David had his students look closely at some addition problems this morning. He set up 5-6 large sheets of paper on the tables in the room, each with a problem such as 16,732 + 5,895. Each of the problems came pre-solved, with every step of the solution carefully worked out and on display.
However, each solution used a different algorithm.* In each case, the students had to figure out how the solving strategy worked and respond to the strategy. Was it easy to understand? Was it clear what was going on? Had they tried it before? Would they consider trying it in the future? Would it work better with some kinds of numbers than others? They moved around the room, analyzing each algorithm and writing briefly about it as well.
*Of course, you remember what an algorithm is--it's an alligator that wins So You Think You Can Dance, right? Right? I mean, think about it, Al's got rhythm? Rhythm? Al? Al-go-rithm? -- Oh, all right. An algorithm is a series of steps which, if performed accurately and sequentially, will always lead to a correct answer.
The kids were very familiar with some of the algorithms on display, less so with others. The "traditional" algorithm was in there: you know, the one with carrying, which we actually call regrouping or trading because it's a more descriptive term thankyouverymuch. ("Seven plus five is twelve, carry the one" should actually be "Seven plus five is twelve, that's a ten and two more; the ten goes upstairs...") There were a couple of other models in which numbers were written in expanded form (that is, 2,368 becomes 2000 + 300 + 60 + 8); there was one which used a series of partial sums...all of them were useful in at least some circumstances. "Think about them," David instructed the students, "compare them, see which ones make sense to you."
A few of the responses, written and oral:
"This one's interesting. It uses expanded form, but then it puts the numbers back together in a very mathy way." (This student and I are going to write a song, perhaps: "A Very Mathy Way." Listen for it at the Grammys.)
*"I recommend this one if you're in calculator school. If not, try different ways."
*"This part over here--that's just like a sloppy copy when you're writing a story. You don't have to look at it too closely."
*"Easy and quick if you fully understand the base ten system." A nice companion for "An example of using the base ten system in an advanced way." Different algorithms, and (I think) different responders as well.
*"[The person who solved it this way] is a strong mathematician that made a hard question easy."
*"Our group thinks it's too messy. There's too many lines and numbers." (But do you understand what's going on? I inquired.) "Oh, we understand it. We just think it's messy."
*"EASY" (Upper case letters and lack of punctuation in the original)
*"This one doesn't make very much sense. But it does make a little bit of sense."
*"I am going to try it because it looks cool and fun."
Showing posts with label addition. Show all posts
Showing posts with label addition. Show all posts
Wednesday, September 28, 2011
Monday, February 15, 2010
Obi-Wan, Mini Hot Dogs, and the Ninety-Dollar Jar of Olives
Hungry? You’ve come to the right place. Some of our second graders have been spending a few of their math periods lately planning menus for a special dinner. If they have leftovers (and judging from the amount of food some of them are planning to buy they’ll have lots), I’m sure they’d love to have you drop by.
Each of the children was instructed first to draw up a guest list consisting of seven fictional characters they’d like to get to know better. Their lists skewed heavily toward fantasy fiction: Lord of the Rings, Harry Potter, the Star Wars crowd. Obi-Wan Kenobi, in particular, will be kept busy nearly every evening for a week with invitations to these foodfests.
Next, they sketched out seating arrangements. One child set up four tables of two people each. Another planned for two tables, each seating four. Someone else constructed an octagon. (Eight people? Yes--seven guests and a host.) Then they planned a menu. “Not just desserts,” I cautioned them when it seemed clear that this was the way the wind was blowing. “A main course, or even two, so people can have a choice. An appetizer, perhaps. Soup, salad. Fruits and vegetables. Drinks.” “Oh, okay,” they said, and proceeded to put together menus that were...well, idiosyncratic, or what else would you call it when red snapper and turkey are served side by side?
