Thursday, April 30, 2009

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...

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.

Tuesday, April 21, 2009

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!

Monday, April 20, 2009

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!

Sunday, April 19, 2009

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.

Wednesday, April 15, 2009

Ways to 100

Formal multiplication instruction, at PDS as elsewhere, is generally the province of third, fourth, and fifth grades. But informally, multiplying comes up considerably earlier than that.

Our kindergarteners recently sorted collections of objects into groups of two, three, or more to see how many groups they had and how many were left over--multiplicative thinking at work. Counting quarters, dimes, or nickels uses simple multiplication concepts. So does telling time on an analog clock.

For that matter, any time children read, write, or model two- or three-digit numbers, they're using basic ideas of multiplication. Our decimal system, after all, is built on groups--groups of ones, tens, hundreds, and on and on.

The pictures below show groupings of 100 objects created by first and second graders. You can see the connection to multiplication: ten groups of ten, two groups of 50, five groups of 20, even 25 groups of four (though one of the dice here appears to have fallen off). Work like this can help children considerably when it's time for a formal introduction of the topic.

You may want to enlarge this last one to see what numbers are on the dice...

Tuesday, April 14, 2009

Games, Games, Games

Children at PDS play a lot of games in math class. If you're the parent of a PDS child, you may have heard your child talk about Bears in a Cave, Addition Bingo, Tens Go Fish, Negative One and Out, Digit Place, Uncover, Cross Out Singles, and many more. (You may also have heard them talk vaguely about "the adding game" or "the game with the pattern blocks where you roll the dice--you know, that one." We teachers are not always as consistent with the names of games as we should be.)

Sometimes I'm asked why we have kids play so many games. The questioners, generally speaking, like the idea of games--but they just aren't sure. They wonder whether--and how--the games help develop mathematical skills and mathematical thinking. They worry that games might take away from "real" work, which mostly means computational work with paper and pencil. And while parents are usually pleased that their children have fun playing these games, they often don't have fond memories of math from their own elementary school days. That makes sense. Throughout American history we have looked at school as a nose-to-the-grindstone institution with a heaping helping of drill and perhaps even drudgery. Traditionally, school has been a place where fun goes to die. We are, as a result, naturally a bit suspicious when children seem to be enjoying themselves. It isn't supposed to be that way.

So, why do we play games?

First, precisely because they are fun. While it's certainly true that some children enjoy filling in worksheets, most don't--or enjoy it only in small doses. There's a place for worksheets, of course, but as a rule children are much more motivated to play games. And a motivated student is generally a student who is more likely to learn.

Second, math games are almost always focused on developing a particular math skill. Negative One and Out, for instance, involves rolling dice to form two-digit numbers, which are then progressively subtracted from a starting three-digit number; the object is to get as close to 0 as possible without passing it. This game provides plenty of practice in subtracting, especially in subtracting with regrouping. The game 3-Digit requires children to compare three-digit numbers. Forceout and other geometry games offer practice in visual thinking. Double Compare gives young children experience in adding small numbers. Cover Up develops children's understanding of fractions. As long as games are reasonably fast-paced, children get essentially the same practice by playing them as they would if they did a couple of worksheets--and, as mentioned, the games are typically more compelling.

Third, because games are an excellent way to bridge the gap between concrete and abstract reasoning. First and second grade children, for example, often play a game we call Subtraction Nim. In this version of a (very) old game, pairs of children place 15 counters on the table. They take turns removing 1, 2, 3, or 4 counters (their choice) from the table and recording the subtraction sentence (such as 15 - 2 = 13). The winner is the player who removes the last counter. After children play a few rounds with the counters, we'll have them put the counters away and try it with the numbers alone. In this way, the game helps move children from the concrete to the more strictly numerical.

