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...

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...

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.