Now came the most mathematical part: shopping. We did this in the comfort of our own classroom, using grocery ads from newspaper inserts as well as the circulars that arrived in my mailbox over time. I gave the children record sheets as well. “Find the items you want to buy,” I told them. “Record what the item is and how much it costs to buy just one. Then decide how many of the item you need. What will the total cost be? Figure that out and record it here in this column; then round that to the nearest whole dollar to make estimating the grand total easier.”
The children fell eagerly to work. “Mini hot dogs, mini hot dogs,” one boy repeated over and over, scanning the ads in vain for his favorite entrée. “Pizza!” crowed a classmate, jabbing a pencil at a listing in the frozen foods section of one advertising supplement. “Pizza?” asked a classmate. “Hey, I need that too!”
The kids needed help deciphering some of the prices: supermarkets, we learned, don’t often write prices the way teachers say you should. Instead of $1.59, for instance, they often write a big 1 followed by a smaller 59 and no decimal point or dollar sign at all: “Looks like a hundred and fifty-nine dollars,” one child said disapprovingly. “5 for $2” was confusing enough, but “4/3” to mean “you can get four for three dollars” was enough to make strong second graders cry. (Well, not literally. They were good sports about it and chalked it up as yet another example of the weirdness of adults.)
Determining how many of each item they needed was again idiosyncratic. One child decided that each person would probably eat 5 cookies, no, make that 6, so he settled on 48 as the number to buy. Somebody else, less generous or perhaps expecting that everyone would be full when dessert came around, decided to purchase just one small apple pie for the table. Did you need one carton of orange juice, or were you better off with 10? Different kids answered similar questions in wildly different ways.
Determining the cost of n cartons of orange juice, when n was more than 1, was often tricky as well. Some of these hosts have a beginning knowledge of multiplication, which they used to good effect. “Each cookie costs ten cents,” mused our 48-cookie-buying friend from before, “so 48 of them would cost, um, $4.80.” One child bought two jugs of milk at $2.99 apiece, set up the costs in columns, and laboriously regrouped to arrive at the total, $5.98; another, trying to determine the cost of two $4.98 items, rounded up to $5 for each, added the estimates to get to $10, and finally subtracted the extra four cents for a total of $9.96. We shared strategies and tried out newer, more efficient ones.
What about items that appeared on menus but were nowhere to be found in the ads? I hear you ask. Surely there were some of those. There were indeed, the aforementioned red snapper and mini hot dogs were among them. “Not a problem,” I said. “You have an idea of what similar things cost. Estimate the price. Just put a circle around the price on your record sheet to indicate that it’s an estimate.” Some estimates were IMHO pretty good. Others were...well, one cent for salad dressing was perhaps overly optimistic, and over $100 for a few other items seemed a tad excessive, but hey, who am I to complain?
Once the estimates were done, it was time to find a grand total. Making an estimate (using the rounded-to-the-nearest-dollar figures on the shopping list) was a job for mental arithmetic. Finding the actual total was a job for a calculator. Children compared the answers to make sure their totals were reasonable, then looked again at the total bill.
At first each was flabbergasted by the high cost of groceries (especially given the $90 jar of olives one of them purchased--an estimate, natch), but after a moment flabbergast vanished to be replaced by pride. Yes, indeed, they seemed to be saying, a meal that costs over $500 must certainly be something special. Now if only they’d found more expensive broccoli and purchased twelve loaves of bread instead of just two--
So the menus are done, the shopping lists are ready to go, and all that remains is to check on the availability of the guests. Anyone know if Dumbledore is free the evening of the 23rd? The invitation says six pm sharp, but he can arrive at seven if he doesn’t mind missing the goldfish and gummiworms being served as appetizers...
Each of the children was instructed first to draw up a guest list consisting of seven fictional characters they’d like to get to know better. Their lists skewed heavily toward fantasy fiction: Lord of the Rings, Harry Potter, the Star Wars crowd. Obi-Wan Kenobi, in particular, will be kept busy nearly every evening for a week with invitations to these foodfests.
