In Which the Math Guy Is Reminded (Yet Again) of the Importance of Not Making Assumptions

The second graders were measuring. They'd cut out replicas of their feet (exact size, natch) and were busily determining how many of these footprints (feetprints?) it took to equal the length of a shelf, the width of the room, and other various and sundry distances. Then they were converting the number of feetprints (footprints?) to inches and recording it all on a chart.

I plunked myself down next to a child who was recording the number of feetsprint she had needed to cover the distance across a table. She'd written a 7, which sounded reasonable--seven second-grade-sized footsprint looked about right--but what was this next to it? A zero? Seventy? Surely she was putting 70 in the wrong place of the chart. Or she'd mismeasured. Or--
Wait a minute.

It wasn't just a zero. It was a bubble letter--you know, the puffy letters that kids love to make, especially when time is of the essence. The ones that slow kids' work pace down to a crawl. The ones that drive me faintly crazy. The ones that--

Hold on.

Now she was decorating the thing. Shading in part of the inside ring, drawing something unrecognizable in the middle. Decorating--during math time! Bubble letters--during math time! I mean, gee whillikers!

I opened my mouth to say something gentle, yet pointed. Okay, something not-so-gentle yet pointed. Something about saving the artistry for art and getting back to math, and by-the-way was 70 really a reasonable answer, and if you'd been paying closer attention to the math rather than to the art you'd know...But then I didn't. "Tell me about what you're drawing," I said instead, pointing. Just in case my assumption was wrong and there was method to her madness.

"Oh, that's a quarter," she explained, barely looking up.

"The coin?" I asked. "The thing that's worth twenty-five cents?" I peered closer. Okay, now that she'd mentioned it I could see that the bubble-letter zero did indeed resemble a quarter. Fine and dandy, but that didn't explain why she drawn a coin as part of this measurement project. I opened my mouth again...but instead of the pointed comment I'd intended, I found myself with a different response, again a response that didn't automatically assume that she'd messed up.

"Why a quarter?" I asked.

"Well," she said, "when I measured the table I found it was seven and a quarter of my footsprints." She tapped the seven on the chart, then the quarter beside it. "So I wrote seven, and then I drew a quarter. That's why."

And that's why I'm glad I asked!

at10tion!

Yup, it's the day we've all been waiting for--at least, the day us Math Guys have all been waiting for: Ten Ten Ten, or 10/10/10. (As many of you Eager Readers no doubt know, 10 holds a special place in every math guy's heart. 2 1/2, 8, 92, 753.6, even 3.14159...., they all have their points (some of them even have decimal points (sorry)), but none of them can hold their own next to Ten.) However you write it, it's as decimal a day as it gets (well, okay, 10/10/1010, a thousand years ago this afternoon, was maybe a skoonch better, and there's something quite appealingly, I don't know, clean, about 10/10/10 back in 10 CE, not that anyone knew it WAS 10 CE at the time).

I had a distant relative who I met back in 1978 or so. She showed me her passport and called my attention to her birthdate. I know, I know, ladies of a certain age are not supposed to reveal their ages, but she was so pleased with the day she was born that she couldn't resist. It was, of course, 10/10/10--1910, that is. I promised not to do the subtraction necessary to calculate her age at the time. Here's to you, Cousin Lily.

So have a happy Ten Ten Ten. Ten cheers for this day, and long may it wave.

Or We Could Just Call 'em Number Cube(s)

First grade, a lesson on estimation.

"I'm going to show you a cup with dice inside," I explained. "I won't show it for very long. So, you won't be able to count how many there actually are. Instead, I want you to decide how many it COULD be and how many it COULDN'T be. Got it?"*

*The purpose of this activity is to get kids to establish a zone of possible answers. Very often kids think of estimation as a sport in which the goal is to guess exactly the right answer. It isn't. This project asks kids to identify numbers that they think are reasonable. The zone can be quite large (when working independently later on, one partnership ruled out 1, 2, 3, 4, and 5, and decided that anywhere from 6 up to 40 was perfectly reasonable) or quite small (another pair, in direct contrast, established a zone that went all the way from 10 to 11). Either way, you get a sense of kids' ability to think about large quantities and a sense of the confidence (or overconfidence) they bring to the table when it comes to mathematical thinking.

"Okay," I said when everyone had taken a look. "Would you say the cup is full?"

NO, they chorused. Some said it was mostly empty, others about half full, but all agreed that it absolutely was not full or even close.

"Could you see all the dice at the same time?" NO, again.

I had a number line of sorts on the board. I touched the number 1. "Could there be just one object in the cup?" I asked. NO. "How about 2?" NO. "Three?" NO, NO, NO. "Okay," I challenged, because after all explaining your thinking is an important part of mathematics, "you sound awfully sure. How can you be so sure?"

A boy raised his hand. "I could see if it was three," he said, holding up three fingers. "I couldn't tell how many there were, so I knew it wasn't more than three." He could tell just by looking, because he knew what three looked like. A couple of other children followed by explaining their own reasoning in remarkably similar terms. Great minds and all that.

One child remained with her hand up. "Yes?" I asked.

She smiled. "I knew it couldn't be just 1," she said, "because you said you had dice in the cup, and if there was just one then you would have to say you had a die in the cup."

True, too!