Fourth, because games involve strategic mathematical thinking. Our fourth graders often play a multiplication game known as Midas Dice. In its most basic form (there are more complex variations too), they roll a die three times and fill the results one at a time into an empty multiplication grid, resulting in a two-digit number multiplied by a one-digit number. The winner is the player who forms the greatest product--or the one with the least product--or the one who's able to predict whether he or she has the greatest or the least...or whatever the teacher decides.

Midas Dice obviously provides practice in multidigit multiplication, just as a worksheet of multiplication examples would do. But Midas Dice adds a twist. Say you roll a 5 on your first turn. Where should you put it to improve your chances of getting the greatest product? Most children realize quickly that a 5 will probably be wasted as the ones digit in the two-digit number. But is it better to have a relatively large number in the tens place of that number--or as the standalone one-digit number? And what if you get a 6 on your next roll? As children play the game, they find that it's very much worthwhile to determine which is greater, 43 x 5 or 53 x 4, and to apply what they learned to the next series of rolls; similarly, they find their chances of winning improve as they think through questions of what is and what is not likely to happen. It's harder to develop this kind of thinking through worksheets alone.

Of course, games aren't perfect. Though we emphasize (and usually get) good sportsmanship, sometimes feelings do get hurt when children become overly competitive, and arguments do break out over whose turn it is or whether someone cheated. Dice fall on the floor, fraction bars get knocked askew, children can become silly. Occasionally players don't try very hard, or cede decision-making to their partners, and even the most interesting game begins to pale after a while. Accordingly, we mix up games with pencil-and-paper practice and other activities as well.

Still, games are very much at the heart of what we do in math. They provide an enjoyable way for students to practice math concepts and skills; they offer a built-in way to challenge players to think more deeply about the topics we're teaching; they help with the transition between concrete thinking and more abstract reasoning. We think of games as being about winning AND losing...but in my book at least, using games is a win for everyone.

Photo credits to Gretchen Lytle.

Friday, April 10, 2009

Nickel-and-Diming the Math Guy

The subject of "a million dollars" came up in Lynn and Judy's first and second grade class yesterday, and so we took a few minutes to check children's understanding of money amounts (which will be a focus of work soon after the break). I began by asking whether they thought I might have a million dollars in coins in my pocket. When they said no, I pulled out my life savings of approximately 87 cents and asked them to reconsider, now that they could see the vast amount of metal resting in my palm. Again, they denied that I was anywhere near a million dollars. So, I had them count it together--quarters first, then dimes, then my only nickel, and finally the pennies. "You don't even have one dollar," they informed me. "See, we were right."

We then repeated the process with Judy's coins. Though she had almost twice as much money as I did, the children agreed that Judy, too, was quite some distance from a million dollars.

It was at this point that a boy in the class raised his hand. "If a dime is worth ten cents and a nickel is only worth five cents," he said, "why is a nickel bigger than a dime?"

Amazingly enough, though I always take pains to point out the size difference, I have NEVER been asked this question before. Nor, I discovered, did I know the answer. "I don't know," I admitted. "I think we're going to have to look this one up. After vacation. Remind me, please!"

And that should have been that until after vacation. But the question was burning a hole in my brain. So I looked it up. And if you're curious, you too can find the answer at http://www.infoplease.com/askeds/nickels-bigger-dimes.html.

Now how to put it into language that children will understand...

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...

Monday, April 6, 2009

Out of the Mouths of Babes...Um, Kindergarteners

A Play in One Act.

The scene: Robbie's kindergarten classroom.

The time: The present. Monday morning, to be exact.

The Cast of Characters: The Teacher; Child A; Child B.

The background: Children were working on a spatial reasoning assignment: cover a given space with exactly five pattern blocks--no more, no less. (The picture below shows a couple of first graders working with pattern blocks. Besides the hexagons pictured, there are 5 other pattern block shapes.)















