PDS Math Guy

Analyze That Algorithm!!

2011-09-28T14:53:00.004-04:00

David had his students look closely at some addition problems this morning. He set up 5-6 large sheets of paper on the tables in the room, each with a problem such as 16,732 + 5,895. Each of the problems came pre-solved, with every step of the solution carefully worked out and on display.

However, each solution used a different algorithm.* In each case, the students had to figure out how the solving strategy worked and respond to the strategy. Was it easy to understand? Was it clear what was going on? Had they tried it before? Would they consider trying it in the future? Would it work better with some kinds of numbers than others? They moved around the room, analyzing each algorithm and writing briefly about it as well.

*Of course, you remember what an algorithm is--it's an alligator that wins So You Think You Can Dance, right? Right? I mean, think about it, Al's got rhythm? Rhythm? Al? Al-go-rithm? -- Oh, all right. An algorithm is a series of steps which, if performed accurately and sequentially, will always lead to a correct answer.

The kids were very familiar with some of the algorithms on display, less so with others. The "traditional" algorithm was in there: you know, the one with carrying, which we actually call regrouping or trading because it's a more descriptive term thankyouverymuch. ("Seven plus five is twelve, carry the one" should actually be "Seven plus five is twelve, that's a ten and two more; the ten goes upstairs...") There were a couple of other models in which numbers were written in expanded form (that is, 2,368 becomes 2000 + 300 + 60 + 8); there was one which used a series of partial sums...all of them were useful in at least some circumstances. "Think about them," David instructed the students, "compare them, see which ones make sense to you."

A few of the responses, written and oral:

"This one's interesting. It uses expanded form, but then it puts the numbers back together in a very mathy way." (This student and I are going to write a song, perhaps: "A Very Mathy Way." Listen for it at the Grammys.)

*"I recommend this one if you're in calculator school. If not, try different ways."

*"This part over here--that's just like a sloppy copy when you're writing a story. You don't have to look at it too closely."

*"Easy and quick if you fully understand the base ten system." A nice companion for "An example of using the base ten system in an advanced way." Different algorithms, and (I think) different responders as well.

*"[The person who solved it this way] is a strong mathematician that made a hard question easy."

*"Our group thinks it's too messy. There's too many lines and numbers." (But do you understand what's going on? I inquired.) "Oh, we understand it. We just think it's messy."

*"EASY" (Upper case letters and lack of punctuation in the original)

*"This one doesn't make very much sense. But it does make a little bit of sense."

*"I am going to try it because it looks cool and fun."

First Week of School

2011-09-15T22:41:00.003-04:00

As you very likely know, PDS opened last Wednesday. I must say it's been very nice to be back with the kids again. There's something quite wonderful about children waving enthusiastically at you from halfway across the room and yelling out"Hi, Math Guy!" Makes a man feel worthwhile!

This blog is also back from vacation, so perhaps it will have some children yelling, "Hi, PDSMathGuy blog!" from across the room at it...Or texting "Hi, PDSMathGuy blog!" Or emailing it. Or perhaps not.

In any case, a few little stories, vignettes from the first few days of school to get you smiling (I hope):

***I am presenting a lesson to first graders about odd and even numbers. Part of the followup includes a grid of numbers: 1, 2, 3, and so on. Kids are supposed to determine which numbers are even and which are odd, then color odd numbers blue and even numbers red. One child digs in the colored pencil box and pulls out a blue one, which--alone among all the pencils in the box, it seems--has been sharpened on both ends. "It's a good thing I'm supposed to color the odd numbers blue," she tells me. "This is a blue pencil. And it's very odd."

***A second grade class. A child is paying extremely close attention to her teacher, who is showing the children how to play a game. You can see the child's eyes locked onto the board, her body still, her jaw determined. When she gets up and heads for her seat, there is a Weary Expression on her face. Dropping down on her chair like a rag doll, she shakes her head and looks up at me. "Hard...work," she says emphatically. "Hard...work!"

***Another second grader. The question is to come up with two numbers you can add to make 27. I'm asking him to try the problem in his head, without manipulatives and without writing numbers on a sheet of paper. "Okay, 15," he says proudly. Well, that's a fine start, I tell him, but I'm asking for two numbers and an addition expression, so just saying 15... He nods. "Well, 15," he says, "plus whatever number you put with 15 to make 27."

***Yet another second grader. "What are some things you know about math?" I ask. "Well, ONE thing I know is that math is fun!" he responds immediately. Then, thinking a moment and remembering who he's talking to, he adds, "And I bet you would agree with that!" (P.S. He's right. I would!)

