A Mathematical License Plate

This SUV was parked at my motel at a convention I attended in Salt Lake City last year. I expect it belonged to a fellow Math Teacher. It would be a major shock if it didn't.

Perhaps someday I shall have my own mathematical license plate. What about:

E 2 THE X

or

3 X 3 EQ 9

or

SQRT NEG1, for those days when I feel imaginary....

Other ideas? Send 'em along. I'm open to suggestions.

37 Cities, 32 States, and Lord Knows How Many Unnecessary Miles

Cheerful Charlie's Tour of the USA is at an end. Between early October and early June he visited, as the title says, 32 states plus the District of Columbia and a total of 37 cities--large ones like San Diego and Seattle, Minneapolis and Houston; smaller ones like Cedar Rapids, Iowa and Springfield, Massachusetts; and small ones indeed such as Wall, South Dakota and Virginia City, Nevada. The students got used to enormous cross-country jumps and routes without rhyme or reason--just excess mileage and wasted gas. Still, they dutifully marked in the origin of each postcard he sent, used concepts of scale and ratio to estimate the distance from one city to the next, and made helpful suggestions about ways to improve his efficiency.

We are still calculating the total mileage. But it ain't gonna be pretty.

Here is the Official T-Shirt of Cheerful's travels. Please note that the list of cities should be quite accurate, as Cheerful sent me his list for proofreading (and boy oh boy did it need it). He did NOT tell me, however, that he was going to include the three lines at the top, so I had no opportunity to do the proofing. The mistakes--OF COURSE--are his and his alone.



You can click on the image for a closer look...

Fifteen Tires Were on Top of a Hill...

Robbie recently had her kindergarteners write story problems. They thought of a situation that involved plus ("joining") or minus ("taking away"), then wrote the story, wrote the number sentence for the problem, and finally made a box with clay figures and other decorations showing the problem. Today, they demonstrated their work for the class.

There were 15 tires on top of a hill...



Then five of them rolled down the hill....



How many were left at the top of the hill?

Hope you got 15 - 5 = 10!

Sally the snake sees 5 dragonflies. 2 fly away. How many are left?



Did everybody get 3? If not, ask your nearest kindergartener for help.

Other problems involved bats or butterflies flying off into the great big world, horses venturing out from the safety of barns, and wolves scaring away some (but not all) of a group of farm animals. I'm no psychologist, but it's pretty easy to tell that kindergarten is drawing to a close!

How to Annoy a First Grader

I'm sure there are other ways too, but one really good way is to ask children to make an estimate.

First, present a "how many" question where the answer's clearly more than 10 or 15 or so: how many cubes in a bag, how many times they can hop in one minute, how many pages in a book, that kind of thing.

Then, ask them to estimate the total, but insist that they give you a "round" number--that is, a multiple of ten (10, 20, 30...).

From a math perspective, asking for a round number makes plenty of sense. Part of the purpose of an estimate is to use numbers that are easy to work with. "If this bag has about 20 cubes, and this one has about 60 cubes, about how many are there in both bags together?" is easier to deal with than "If this bag has about 19 cubes, and the other one has abut 63 cubes..."

But from a kid's-eye perspective, it's frustrating (or "fruster-rating," as some children say) to have to give a round number. That's because children of this age tend to view the purpose of estimation as "guessing the right answer," not simply coming up with a number you can use when you don't need, or can't get, an exact answer. By limiting their choices to multiples of ten, I make it very difficult to choose the correct total.

And they hate that. Recently I insisted that kids give me a round number for an estimate. "How many say it's about 10?" I said. "About 20? About 30? Who says about 40? Raise your hand..." Several of the children refused to vote. (Insurrection!) And when the true total was revealed to be 42, one child said to me reproachfully "No fair! You didn't let us pick that one!"

So enforcing a round number estimate is one good way to annoy a first grader. Here's another way, related to the first. Today we were working on probability. Partners were given an envelope with five cards. They recorded the number of red cards and the number of black cards, and then made estimates of how many of each color they would get if they pulled a card from the envelope 25 times (replacing the card after pulling it, of course). Next, they tried it out and recorded the results. Finally, they needed to decide if their initial estimate was "close" or "not very close."

One pair predicted 22 blacks and 3 reds. Not a bad prediction, given that they had 4 black cards and just 1 red one in their envelope. These children were not just interested in the results; they were invested. "Come on, BLACK!" they'd say, pulling out a card and discovering that it was...the two of spades. (Fist-pumping ensued.) Then, after a while, one of them commented "We need another couple of reds," and lo and behold, whaddaya know, the next card out of the envelope was the five of hearts! (More fist pumps.) And amazingly enough, after 25 pulls they had--wait for it--22 blacks and 3 reds. An astonishing coincidence, to be sure.

"The page just says 'close' or 'not very close,'" they complained to me after they were finished. "Where's the one for 'we got it exactly right'?"

"Oh, there isn't one," I said. "You can mark 'close.' The point of a prediction like this is to be close, that's all. That's what we care about."

"Yeah," they said, "but we got it exactly right."

"So you did," I agreed, "but when you make an estimate or a prediction you are just trying to get near the real total. Your estimate was a good one. But it would have been just as good if you had predicted 21 blacks and 4 reds. Or even 20 blacks and 5 reds. Just circle 'close.'"

Fist-pumping was now over. The two exchanged unhappy glances, returned to their seats, and circled 'close.' Against their wills, of course.

Oh well-they'll get there eventually. I hope! In the meantime, feel free to annoy your own personal first grader all you like with these methods...

Probability and Percentages

Probability reared its random head in the 3-4s today. We investigated vocabulary such as impossible, unlikely, equally likely, likely, and certain, in addition, of course, to random, defined by one third grader as "not moving your hand around and around and around trying to find exactly the right one." We also introduced various ways of using numbers to write probabilities. If there is just one red card in a group of 5, then the chances of getting a red card (at random, of course), are "1 out of 5," or "1 in 5," or "1/5."

Or "20%." Percentages can be tricky and often require some serious numbercrunching to carry out. At the same time, they can be extremely useful in comparing two probabilities (it's hard to tell by looking whether 3 out of 7 is better or worse than 12 out of 29) and in getting a rough idea of how likely an event actually is (a percentage is easier to interpret than a fraction like 57/243). So we did some fairly straightforward percentages, using what students already know about fractions and division. If the probability of drawing a red card is 1/5, that's also 20%, because 20 is one fifth of 100. And if there were two red cards out of 5, the probability would be 40%--double the previous one. We also looked at more complicated situations such as 1/7, dividing 100 by 7 to get an approximate equivalent of 14%. Not good odds, the classes agreed.

Later, on their own, they found the probabilities of various events, expressing them in both fraction form and as a percentage. Conversion was easy enough when the denominator of the fraction was 10; most students recognized right away that 7 out of 10, say, was 70%. Other denominators were a bit more complicated. Still, the students persevered, and in the end 110% of them thoroughly understood percentages...wait...

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!

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.