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.

Shrubbery Ice Cream?

This doesn't actually have much to do with math, but it's my blog and so I'm including it.

Bill and Rachel's 1-2 will visit a diner on Market Street tomorrow as part of their social studies curriculum. Today, the children pre-ordered items from the diner's menu. The total cost had to be $3.25 or less (that's where the math comes in). Most kids opted for milk shakes, hot chocolate, or scoops of ice cream, though one independent thinker chose a bagel with cream cheese as part of his meal.

I'm generally very good at deciphering invntd spelilng, but a child had written a list of possible ice cream flavors, and I was stumped by one of them. "Chocklit" I understood just fine, and "vanila" was obvious enough, and I also spied the eminently readable "coffy," along with a few others. But what, I wondered, was "shrubary"?

"Shrubbery," I soon decided, for what else could it be? It put me in mind of Monty Python's star-crossed search for the Holy Grail. "You must bring us some shrubbery ice cream!" cry out the knights who say Ni! "A premium brand. With hot fudge topping." But what shrubbery ice cream might look like, let alone taste like, was beyond my imagination. So I asked Bill.

"That one? Oh, that one says strawberry," he told me.

"Strawberry is the most flavorish kind of ice cream," said one of the children, overhearing. 

Shrubary...Strawberry! Of course. Feeling like a batter who'd just shruck out, I wished them all a fine time on Market Shreet and slowly shrolled away.

A Risk-Averse Generation

My good friend Cheerful Charlie had a summer job opportunity, I told the third and fourth graders recently. He could choose four different payment plans, which could lead to different amounts of cash for his eight-week period of employment. Plans A and B would give him a fixed amount of money; Plans C and D involved some element of chance. Students were asked to study the plans, do some calculations, and write a letter to advise Cheerful of his best strategy.

Most of the children recognized that plans C and D might bring in a lot of money. With luck, Cheerful could make over $1700 on Plan C--and a whopping $2400 with Plan D. By comparison, Plan B, the better of the two "fixed" plans, would earn Cheerful just $1275.

But almost unanimously, the letters warned Cheerful away from C and D. In most cases, it was a gut feeling that having a guaranteed income was better than taking a chance. "Plans C and D are a bit too random," wrote one girl. "If you take C or D you're taking a risk," noted a boy. "Plan C is a gamble," explained a third grader, "because it's a different amount each time."

A few children went a bit further by determining the probabilities for each plan. "In Plan C you only have a 2/8 chance to get [the best possible result]," wrote one. A classmate calculated, correctly, that Cheerful's expected income for Plan C was just $650. Plan D, which involved a fair coin and the possibility of earning either $300 or $0 for the week, was not much better. "Tails is not luckier than heads," one student admonished Cheerful. Another cautioned him not to be seduced by the possible $300 weekly payouts. "You're thinking, go for Plan D," he wrote. "Don't! You could end up getting zero dollars!"

It'll be interesting to see if this risk aversion lasts. The popularity of casinos and lotteries demonstrates that many Americans are eager to Plan-C-and-D themselves to easy riches. As someone who thinks of state lotteries as a tax on the mathematically unaware, I'm pleased that our students were so clear about the drawbacks to this approach. Of course, all bets may be off when these guys are old enough to take a trip to Foxwoods or Atlantic City...

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.

Fractions + Transformations = ?

This is a recipe.

Start with a small 4x4 square.

Sketch a continuous series of line segments (no curves) to divide the square neatly in half.
(Be interesting, please: no fair drawing a straight vertical or horizontal line or a simple diagonal.)

Prove that the two sections do indeed take up the same area.


Check to see if the figure has rotational symmetry. (That is, if it looks exactly the same when it's rotated any distance less than 360 degrees.)

Color the two sections contrasting colors.

Repeat the process 3 times. You may transform the original design by a) rotating (turning) the design 90, 180, or 270 degrees, or b) reflecting (flipping) the design as if it were appearing in a mirror. You may also keep the design oriented exactly the same as the original.


Arrange the four squares into a larger square.

Repeat this larger square four times. Place these together to create a sixteen-square unit.

Write a description of what you did.


The pictures show the results.

And the answer to the equation? Well, we could say "Fractions + Transformations = An Example of Applied Mathematics." Or, we could also say simply "Fractions + Transformations = Art." Your choice!

