The second graders were measuring. They'd cut out replicas of their feet (exact size, natch) and were busily determining how many of these footprints (feetprints?) it took to equal the length of a shelf, the width of the room, and other various and sundry distances. Then they were converting the number of feetprints (footprints?) to inches and recording it all on a chart.
I plunked myself down next to a child who was recording the number of feetsprint she had needed to cover the distance across a table. She'd written a 7, which sounded reasonable--seven second-grade-sized footsprint looked about right--but what was this next to it? A zero? Seventy? Surely she was putting 70 in the wrong place of the chart. Or she'd mismeasured. Or--
Wait a minute.
It wasn't just a zero. It was a bubble letter--you know, the puffy letters that kids love to make, especially when time is of the essence. The ones that slow kids' work pace down to a crawl. The ones that drive me faintly crazy. The ones that--
Hold on.
Now she was decorating the thing. Shading in part of the inside ring, drawing something unrecognizable in the middle. Decorating--during math time! Bubble letters--during math time! I mean, gee whillikers!
I opened my mouth to say something gentle, yet pointed. Okay, something not-so-gentle yet pointed. Something about saving the artistry for art and getting back to math, and by-the-way was 70 really a reasonable answer, and if you'd been paying closer attention to the math rather than to the art you'd know...But then I didn't. "Tell me about what you're drawing," I said instead, pointing. Just in case my assumption was wrong and there was method to her madness.
"Oh, that's a quarter," she explained, barely looking up.
"The coin?" I asked. "The thing that's worth twenty-five cents?" I peered closer. Okay, now that she'd mentioned it I could see that the bubble-letter zero did indeed resemble a quarter. Fine and dandy, but that didn't explain why she drawn a coin as part of this measurement project. I opened my mouth again...but instead of the pointed comment I'd intended, I found myself with a different response, again a response that didn't automatically assume that she'd messed up.
"Why a quarter?" I asked.
"Well," she said, "when I measured the table I found it was seven and a quarter of my footsprints." She tapped the seven on the chart, then the quarter beside it. "So I wrote seven, and then I drew a quarter. That's why."
And that's why I'm glad I asked!
Showing posts with label measurement. Show all posts
Showing posts with label measurement. Show all posts
Thursday, October 28, 2010
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!
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...
Friday, May 29, 2009
Things Your Children Probably Shouldn't Be Telling Us, Part 1
So we were working on time in the 1-2s this week. Today, the kids named and ordered various units of time, from milliseconds and seconds up to centuries and millenniums, and explained what they knew of the relationships between them, using nice NUMBER SENTENCES (60 sec = 1 min, 1 day = 24 hours, etc). On the whole, they did quite well, though why "half an hour" and "5 minutes" don't count as separate units of time was a bit mysterious for a few of our first graders. Next year--
Anyhow, after this lead-in, I asked children to fill out a sheet about time units. The structure was simple enough. "It takes about one SECOND to..." was the first one, and kids were supposed to think of an activity that takes about one second. Then they followed it with one minute, one hour, and one day.
The responses were fun and revealing of children's understanding: one second to "squash a bug," "pick up a feather," or "say four letters of the alphabet," one hour to "clean my room" or "draw a perfect picture" (quick, tell Picasso!), one day to "make a really good sculpture."
My personal favorite, though? "It takes about one minute to do my homework." Given that mathups, reading, and spelling alone are supposed to take at least 20-25 minutes each night, this is the sort of statement that is perhaps better left unsaid. Ah well--by high school I'm sure this child will have figured that out!
Anyhow, after this lead-in, I asked children to fill out a sheet about time units. The structure was simple enough. "It takes about one SECOND to..." was the first one, and kids were supposed to think of an activity that takes about one second. Then they followed it with one minute, one hour, and one day.
The responses were fun and revealing of children's understanding: one second to "squash a bug," "pick up a feather," or "say four letters of the alphabet," one hour to "clean my room" or "draw a perfect picture" (quick, tell Picasso!), one day to "make a really good sculpture."
My personal favorite, though? "It takes about one minute to do my homework." Given that mathups, reading, and spelling alone are supposed to take at least 20-25 minutes each night, this is the sort of statement that is perhaps better left unsaid. Ah well--by high school I'm sure this child will have figured that out!
Thursday, May 28, 2009
Finding the Center
One of the perks about being a member of NCTM (http://nctm.org, the National Council of Teachers of Mathematics) is that you get a subscription to a journal called, what else, Teaching Children Mathematics. This journal has a monthly feature called "Problem Solvers," which presents an open-ended problem and encourages teachers to try it with their classes. Teachers are then invited to write up their experiences and send 'em in. From time to time I've tried these problems out, and once I even got around to sending in my reflections.
