Birthday

"Know what?" crowed the little boy. "It's my birthday!"

"Congratulations!" I said. "How old are you now? Twelve?"

"Noooo!" he chortled. "I'm not twelve. I'm THIS much!" He held up four pudgy little four-year-old fingers.

"Oh, you're FOUR!" I said.

"Yeah!" he agreed.

"Congratulations," I said again. "Four is a very big number."

He went off to explain the situation to a few other adults, including some, like me, whom he knew reasonably well, and some whom he didn't know at all. "Know how old I am?" I heard again and again. "THIS much!" And out would go the four pudgy four-year-old fingers while the people--those who knew him and those who did not--reacted with appropriate surprise and astonishment.

When he'd finished working the room, he began all over again. "Guess what?" he said to me. "It's my birthday!"

"He's told me that three times already," remarked the teenager who was standing next to me.

"Well, it's very important news," I reminded her, and bent down, the better to talk to the birthday boy. "Congratulations," I said again.

"Do you know how old I am?" he demanded, and out went the four fingers: "I'm THIS many!"

"It certainly is a lot," I agreed.

"And do you know how old I was yesterday?" he asked.

An additional wrinkle. Fortunately, it sounded like a subtraction problem, and I'm pretty good at subtracting. I pondered. "You were three," I guessed.

He looked at me. He looked at his fingers. He looked at me again. A look of utter astonishment began to creep across his face.

"You're right!" he said. "But...how did you KNOW????"

SummerMath, Part 3: The Bikepath, the Ballpark, and Beyond

My family went to the ballgame the other day, attracted by among other things a "Henry Hudson Bobblehead" giveaway (see picture). My son is quite eager to show off his Hudson Valley roots with this, um, iconic image when he heads west for his next college semester, and as for the rest of us, well, how could we pass up such a quality and historic freebie??

(Ol' Henry)

Anyhow, the game put me in my mind of Sports and Math. I spent most of my childhood free time engaged in one of five activities:

1) eating
2) reading the classics, mainly the Hardy Boys books
3) writing short stories with meandering plots and lots of unnecessary characters
4) playing board games and card games (see SummerMath Parts 1 and 2)
and
5) playing, watching, reading about, or thinking about baseball.

Baseball and math are closely linked, and in fact I learned quite a lot about math from my interest in baseball. My 1972 Sports Illustrated baseball board game (see 4 and 5 above) helped inform me about probability. I can remember the power I felt when I realized that I could use what (little) I knew about ratios to compare teams' won-lost records in my head--was it better to be 38-37 or 37-36, and how could I prove it? And I developed some facility with division, if not comprehension of WHY it worked, by virtue of calculating my batting average every day back when I was ten or so. (My batting average was very good. I counted it as a hit, of course, if someone muffed a ball I'd put in play. Or if the umpire mistakenly called me out when I was CLEARLY safe at first--don't laugh, it happened all the time. Or if I hit a line drive or a deep fly ball that somebody managed to corral, but which clearly SHOULD'VE been a hit--why should I be penalized just because my opponents had good hands? That was in addition to the occasional, you know, REAL hits I got. As I said, my batting average was very good.)

In any case, there are lots of ways to combine math with sports, for those of you whose children like to watch baseball, play soccer, ride bikes, or mess around with balls and such in the back yard after dinner. Here are some ideas of questions you can ask and projects you can do:

*Counting and estimating. "I wonder how many pitches the pitcher will throw this inning. Do you think it'll be more than 15 or less than 15?" "Take ten shots on goal from right here. Let's see how many go in...Now let's move you back a few feet. How many do you think will go into the net now?" "Good job! We just did 6 throws back and forth in a row without dropping a single one. Think we can beat that record? Let's keep track."

*Adding and subtracting, multiplying and dividing. "The scoreboard says the Renegades are winning 7 to 2. How many runs are they winning by?" "That's your third basket. Each basket is worth 2 points. How many points do you have so far?" I'll just add that I have taught many primary graders who could count rapidly by twos, fives, and tens when they came to my class, and a few who could rattle off threes, fours, and nines; but the only one I ever had who could count fluently by sevens was the one who lived and died with the NY Giants. Sevens...football...hmm!

*Measuring. "You sure hit that one a long way! I wonder how far it went.." You can measure with "nonstandard units," such as steps or rake lengths, which tends to be a little more meaningful for younger children, or with standard units--feet, yards, meters. "14 rake lengths--that's a lot. Whoa, that one went even further! Would you say 15, or 20, or even more?" How long does it take to run around the yard or the perimeter of the park? Time your child; let your child time you. Write it down. Try it again another day. Look at the map of one of the local bike paths. "It's 10 and a half miles long! How far do you think we'll get before I'll be ready to turn around?...I see another mileage marker up ahead--4 miles and still going!"

*Graphing. These take a little more time and energy, but they're great for kids who really love sports, especially team spectator sports. Work with your child to make a bar graph showing his or her favorite team's wins and losses.

(A sample bar graph)

Update it daily; use the internet or the newspaper to get the scores.






Or, make a line graph showing the number of runs your team scores on a daily basis. Look how the line moves around. What has the trend been? More runs over time, or fewer or about the same? How could you show the number of runs they gave up each day on the same graph?

(A sample line graph)

Can you make a graph showing how many times you go swimming/bicycling/hiking this month? We'll write the words down here; put up a blue sticker for the water whenever we swim, a red sticker for the color of your bike to show each time you go for a ride, a green sticker for the color of the leaves to stand for a hike.


(A sample picture graph)

Which has the most so far? The fewest? How many more bike rides have you taken than hikes?

Of course, I don't mean to reduce sports and physical activity to numbers. Nor is the point for kids to quantify their outside play. Be sure that timing and measuring are just for fun, a nice way of bringing a little math into children's lives, not an opportunity for frustration and embarrassment because they can't seem to beat their old record; be sure that a graph is a cute little add-on, not another chore that has to be done or the sole reason for taking a bike ride or going out for a hike. Sports are their own reward. Though, now that I think about, the ability to hit .658 (and calculate it properly!) might be its own reward, too...

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!

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...