:
*Remove the face cards (and sometimes the tens). Remind players that ace counts as 1.
*Deal each player a card face up. High card bets (or folds). Other players follow (or fold). (I generally don't do raises, but you can if you like.)
*Next, deal a second card face down. High card showing bets again; others follow.
*Finally, deal a third card face down. High card showing bets again; others follow.
Who gets the dough? Here are some possible ways to do it.
*Multiplication practice. Choose two of your three cards. Find the product (what you get when you multiply them). Greatest product wins the pot. Alternatively, play high/low in which players who have low cards still can win. Before revealing their cards, players announce whether they're going for high or going for low. Those who announce they're going for high reveal their products; highest product gets half the pot. Those who announce they're going for low do the same; lowest product gets the other half the pot. Sneaky, huh?
*Greatest 3-digit number. Or greatest 2-digit number chosen from the 3 cards. Or high/low. Which way should you order 4, 7, and 2 if you're going for high? Which way for low? Which gives you a better chance of winning?
*Greatest sum. Make a 2-digit number and a 1-digit number (so if your cards are 4, 7, 2 you can do 47 and 2, or 24 and 7, or...). Add them, mentally or with paper and pencil. Greatest sum wins; or do high/low...
*Make it 5 cards instead of 3. Your goal is to have the 5 cards that add to a total nearer 25 than anyone else. This one's especially interesting because what looks like a "good" hand early on may prove to be a "bad" hand as those nines and tens don't stop coming
Or other variations that you and your children come up with.
As before, these games should be considered an opportunity for some fun rather than a chore. They're games, after all. Be aware of when your child starts to squirm, or when the brain begins to turn off, or when the beautiful day outside is becoming more appealing than the king of hearts. But if you don't overdo it and play your cards right (hardy-har-har), these games can be great ways to help your child have fun--and practice a little math in the bargain.
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