Anyway, a recent Problem Solvers challenge caught my eye: How would you go about finding the geographic center of the United States (minus Alaska and Hawaii)? O-ho! I thought. This will be an interesting problem to do with all the grade levels I work with! But then field trips and special events got in the way, and so did division and fractions and 3-d geometry and other such valuable topics--so in the end I managed to do the problem only with a few 4th graders and a few 1st graders.
At some point I'll talk more about the 4th graders, who generally did quite well--they showed some sophisticated thinking about the assignment, and made use of a number of different mathematical skills to come up with an answer. This post, though, will be about the 1st graders, whose work was...um...
*************************************************
For the children , this was one of the easiest questions I'd asked all year. “It’s right here,” said one girl, touching the middle of the border between Kansas and Nebraska. The others nodded agreement and, not to be outdone, put their fingers on roughly the same spot themselves. That part of the Great Plains has never been so crowded.
This was a good estimate—a very good estimate, in fact, but I was looking for an explanation of how they'd figured it out too. When no clear explanation was forthcoming--in fact, when there was no explanation of any kind--I asked whether there were any tools they could use to show me what they were thinking. When I said tools I had in mind, oh, rulers, or some other kind of measuring device. They did not.
“A jackhammer?” suggested one boy.
"You could use a compass," said the girl who had made the initial estimate. "You would walk with the compass. You can start anywhere, like in California. Then you walk this way.” She put her finger near San Francisco and slid it eastward on the map. “When you get there, you stop.”
“How do you know when you’re there?” I asked.
She shrugged. “Because you’ll get to that place, and then you’ll be there.” She was too polite to say Duh!, but you could hear it all the same.
“What do the rest of you think?” I inquired. A chorus of “I agree”s and “Uh-huhs” rose from the other children. I believe this is called proof by intimidation.
I decided we'd better back up. “How about this table?” I asked. “Where’s its center? And how do you know?” Several hands slapped down in a place reasonably close to the center, if not the exact spot. The center, they explained, had to be in the middle of the lines that divided the table in half. Duh! Again, politeness reigned, but I knew what was what.
“So now we know about the center of the table,” I said. “I wonder if that might help us find the center of the country.” I opened up the map again. “What do you think?”
There was brief discussion. One child pointed out that the United States wasn’t a nice regular shape, such as a circle or a square, so it didn’t really have a center. Another argued that the whole world would have a center, “because that would be a sphere and then you could find the middle of it.” But they all deferred to a girl who cut to the chase. "The center would be right here," she said, stabbing a forefinger at a spot in the middle of Kansas, just south of the original place chosen. “That’s the center.”
Back we’d come to our starting point. “How do you know?” I asked once more, feeling like the twenty-first century version of a broken record and hoping she'd say something about lines that divided the country in half...
Nope. The child looked at me with something resembling pity. “You go up in a plane,” she said, “and then you can see where the center is and you go there.” Duh!
From which I conclude one or more of the following:
*First graders seriously underestimate the size of the country.
*First graders see no reason to calculate the exact position of a center when eyeballing it will do.
*Sometimes it’s really hard to explain your thinking, especially just before lunch on a Monday morning.
Oh well--onward!
P. S. If you'd like to know more about the geographic center, here's a rundown: http://en.wikipedia.org/wiki/Geographic_Center_of_the_Contiguous_United_States.
Anyway, a recent Problem Solvers challenge caught my eye: How would you go about finding the geographic center of the United States (minus Alaska and Hawaii)? O-ho! I thought. This will be an interesting problem to do with all the grade levels I work with! But then field trips and special events got in the way, and so did division and fractions and 3-d geometry and other such valuable topics--so in the end I managed to do the problem only with a few 4th graders and a few 1st graders.
At some point I'll talk more about the 4th graders, who generally did quite well--they showed some sophisticated thinking about the assignment, and made use of a number of different mathematical skills to come up with an answer. This post, though, will be about the 1st graders, whose work was...um...
*************************************************
For the children , this was one of the easiest questions I'd asked all year. “It’s right here,” said one girl, touching the middle of the border between Kansas and Nebraska. The others nodded agreement and, not to be outdone, put their fingers on roughly the same spot themselves. That part of the Great Plains has never been so crowded.