In the Fullness of Time

2010-12-07T22:20:00.002-05:00

It was a He-Man action figure, I think, that started the trouble on that spring day early in my teaching career. If you're old enough to remember they heyday of He-Man and his, um, business associates (the name Skeletor ring a bell?), congratulations. (And you do need to have some age on you, as these figures were popular in my kindergarten classroom around the same time as the A-Team and just before Cabbage Patch Kids (bless their ***** little hearts) and Transformers.) If you're not old enough to remember the excitement, if you grew up outside the cultural influence of the USA, or if your memory is like a sieve, never mind. It was a toy, that's all you need to know; a coveted toy, a toy with a definite cool factor, the kind of toy that conferred automatic social standing on its owner, even in kindergarten.

Anyhow, there was a stray He-Man figure, a slightly dusty one that someone had found wedged halfway under a shelf near the block corner. He immediately claimed it as his own property, based on two incontrovertible legal principles (incontrovertible among five-year-olds, at least): first, he'd found it; and second, just in case that didn't carry any weight with others, he was almost certain he'd lost that very figure some time earlier, and lost it in a part of the room that was surprisingly near the shelf in question. At first no one challenged the incontrovertibility of these claims, and he was happily showing it off to all his friends, and growing no doubt in social standing all the while...when up came another child, who promptly rocked the boat by claiming this very He-Man as his own property.

It soon became apparent that this was going to be one of those situations: a he-said, he-said situation, long before Anita Hill and Clarence Thomas came along to popularize that expression, or one almost exactly like it. A glance at the toy in question was of no help in resolving ownership. One He-Man toy looked very like another, after all, extruded as they were from the same plastic molding machine and colored by the same shade of dyes, and other than the dust there were no particular marks that distinguished this particular figure as belonging to either of the boys. No parent had scrawled his or her child's initials on the figure's lower back; no child had affixed a slice of black tape to He-Man's mighty foot for easy recognition; no serial numbers appeared on He-Man's warrious helmet.

"It's mine," said one of the boys, lower lip trembling, "I know it is. I remember."

"It's mine," said the other boy, a catch in his voice. "I recognize it."

Both looked pleadingly at the adult, a.k.a me. A Solomonic situation, I remember thinking, and briefly toyed with the notion of threatening to cut the figure in half. But no, I ducked, and simply suggested that they try to work out a fair way to solve the problem. That, after all, was high on the list of values at the school, the ability to talk about problems and come to an agreement that would be, um, agreeable to both parties. (I note here that few of my former students seem to have gone into politics.) "Take a few minutes," I said, "talk about it, see what you come up with. When you have a plan that you think will work, let me know."

Off they went. And a few minutes later they were both back. The quivering lower lip was still, the tremor in the voice had ceased. "We worked it out," they told me in unison. And they had. "First I get it for ten years..." said the boy who had found it below the shelf.

"And then I get it for ten years," added the other boy.

It was a fair deal, on its surface, but I couldn't help feeling there was a flaw in their plan somewhere or other...

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

2010-10-28T15:04:00.004-04:00

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!

2010-10-10T17:35:00.005-04:00

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)

2010-10-09T21:10:00.003-04:00

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!

You're Wrong HA HA HA

2010-09-18T17:13:00.003-04:00

Ellen asked her 4th graders to write down what helped them in solving math problems, and what did precisely the opposite. This is a great assignment, as it requires kids to think about their own study habits and student skills: metacognition at its finest.

It was also fun to read the replies. Most students said that a quiet room helped. (More students, it strikes me, than there are students who actually help KEEP the room quiet, but it's the thought that counts.) Many said that having manipulatives to work with was helpful. And a few were very specific: What helps, wrote one student, is "people keeping their feet to themselves."

More interesting, however, were the things that children said did NOT help. Here they were usually quite detailed and focused in their complaints.

It makes it hard, said one child, when "people [are] invading my personal space." (Presumably that includes invading it with feet; see above.)

"Being told that your idea is terrible," suggested a classmate.

"When people are unnice," commented a third. (I agree. I much prefer it when people are unmean. Not to mention unloud.)

Another student wrote: "To hear any bragging or any kind of DISTRACTION." [caps in the original]

Another hated hearing "I wish you'd catch up with me so we can work together, since I'm so much farther than you." (A subtle slam, couched in nice enough words but with a very unnice message.)