AWARDS!!!

The moment you've all been waiting for...the recap of the ***MATH POETRY CONTEST***.

We had winners in the following coveted categories:

The "I Got Plenty of Nothin'" Award for best use of the number 0 in a math poem.

The "I Can Count to Two! Can You Count to Two, Too?" Award for best use of the number 2 in a math poem.

The "Four Color Trapezoid with Four Wheel Drive" Award for best use of the number 4 in a math poem.

The "Devon and Kevin Go to Heaven" Award for best use of the number 7 in a math poem.

The "Sideways Infinity" Award for best use of the number 8 in a math poem.

The "Kind of Sort of Upside Down 6" Award for best use of the number 9 in a math poem.

The "Head, Shoulders, Knees, and Toes, Minus the Head, Shoulders, and Knees" Award for best use of the number 10 in a math poem.

The "Through the Looking Glass" Award for best use of negative numbers in a math poem.

The "JVLIVS CAESARIS" Award for best use of Roman numerals in a math poem.

The "What Comes After a Gazillion and One?" Award for best use of Large Numbers in a math poem.

The "Hey Jude" Award for best use of repetition best use of repetition of repetition in a math poem math poem.

The "Honey, Do You Love Me?" Award for best mention of bees or beehives in a math poem.

The "A plus, 100%, Red-Letter" Award for best mention of the Math Guy's Correct Box in a math poem.

The "Sixteen Going on Seventeen" Award for best use of numbers 13 through 19 in a math poem.

The "Pass the Pepper" Award for best mention of food in a math poem.

The "Boxcars and Snake Eyes" Award for best use of doubles facts in a math poem.

The "Elementary, My Dear Watson" Award for providing the reader with clues to the poet's favorite number.

The "It's-Not-Easy-Being-Green" Award for best references to nature in a math poem.

The "Age Before Beauty" Award for best mention of ages in a math poem.

The "Count von Count" Award for best use of the numbers 1, 2, and 3 IN THAT ORDER in a math poem.

The "Help Help I'm Being Invaded by Rabbits" Award for best use of multiplication in a math poem.

There were multiple winners of some of these awards. Winners received hot-off-the-presses suitable-for-framing certificates of merit. Also, pencils. Congrats to all who participated!

Seventy-four

Sometimes children know more than we give them credit for knowing. Sometimes, they don't know as much as we think they do. And sometimes, we're not even on the same planet.

I started my teaching career in a kindergarten classroom about a million years ago [ED: Check this figure]. That fall, some of the kids became very interested in bean estimates--that is, putting some dried kidney beans in a small glass container and then trying to guess how many there were. At first, we stuck with relatively small numbers--up to 20 or so. But before long, the children wanted to try their luck with larger numbers.

Well, why not? I remember thinking. I knew that most teachers would say it was pointless to go much above twenty with beginning-of-the-year kindergarteners. Conventional wisdom held (and still holds) that it's difficult for children that young to comprehend numbers such as 500, 200, or even 50. But these kids were interested. And maybe they were smarter than your average five-year-old where numbers were concerned. Or maybe the conventional wisdom was wrong.

So one day I let a child pile a few handfuls of beans into the container and then get estimates. [Actually, in this context, "guesses" is a better term--most children gave the first large number that popped into their heads.] To check the guesses, we poured the beans onto the floor at meeting time, and I modeled separating them into groups of ten, with ones left over. They seemed to understand this just fine. "Let's count by tens," I said, pointing to the piles in turn, and they chorused along with me, ten, twenty, thirty, all the way up to seventy. "Now we have to switch and go by ones," I instructed them, and touched the ones in turn, counting aloud: seventy-one, seventy-two, seventy-three, seventy-four.

"There," I said, sitting back. "Seven tens is seventy, and four more ones makes seventy-four. That's a lot of beans!" The children nodded soberly. It was a lot of beans. "Seventy-four beans," I repeated. "We should write that number down so we don't forget. I wonder if anybody knows how to write it. "

Several hands waved. What a capable class, I remember thinking. Understanding the decimal system so well at such a tender age! I chose the child who had filled the container, and she stepped up to the board and picked up the chalk. "Seventy-four, right?" she asked.

"Seventy-four," I confirmed.

So she wrote, and stepped away to admire her handiwork, and with a sinking heart I saw what she had written--

7D4.