This was a good estimate—a very good estimate, in fact, but I was looking for an explanation of how they'd figured it out too. When no clear explanation was forthcoming--in fact, when there was no explanation of any kind--I asked whether there were any tools they could use to show me what they were thinking. When I said tools I had in mind, oh, rulers, or some other kind of measuring device. They did not.
“A jackhammer?” suggested one boy.
"You could use a compass," said the girl who had made the initial estimate. "You would walk with the compass. You can start anywhere, like in California. Then you walk this way.” She put her finger near San Francisco and slid it eastward on the map. “When you get there, you stop.”
“How do you know when you’re there?” I asked.
She shrugged. “Because you’ll get to that place, and then you’ll be there.” She was too polite to say Duh!, but you could hear it all the same.
“What do the rest of you think?” I inquired. A chorus of “I agree”s and “Uh-huhs” rose from the other children. I believe this is called proof by intimidation.
I decided we'd better back up. “How about this table?” I asked. “Where’s its center? And how do you know?” Several hands slapped down in a place reasonably close to the center, if not the exact spot. The center, they explained, had to be in the middle of the lines that divided the table in half. Duh! Again, politeness reigned, but I knew what was what.
“So now we know about the center of the table,” I said. “I wonder if that might help us find the center of the country.” I opened up the map again. “What do you think?”
There was brief discussion. One child pointed out that the United States wasn’t a nice regular shape, such as a circle or a square, so it didn’t really have a center. Another argued that the whole world would have a center, “because that would be a sphere and then you could find the middle of it.” But they all deferred to a girl who cut to the chase. "The center would be right here," she said, stabbing a forefinger at a spot in the middle of Kansas, just south of the original place chosen. “That’s the center.”
Back we’d come to our starting point. “How do you know?” I asked once more, feeling like the twenty-first century version of a broken record and hoping she'd say something about lines that divided the country in half...
Nope. The child looked at me with something resembling pity. “You go up in a plane,” she said, “and then you can see where the center is and you go there.” Duh!
From which I conclude one or more of the following:
*First graders seriously underestimate the size of the country.
*First graders see no reason to calculate the exact position of a center when eyeballing it will do.
*Sometimes it’s really hard to explain your thinking, especially just before lunch on a Monday morning.
Oh well--onward!
P. S. If you'd like to know more about the geographic center, here's a rundown: http://en.wikipedia.org/wiki/Geographic_Center_of_the_Contiguous_United_States.
Sunday, May 3, 2009
George's Excellent Adventure
Sometimes the best lessons are the ones you don't plan.
Friday morning, Ellen poked her head into the office as I was preparing for a fraction lesson with the 1-2s. "Elizabeth found a Where's George dollar in her lunch money," she said. "Okay if we take a few minutes to enter it at the beginning of math time today?"
Where's George, I should explain, is a lovely internet project that tracks paper money as it moves across the country (www.wheresgeorge.com). Since the 3-4 classes handle lots of money in their capacity as Pizza People, they occasionally run into Where's George bills, which are recognizable by special markings. We log onto the site, enter the bill's serial number, note our location, and press Enter. If the sound on my laptop is turned on, we'll then hear a cash register noise and the bill's previous location(s) will appear. Most of the bills we've found thus far have come from nearby places such as Pennsylvania and Massachusetts, Brooklyn and Kingston, but we've had bills from Missouri, Tennessee, and Texas as well. It's fun, and suspenseful, and teaches a bit about geography--and you never know when someone will find "our" bill and put it in again.
When we entered Elizabeth's bill, the screen showed that the bill was now 1128 miles from its original location. I had a sudden brainstorm. Instead of scrolling down and telling the class where the bill had come from, I'd have them narrow the possibilities by using math--specifically, their measuring and estimation skills. They'd been working on maps all year long, after all, filling in states that Cheerful Charlie had visited in his round-the-US tour. Ellen got one of the students' maps, and we hung it up. We determined that 1128 was very close to 1100, in double-round numbers, and at 200 miles to the inch, the class quickly calculated that the starting point was about 5.5 inches away from us.
It was clear to most of the students that the possibilities would form the arc of a circle, and so we did a little measuring. We ended up with a curve beginning at the western end of Michigan's Upper Peninsula and then zagging through Wisconsin, Minnesota, Iowa, Missouri, Arkansas, and Mississippi--all of them marked on the students' maps--before catching a piece of south-central Florida and disappearing over the Atlantic Ocean. "Why can't the bill have started here?" I asked, indicating where the arc crossed the Gulf of Mexico. That was obvious. "It's too wet for money in the ocean," a third grader answered (unless, he added, there were islands he "didn't know about"). As for why we didn't go north of Michigan, that was obvious too: Canada has its own money.