"To say 'I'm right and you're wrong HA HA HA," wrote another student. (Bad enough to be wrong, worse for someone else to be right, but I agree, the HA HA HA really puts it over the edge into no-jury-would-ever-convict territory.)

And this one, which sounds like a couple of lines from a 50s song, maybe sung by one of those girl groups I can never remember the name of:

"I don't like to be pressured
I don't wanna be bugged."

(If you'd like to take a stab at completing the lyric, please note that "pressured" rhymes nicely with "Eschered"--presumably the act of being turned into an impossible figure or a slightly bizarre tessellation--and "bugged" with "hugged," "chugged," "plugged," "tugged," and "mugged." Among others. Good luck!)

The World Cup, Plus Math

2010-06-21T21:41:00.005-04:00

Yes, I have been following the World Cup tournament. Though I cannot say I know all that much about soccer, I appreciate the "beautiful game" as much as anyone in my knowledge range. I particularly enjoy beautiful things like, oh, the delightful drone of the airhorns, the charming attempts by France's media, athletes, and managers to blame everybody else for the French team's insipid play, and the ever-admirable arguments over the eyesight, credibility, and integrity of various officials. And the goals are pretty cool too. (Though I have to say that as a one-time [very low-level] water polo goalie I appreciate some of those saves even more. Man oh man alive, the things those guys can do...]

But what attracts us Math Guys to the World Cup, of course, isn't flags, or controversy, or those funky yellow shoes the Honduras players are wearing. No, the real draw, OB-viously, is the mathematics of the tourney. There really are a lot of interesting mathematical puzzles and problems surrounding the World Cup, and on the chance that your child--or you yourself--might be interested in a few World Cup Math Queries, here's your chance.

Some background: In the first round, teams are divided into groups of 4. Each team plays each other team in its group once, for a total of 3 games. You get 3 points for a win, 0 points for a loss, and 1 point for a tie.

A warm-up or two first.

Question 1. What is the greatest number of points a team can get during the first round of the tournament?

Question 2. What is the fewest number of points a team can get during this round?

Too easy? All right. Here's a slightly harder one.

Question 3. A team can get various single-digit point totals during the first round. For instance, 2 ties will result in 2 points. However, there is one single-digit point total that a team canNOT get during this round. What is it?

Question 4. If I told you my team finished round 1 with just 1 point, you'd be able to tell me its record. (You'd also be able to tell me my team was going home.) What would its record be? That is, how many wins, how many losses, how many ties?

Question 5. There's one point total, though, that's ambiguous. That is, there are two possible won-loss-tie records that will result in that number of points. What is that total? And what are the two different records that will get you there?

I alluded a couple of questions ago to the fact that some teams GO HOME. Yes. Only the top two teams from each group survive to play another day. I think we call this Darwinism. If teams are tied for second/third place there's a series of tiebreakers to resolve who moves on. But just two teams advance. (Oh! Now I understand the message of the Australian patriotic song "Advance, Australia Fair"! It's a song urging the Socceroos to move forward in the tournament! See how neatly all this fits together?)

So. Question 6. What is the greatest number of points a team can get in the first round and FAIL TO ADVANCE? (Yes, it involves losing a tiebreaker.)

And Question 7. How about the least number of points a team can get in the first round and ADVANCE ANYWAY? (If you're reasoning by analogy from number 6, you might think again.)

Perhaps the most interesting challenge, though, is trying to reconstruct the individual results from the group standings. Surprisingly often, it can be done. Here's a current example. This is the UP TO THE MOMENT scoresheet for the four nations of Group G after each has played two games:

Country Points
Brazil 6
Portugal 4
Ivory Coast 1
N. Korea 0

Question 8: What were the four games already played, and how did they turn out?

(Spoiler alert.) Okay. If you've been paying attention you know that Brazil has 2 wins (3 + 3 = 6). Ivory Coast has a tie. North Korea has no points, so it's lost twice, and Portugal, with 4 points, must have a win (3 pts) and a tie (add 1 more). Now, follow along:

1. Brazil has only wins, but Portugal has no losses. Therefore Brazil hasn't played Portugal yet.

2. Since Brazil hasn't playd Portugal, then its two games were against North Korea and Ivory Coast.

3. Brazil won both of these games, since Brazil only has wins.

4. Portugal also has two games to be accounted for, and since the Portuguese haven't played Brazil they also must have played NK and IC.

5. Portugal has a win, which matches up nicely with North Korea's loss (the one that wasn't to Brazil);

6. and since Portugal and Ivory Coast are the only teams with ties, that game must have been a draw.