[The Curtain Opens]

The Teacher (looking over Child A's work): Nice job! I see you did it with two trapezoids and three triangles.
Child A (pleased): Yeah. I know 2 + 3 makes 5, so it has to be 5 altogether.
The Teacher: Do you think there's another way to do it, or do you think this is the only possible way to cover the shape with 5 pattern blocks?
Child A (hesitantly): I think there's probably another way...
Child B (across the table, overhearing): There IS another way! There's ALWAYS another way!
[The Curtain Falls, to Thunderous Applause]

In truth, there isn't always another way, and some "other ways" are inefficient or unnecessarily complicated. But quite often there are multiple approaches, and this is good to keep in mind. Math education has suffered from the widespread idea that there is one path to enlightenment, scuse me, the right answer, and that this path is mighty narrow. It's nice to see a five-year-old who already has formed a dissenting opinion. Here's hoping she keeps this perspective as she moves on through her education.

Sunday, April 5, 2009

Poetry Contest Update

The poetry contest is a wrap. The Correct Box has been opened and the poems are being given the once-over by the judges, who are enjoying them immensely.

Here are the full stats.

Number of students who entered the contest --- 72
Number of poems submitted --- 78
Number of entries submitted WITH THE AUTHOR'S NAME ATTACHED --- 78*

Number of entries placed in the Correct Box --- 78
Number of entries placed in the Incorrect Box --- 0
Number of entries emailed to contest judge Cheerful Charlie --- 1**

The most popular favorite number [the favoritest number?] --- 9 
The greatest of the favorite numbers --- 103
The range of the favorite numbers ---  203
The median of the favorite numbers --- 8
The harmonic mean of the favorite numbers --- oh, never mind

The most popular favorite number sentence --- 4 + 4 = 8

*This is truly remarkable.
**This entry was emailed AND placed in the Correct Box. Someone was being extra careful.

Results will be announced at Lower School Assembly on Thursday, April 9. Tune in to see who has won the coveted Pass-the-Pepper Award for creative mentions of food in a math poem, the prized Julius Caesar Award for use of Roman numerals in a math poem, and the much-sought-after Sideways Infinity Award for clever uses of the number 8 in a math poem, among others. Be there or be []. 

Thursday, April 2, 2009

Bad at Fractions

I just about always wear a collared shirt with buttons to school, so several kids noticed when I showed up in a T-shirt today. "I don't think I've ever seen you in a half-sleeve T-shirt," one fourth-grade girl commented. "I have," a classmate said proudly. "Really?" I asked. "Here at school?" "No," he said. "In a restaurant."

There was a reason for the shirt. The third and fourth graders are working on fractions, and the shirt's message is, well, fractional. It proclaims:

5 out of 4 people are bad at fractions.

I used the shirt's message as a very informal way of checking students' understanding of fractions and fractional thinking. My hope was that they'd lodge a complaint, and fortunately I was right.

"Your shirt's wrong," one student stated flatly after she read it. "It should be '4 out of 5 people are bad at fractions,' not 5 out of 4."

"Yeah," a classmate agreed. "It doesn't make sense this way. If there are only 4 people, you can't take 5."

"It can't be more than the whole," someone in another class pointed out. "It's 1 and one fourth, but that doesn't make sense when you're talking about people."

"The shirt is bad at fractions," somebody said. "It's a bad-fraction shirt. It's complaining about people being bad at fractions, but the person who made it is the one that's bad at fractions."

"They're trying to disguise the fact that they're bad at fractions," noted a fourth grader.

"So I guess I should take it back to the store and exchange it for a shirt that's mathematically correct," I said. "What do you think?"

A few nodded slowly, but the bulk of them shook their heads. "It's a joke," someone explained. "People will see the shirt in the store and say, 'Oh, that's wrong!' and then they'll buy the shirt to make other people confused."

That settled, we moved on to the rest of the lesson.

There's a lot more to fractions, obviously, than determining what's wrong with a T-shirt statement. Still, it's kind of fun to use something as mundane as a T-shirt to do a brief informal assessment--and nice to know that the kids could see the error, and even, perhaps, the irony.

Photo credit to Rhiannon P. in Jan's class.