I scrolled down on the webpage and revealed the answer: the bill had originated in Florida. ("I knew it!" half the class exclaimed.) I named the town, which I'd never heard of. But Ellen had: her brother lived there. She asked if there was any way to find out who started the bill on its travels. Well, yes, there was; I clicked on the profile button and found a first name, Bob.
It wasn't Ellen's brother. But that was all right. Bob had provided us with a nice map of the US, each state filled in with one of six colors. Now I had my second brainstorm. We'd done a little real-life estimating and measuring with scale; it was time for some real-world data analysis.
"What do you think this map shows?" I asked, turning the computer so the students could see. Temperatures, guessed one boy. Good thought, but no. How many people live in each state? asked a girl. Close, I said. Think about what website this is, Ellen suggested, and suddenly hands were flying up all over the meeting area. Bob, they realized, had marked dozens and dozens of bills and sent them into the wild. The colors showed how many of those bills had turned up in each state.
[Here is Bob's Hit Map, by the way:]
Right on the money! (So to speak.) The only question now was which colors stood for the most bills and which for the fewest. To help, I had them identify a few key states on Bob's map, and then I gave them a little extra information. California, I explained, had the most people of any state. Texas, New York, and Florida were next. Wyoming took up a lot of space, but it had fewer people than any other state.
Working as a group, the class swiftly came up with a sensible schematic for the colors. Red, the color of Florida, California, and New York, would be the most. Bright green, it seemed apparent, would be next, judging from what the children knew of population and distance, and so on, down to lowly Wyoming, the only state that was colored gray.
The guesses were in. It was now time for the Great Unveiling. I had everyone's full attention: they were deeply invested in the outcome by now. And the results were entirely satisfactory. The class had four out of six colors right; the only error had been reversing the orderof the fourth- and fifth-most colors.
Not bad, not bad at all, I told them, and we moved on to the regularly scheduled lesson on division.
Friday morning, Ellen poked her head into the office as I was preparing for a fraction lesson with the 1-2s. "Elizabeth found a Where's George dollar in her lunch money," she said. "Okay if we take a few minutes to enter it at the beginning of math time today?"
Where's George, I should explain, is a lovely internet project that tracks paper money as it moves across the country (www.wheresgeorge.com). Since the 3-4 classes handle lots of money in their capacity as Pizza People, they occasionally run into Where's George bills, which are recognizable by special markings. We log onto the site, enter the bill's serial number, note our location, and press Enter. If the sound on my laptop is turned on, we'll then hear a cash register noise and the bill's previous location(s) will appear. Most of the bills we've found thus far have come from nearby places such as Pennsylvania and Massachusetts, Brooklyn and Kingston, but we've had bills from Missouri, Tennessee, and Texas as well. It's fun, and suspenseful, and teaches a bit about geography--and you never know when someone will find "our" bill and put it in again.
When we entered Elizabeth's bill, the screen showed that the bill was now 1128 miles from its original location. I had a sudden brainstorm. Instead of scrolling down and telling the class where the bill had come from, I'd have them narrow the possibilities by using math--specifically, their measuring and estimation skills. They'd been working on maps all year long, after all, filling in states that Cheerful Charlie had visited in his round-the-US tour. Ellen got one of the students' maps, and we hung it up. We determined that 1128 was very close to 1100, in double-round numbers, and at 200 miles to the inch, the class quickly calculated that the starting point was about 5.5 inches away from us.
It was clear to most of the students that the possibilities would form the arc of a circle, and so we did a little measuring. We ended up with a curve beginning at the western end of Michigan's Upper Peninsula and then zagging through Wisconsin, Minnesota, Iowa, Missouri, Arkansas, and Mississippi--all of them marked on the students' maps--before catching a piece of south-central Florida and disappearing over the Atlantic Ocean. "Why can't the bill have started here?" I asked, indicating where the arc crossed the Gulf of Mexico. That was obvious. "It's too wet for money in the ocean," a third grader answered (unless, he added, there were islands he "didn't know about"). As for why we didn't go north of Michigan, that was obvious too: Canada has its own money.
I scrolled down on the webpage and revealed the answer: the bill had originated in Florida. ("I knew it!" half the class exclaimed.) I named the town, which I'd never heard of. But Ellen had: her brother lived there. She asked if there was any way to find out who started the bill on its travels. Well, yes, there was; I clicked on the profile button and found a first name, Bob.