Therefore, the four games thus far:
Brazil over Ivory Coast
Brazil over North Korea
Portugal ties Ivory Coast
Portugal over North Korea
--Gimme a Q! Gimme an E! Gimme a D!

One more before I let you go. It's a bit tricky.

Question 9. WITHOUT LOOKING UP THE RESULTS, give the result AND THE FINAL SCORE of each of the four games played thus far by the teams of Group E.

Country Points Goals Scored Goals Allowed
Netherlands 6 3 0
Japan 3 1 1
Denmark 3 2 3
Cameroon 0 1 3

My friend Cheerful Charlie says the problem can't be solved, and he's (sort of) right; if you only look at the points column, you CAN'T tell who played (and beat) who. But if you look closely at the goals columns as well, you'll realize that there is a unique solution...one which, fortunately enough, happens to match reality. Good luck!

**Want answers? Want to know how you were supposed to know? Disagree with me and want me to (ahem) prove you're wrong? Fine. Send me an SASE containing your questions and comments and $1000 and... No, just email me at scurrie at-sign poughkeepsieday dot org and I'll be happy to respond. As soon as I'm done blaming France's troubles on the Slovakian equipment manager and a God-awful offsides call by a crooked linesman from Qatar, that is.**

When East Is West, and West Is East, and Ever the Twain Shall Meet

2010-06-21T00:54:00.003-04:00

The current xkcd comic raises an interesting question.

http://xkcd.com/753/

{Not that the non-JFKs among us are immune from this sort of thing. In the US alone:

+Large portions of West Virginia are actually east of much of Virginia, and some parts of Virginia are west-er than any part of "West" Virginia.

+Spartanburg, South Carolina is north of Wilmington, NORTH Carolina.

+The East River flows south as it makes its way through New York City, and the easternmost point in the continental US is called, what else, West Quoddy Head.

+South Bend is in extreme northern Indiana, West Bend a long way from anything we might call "western" Wisconsin, and for the fun of it I'll throw in the fact that Jo Daviess County in the northwestern corner of Illinois is traditionally lumped in with Carbondale, Decatur, and Cairo as part of the region known as "Downstate."

Okay, okay, I know there are reasons for all these names, North Carolina is (mostly) north of South Carolina, the East River is east (of Manhattan), Downstate Illinois basically means not-Chicago, yeah, yeah, yeah, what-ever. I still say we need truth in advertising. What's west should never be east, and what's down should not be up, and it should be possible to distinguish north from south easily and quickly and confidently. The spatial reasoning skills of our youngsters are at stake! Won't someone please think of the children?}

The Seven-Sided Hexagon; or, Epic Geometric Fail

2010-06-14T11:07:00.007-04:00

So there I was in the local Rite Aid the other day, picking up grape juice and glue (don't ask), and my path toward these goodies took me down the Seasonal aisle. Where I discovered, much to my surprise, that though it was already early June the Back to School sales had not yet started and there was not a single Halloween mask anywhere in sight. (I wonder if the Home Office knows about this?)

More amazing still, I spotted a few assorted pieces of summer merchandise. Notably, a large box that contained a "Hexagonal Canopy." Which had a helpful explanatory diagram on the side of the box, for those who weren't quite sure what "hexagonal" meant:

Oops.

Birthday

2010-06-06T21:08:00.002-04:00

"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????"

Boys, Girls, and Paper Airplanes

2010-03-31T23:54:00.007-04:00

PDS is on vacation this week, as many of you loyal readers know, but I am not on vacation exactly; among other things, my Vassar College course continues, and so I was in the classroom on Tuesday over at the Old Observatory.

As many of you loyal readers once again know, the class I'm teaching is Math and Science Methods, an elementary education course. We are now done with math (aww) and moving on to science. On Tuesday, we started with an interesting discussion about science education--the students' recollections of science, their associations with it, and so on.

A few notable highlights. On the whole these are relatively science-savvy students. One attended an elementary school near the Pacific Ocean in which the curriculum was based around marine science. Another had thoughts of going to medical school. A third took organic chemistry last fall--it's a notoriously hardcore science course, but one which she took (and I quote from the questionnaire I handed out at the beginning of the semester) "for fun." Others took AP Biology, have good memories of hunting for rocks, and so on. And when I asked them whether they had ever felt in any way that science was off-limits to them because of their gender (all the students enrolled in this class are women), they almost unanimously said they had not. I was pleased, if surprised. "That's very good to hear," I told them. "I don't know that the answer would've been the same in my generation."

