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...

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.

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...

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...

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.

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 []. 

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.