It wasn't Ellen's brother. But that was all right. Bob had provided us with a nice map of the US, each state filled in with one of six colors. Now I had my second brainstorm. We'd done a little real-life estimating and measuring with scale; it was time for some real-world data analysis.
"What do you think this map shows?" I asked, turning the computer so the students could see. Temperatures, guessed one boy. Good thought, but no. How many people live in each state? asked a girl. Close, I said. Think about what website this is, Ellen suggested, and suddenly hands were flying up all over the meeting area. Bob, they realized, had marked dozens and dozens of bills and sent them into the wild. The colors showed how many of those bills had turned up in each state.
[Here is Bob's Hit Map, by the way:]Right on the money! (So to speak.) The only question now was which colors stood for the most bills and which for the fewest. To help, I had them identify a few key states on Bob's map, and then I gave them a little extra information. California, I explained, had the most people of any state. Texas, New York, and Florida were next. Wyoming took up a lot of space, but it had fewer people than any other state.
Working as a group, the class swiftly came up with a sensible schematic for the colors. Red, the color of Florida, California, and New York, would be the most. Bright green, it seemed apparent, would be next, judging from what the children knew of population and distance, and so on, down to lowly Wyoming, the only state that was colored gray.
The guesses were in. It was now time for the Great Unveiling. I had everyone's full attention: they were deeply invested in the outcome by now. And the results were entirely satisfactory. The class had four out of six colors right; the only error had been reversing the orderof the fourth- and fifth-most colors.
Not bad, not bad at all, I told them, and we moved on to the regularly scheduled lesson on division.
Monday, April 27, 2009
Ozzes and Libs
Back in the halcyon days of my youth, it was taken for granted that the US would very soon be shifting over to the metric system from the cumbersome "English" system of measurements then in use, featuring feet and inches, pints and quarts, and as Lucy Van Pelt of "Peanuts" fame put it, ozzes and libs. The forward-thinking teachers at my forward-thinking elementary school prepared us by using Cuisenaire rods to help us think in centimeters and decimeters (a white rod = 1 cm, an orange rod 1 decimeter). Forward-thinking radio stations began giving the temperature in degrees Celsius along with degrees Fahrenheit. (Though giving the Celsius BEFORE the Fahrenheit might have been more successful.)
Even baseball, rarely identified as a forward-thinking sport under any circumstances, got into the act. No, they didn't redefine the distance between the bases as 27.43 meters, or go to ten-out innings and ten-inning games (and a good thing too, ballgames being slow enough as they are), but the forward-thinking Cincinnati Reds posted the distance to the outfield fences in Riverfront Stadium in meters as well as feet, and it seemed only a matter of time before other teams did the same. Yes indeedy, the metric system was on the move.
Well, the metric system may have been on the move, but like Godot and the Robert E. Lee it never quite arrived. True, it's made a few inroads. You can buy 2-liter pop bottles in stores all across the country, for example, and metric is spoken among all scientists--even those from the US. Still, very few Americans think in metric, and the reality is that metric measurements are not a part of very many people's ordinary lives. Like it or not, we still measure the distance to work in miles and the capacity of our gas tanks in gallons. If we hear that the temperature is 28 degrees, we dress our children in coats, not shorts and sandals. When it comes to snow, we know that ten inches is a lot; we're not sure what to make of "254 mm". We buy bologna by the pound and extension cords by the foot. In the race for American hearts and minds, the metric system is behind by, oh, 72.5 kilometers or so.
I won't debate whether this is good or bad (well, I won't debate it today, at least). It does present a bit of a problem for math teachers, however. In Germany or South Korea or Chad or practically anywhere else on the globe, children learn metric measurements; it's simple as that. In the US, we have to teach two systems. We have to teach customary measurements, because that's how Americans measure, and it's how Americans think. We have to teach metric measurements, too, though, because they will be needed for science, because they're in use elsewhere, and because--hey, you never know--we might actually convert to metric someday. So teaching measurements is a trickier business here than elsewhere.
Elementary schools typically deal with this problem by introducing the familiar "English" units first. Then it's time for a brief glimpse at the corresponding metric measurements. Immediately after investigating feet and inches, say, children then spend a short(er) period getting to know meters and centimeters. Then it's on to ozzes and libs, followed by grams and kilograms. And so on. Science instruction helps extend metric understanding, but the bulk of math instruction focuses on customary units. Combined with the use of the English system in everyday life, kids usually come away with a pretty good sense of how long a foot is or what it's like to be outside on a 70-degree day. They don't, however, get the same experience with metric measurements.