Next, they sketched out seating arrangements. One child set up four tables of two people each. Another planned for two tables, each seating four. Someone else constructed an octagon. (Eight people? Yes--seven guests and a host.) Then they planned a menu. “Not just desserts,” I cautioned them when it seemed clear that this was the way the wind was blowing. “A main course, or even two, so people can have a choice. An appetizer, perhaps. Soup, salad. Fruits and vegetables. Drinks.” “Oh, okay,” they said, and proceeded to put together menus that were...well, idiosyncratic, or what else would you call it when red snapper and turkey are served side by side?
Now came the most mathematical part: shopping. We did this in the comfort of our own classroom, using grocery ads from newspaper inserts as well as the circulars that arrived in my mailbox over time. I gave the children record sheets as well. “Find the items you want to buy,” I told them. “Record what the item is and how much it costs to buy just one. Then decide how many of the item you need. What will the total cost be? Figure that out and record it here in this column; then round that to the nearest whole dollar to make estimating the grand total easier.”
The children fell eagerly to work. “Mini hot dogs, mini hot dogs,” one boy repeated over and over, scanning the ads in vain for his favorite entrée. “Pizza!” crowed a classmate, jabbing a pencil at a listing in the frozen foods section of one advertising supplement. “Pizza?” asked a classmate. “Hey, I need that too!”
The kids needed help deciphering some of the prices: supermarkets, we learned, don’t often write prices the way teachers say you should. Instead of $1.59, for instance, they often write a big 1 followed by a smaller 59 and no decimal point or dollar sign at all: “Looks like a hundred and fifty-nine dollars,” one child said disapprovingly. “5 for $2” was confusing enough, but “4/3” to mean “you can get four for three dollars” was enough to make strong second graders cry. (Well, not literally. They were good sports about it and chalked it up as yet another example of the weirdness of adults.)
Determining how many of each item they needed was again idiosyncratic. One child decided that each person would probably eat 5 cookies, no, make that 6, so he settled on 48 as the number to buy. Somebody else, less generous or perhaps expecting that everyone would be full when dessert came around, decided to purchase just one small apple pie for the table. Did you need one carton of orange juice, or were you better off with 10? Different kids answered similar questions in wildly different ways.
Determining the cost of n cartons of orange juice, when n was more than 1, was often tricky as well. Some of these hosts have a beginning knowledge of multiplication, which they used to good effect. “Each cookie costs ten cents,” mused our 48-cookie-buying friend from before, “so 48 of them would cost, um, $4.80.” One child bought two jugs of milk at $2.99 apiece, set up the costs in columns, and laboriously regrouped to arrive at the total, $5.98; another, trying to determine the cost of two $4.98 items, rounded up to $5 for each, added the estimates to get to $10, and finally subtracted the extra four cents for a total of $9.96. We shared strategies and tried out newer, more efficient ones.
What about items that appeared on menus but were nowhere to be found in the ads? I hear you ask. Surely there were some of those. There were indeed, the aforementioned red snapper and mini hot dogs were among them. “Not a problem,” I said. “You have an idea of what similar things cost. Estimate the price. Just put a circle around the price on your record sheet to indicate that it’s an estimate.” Some estimates were IMHO pretty good. Others were...well, one cent for salad dressing was perhaps overly optimistic, and over $100 for a few other items seemed a tad excessive, but hey, who am I to complain?
Once the estimates were done, it was time to find a grand total. Making an estimate (using the rounded-to-the-nearest-dollar figures on the shopping list) was a job for mental arithmetic. Finding the actual total was a job for a calculator. Children compared the answers to make sure their totals were reasonable, then looked again at the total bill.
At first each was flabbergasted by the high cost of groceries (especially given the $90 jar of olives one of them purchased--an estimate, natch), but after a moment flabbergast vanished to be replaced by pride. Yes, indeed, they seemed to be saying, a meal that costs over $500 must certainly be something special. Now if only they’d found more expensive broccoli and purchased twelve loaves of bread instead of just two--
So the menus are done, the shopping lists are ready to go, and all that remains is to check on the availability of the guests. Anyone know if Dumbledore is free the evening of the 23rd? The invitation says six pm sharp, but he can arrive at seven if he doesn’t mind missing the goldfish and gummiworms being served as appetizers...