That's about how we do it at PDS, too: customary units first, metric in science and as a follow-up. Sometimes I have qualms about this approach. The rest of the world uses metric, after all. Besides, while it's not perfect, the metric system does make logical sense; it's certainly easier to convert centimeters to meters than to convert inches to feet. And maybe my forward-thinking teachers were right, if a bit off in their estimation of time, and the children of today will be using metric units for practically everything when they're adults. Perhaps, I think now and then, we should put less emphasis on miles and more on milliliters.
But the reality is that we already are pressed for time. There's a ton (okay, okay, 909 kg) of stuff to cover in the curriculum, with measurement being only one of many topics worth pursuing. Besides, as long as the metric system isn't in widespread use here in the US, instruction in metric units isn't going to be terribly meaningful to children. There are good reasons for focusing on the units that children hear and see in everyday life. ("See that bird? About 50 meters away?" "Huh?") And so, for now at least, your children will spend a good chunk of their measuring time at school looking at pints and quarts, inches and yards, degrees Fahrenheit, and of course, our old friends ozzes and libs.
Even baseball, rarely identified as a forward-thinking sport under any circumstances, got into the act. No, they didn't redefine the distance between the bases as 27.43 meters, or go to ten-out innings and ten-inning games (and a good thing too, ballgames being slow enough as they are), but the forward-thinking Cincinnati Reds posted the distance to the outfield fences in Riverfront Stadium in meters as well as feet, and it seemed only a matter of time before other teams did the same. Yes indeedy, the metric system was on the move.
Well, the metric system may have been on the move, but like Godot and the Robert E. Lee it never quite arrived. True, it's made a few inroads. You can buy 2-liter pop bottles in stores all across the country, for example, and metric is spoken among all scientists--even those from the US. Still, very few Americans think in metric, and the reality is that metric measurements are not a part of very many people's ordinary lives. Like it or not, we still measure the distance to work in miles and the capacity of our gas tanks in gallons. If we hear that the temperature is 28 degrees, we dress our children in coats, not shorts and sandals. When it comes to snow, we know that ten inches is a lot; we're not sure what to make of "254 mm". We buy bologna by the pound and extension cords by the foot. In the race for American hearts and minds, the metric system is behind by, oh, 72.5 kilometers or so.
I won't debate whether this is good or bad (well, I won't debate it today, at least). It does present a bit of a problem for math teachers, however. In Germany or South Korea or Chad or practically anywhere else on the globe, children learn metric measurements; it's simple as that. In the US, we have to teach two systems. We have to teach customary measurements, because that's how Americans measure, and it's how Americans think. We have to teach metric measurements, too, though, because they will be needed for science, because they're in use elsewhere, and because--hey, you never know--we might actually convert to metric someday. So teaching measurements is a trickier business here than elsewhere.
Elementary schools typically deal with this problem by introducing the familiar "English" units first. Then it's time for a brief glimpse at the corresponding metric measurements. Immediately after investigating feet and inches, say, children then spend a short(er) period getting to know meters and centimeters. Then it's on to ozzes and libs, followed by grams and kilograms. And so on. Science instruction helps extend metric understanding, but the bulk of math instruction focuses on customary units. Combined with the use of the English system in everyday life, kids usually come away with a pretty good sense of how long a foot is or what it's like to be outside on a 70-degree day. They don't, however, get the same experience with metric measurements.
That's about how we do it at PDS, too: customary units first, metric in science and as a follow-up. Sometimes I have qualms about this approach. The rest of the world uses metric, after all. Besides, while it's not perfect, the metric system does make logical sense; it's certainly easier to convert centimeters to meters than to convert inches to feet. And maybe my forward-thinking teachers were right, if a bit off in their estimation of time, and the children of today will be using metric units for practically everything when they're adults. Perhaps, I think now and then, we should put less emphasis on miles and more on milliliters.
But the reality is that we already are pressed for time. There's a ton (okay, okay, 909 kg) of stuff to cover in the curriculum, with measurement being only one of many topics worth pursuing. Besides, as long as the metric system isn't in widespread use here in the US, instruction in metric units isn't going to be terribly meaningful to children. There are good reasons for focusing on the units that children hear and see in everyday life. ("See that bird? About 50 meters away?" "Huh?") And so, for now at least, your children will spend a good chunk of their measuring time at school looking at pints and quarts, inches and yards, degrees Fahrenheit, and of course, our old friends ozzes and libs.
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