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!
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, April 7, 2009
From Lower School to College
I taught this afternoon. No surprise--teaching is what I do, after all. But today's audience wasn't the usual run of five- to ten-year-olds. Instead, they were college students.
This is the third year now that I've had the opportunity to work with the students in the math methods class at Vassar College (taught this year by Professor Chris Bjork in the Old Observatory, pictured below). This semester, I'm presenting two workshops to the students, and they'll be coming to visit at least once during a math class at school. It's a nice way to bridge the gap between theory and practice for the students--and a nice way to connect the PDS and Vassar communities.
Today's workshop was on addition and subtraction. We looked at how and when to introduce these concepts, discussed a little bit of developmental theory, and talked about why it's wise to model operations and algorithms with manipulatives and real-life situations before moving into the realm of the abstract. We played a couple of computation games as well (field tested, of course, on genuine PDS children). The students were a pleasure--they were focused and interested and asked some thoughtful questions.
I'll write more about this visit later, but for now I have two observations about how college students are different from children in elementary school.
1. College students are much more skilled than elementary students at discussing a question with a partner. "Talk to the person next to you about what the answer to this problem might be," I tell the children at school, and the response all too often is "It's seven! It's seven, seven, seven, seven, seven, seven, it's SEVEN." It can take multiple prompts before they remember to explan why they think it's seven.
College students, on the other hand, at least these college students, discuss the question thoughtfully, carefully, and respectfully. They take turns talking (!). They don't shout, and they don't repeat themselves. Score one for the college students.
2. Elementary students, on the other hand, are much more comfortable than college students at sharing the results of their discussions (assuming they've actually had 'em). "Raise your hand if you'd like to summarize what you and your partner talked about," I'll say, and hands typically shoot up all through the room. The same question to college students is met with tentative glances, furrowed brows, and, after a long pause, a hand or two creeping up slowly until it's about even with the student's ear. They get there in the end--but it's slow.
Now if we could just combine the best of both worlds...
This is the third year now that I've had the opportunity to work with the students in the math methods class at Vassar College (taught this year by Professor Chris Bjork in the Old Observatory, pictured below). This semester, I'm presenting two workshops to the students, and they'll be coming to visit at least once during a math class at school. It's a nice way to bridge the gap between theory and practice for the students--and a nice way to connect the PDS and Vassar communities.
Today's workshop was on addition and subtraction. We looked at how and when to introduce these concepts, discussed a little bit of developmental theory, and talked about why it's wise to model operations and algorithms with manipulatives and real-life situations before moving into the realm of the abstract. We played a couple of computation games as well (field tested, of course, on genuine PDS children). The students were a pleasure--they were focused and interested and asked some thoughtful questions.
I'll write more about this visit later, but for now I have two observations about how college students are different from children in elementary school.
1. College students are much more skilled than elementary students at discussing a question with a partner. "Talk to the person next to you about what the answer to this problem might be," I tell the children at school, and the response all too often is "It's seven! It's seven, seven, seven, seven, seven, seven, it's SEVEN." It can take multiple prompts before they remember to explan why they think it's seven.
College students, on the other hand, at least these college students, discuss the question thoughtfully, carefully, and respectfully. They take turns talking (!). They don't shout, and they don't repeat themselves. Score one for the college students.
2. Elementary students, on the other hand, are much more comfortable than college students at sharing the results of their discussions (assuming they've actually had 'em). "Raise your hand if you'd like to summarize what you and your partner talked about," I'll say, and hands typically shoot up all through the room. The same question to college students is met with tentative glances, furrowed brows, and, after a long pause, a hand or two creeping up slowly until it's about even with the student's ear. They get there in the end--but it's slow.
Now if we could just combine the best of both worlds...
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