And yet. A little later I asked them to tell me the first image that popped into their heads when I said a certain word. The word, you'll not be shocked to learn, was scientist. And here were the results. "My brother," said one student. "Bill Nye the Science Guy standing in a lab," said another. The rest all admitted to seeing a figure in a white coat in a laboratory. What kind of figure? Male or female? Male, they admitted, one after the other. Did anyone visualize a woman? I asked. Hesitation all around, then somewhat embarrassed shakes of the head. It may not have been the answer they wanted to give, but it was the truth: in this group of bright, well educated women, many of whom had a strong science background, all of whom attended a college that had a long history of empowering women and fighting stereotypes, every single one thought of a generic "scientist" as a male.

And one more thing. I'd assigned them to take some time to design two paper airplanes and bring them to class: one that would fly far, and one that would fly for a long time (distance for the first, duration aloft for the second). The purpose in part was to have them do the testing and questioning that's so central to science: what if I fold this wing up a bit more? what if I add some tape here? what throwing motion seems ideal? We would have a fly-off, I told them, after which valuable prizes (might) be awarded. (We did have the fly-off, by the way--see the picture below. The miserable weather cleared just enough to enable us to throw the planes off the balcony of the Old Observatory--one of the original buildings at Vassar and not so incidentally the building that served as the laboratory and office of the great astronomer Maria Mitchell, one of the finest scientists of her generation.)

Now, a comment here. I grew up making paper airplanes. I am sure I single-handedly destroyed dozens of trees in the process of making something that would fly, and fly well. (In my case I was less interested in distance than in duration aloft: the planes' ability to do loop-the-loops, arcs, and other tricks.) Most of my friends, as I look back, were into paper airplanes too. I don't remember girls getting involved much, if at all. Certainly my sister had little interest. Neither did my girl cousins. When I went on to become a classroom teacher, the trend continued. In nearly all my years at various grade levels, a group of kids started making whole fleets of paper airplanes at some point during the year. The group was almost always exclusively boys. Once in a while a girl would join in briefly, and was usually welcomed, but didn't stay for long. The only girls over the years who spent much time making and flying the planes were the few who usually sought out boys, rather than girls, as playmates. So I suspected that most of the young women in this college class had little experience with paper planes, and that's another reason why I assigned this as a task.

"So!" I said on Tuesday. "Did you enjoy the process?"

There was much grimacing and wrinkling of noses.

"I take it that means NO," I said. "How many of you spent much time as kids making paper airplanes?"

No hands went up. A few admitted that they had made an occasional plane, but added that their interest level had been low and their frustration level had been lower still. "Hmm," I said. "Now I wonder why that would be?"

One student raised her hand. "Paper airplanes are really a boy thing," she said, and then, realizing that we had once again stumbled into the tangled thicket of gender politics, added quickly, "I mean, I hate to stereotype, but..."

"No, no, go ahead," I said. "We're better off hearing the stereotypes than pretending they don't exist. We can always address them once they're on the table."

"All right," she said, nodding. "See, I went to these websites for information, and they were all, just, I don't know, written for boys. They were, like, 'Here's a great plane to throw at your teacher,' and I..." Her voice trailed off.

"And you never had the inclination to throw a plane at your teacher," I supplied. (Hoping it was true since I was a stationary target.)

"That's right," she said. "And one of the planes they said was best had directions, about 35 steps, and I got to about the twentieth step and it wasn't working, so..." She shrugged, leaving no doubt that while frustration might be a motivating force for some things, enduring all that agony for a $%^$% paper airplane wasn't worthwhile.

"I did the same thing," contributed another student. "I wanted to do my best, but it was so complicated and I found it really frustrating when I couldn't follow the directions. They even had a VIDEO on the website I looked at, and that didn't help either."

"Girls in my school didn't make paper airplanes," the young woman beside her remarked. "We made fortunetellers instead. You know, those things where you--" She pushed her fingers back and forth, miming the motion of turning a fortuneteller this way and that. "Boys sometimes used them," she added, "but the girls made them for the boys who wanted them."

"FORTUNETELLERS!!!" the rest of the class chorused, and then went off into a babble of individual conversations recalling the halcyon days of elementary school folding sessions. "Fortunetellers! They were so cool..." Evidently they had all made fortunetellers, and frequently at that. I, on the other hand, can't remember ever having made one. (I think I tried once and it was too frustrating. Hmm.)

"Yeah, it IS a boy thing," another student commented. "I showed my boyfriend the assignment and I didn't care all that much about it one way or the other, but he was SO EXCITED...."

So I suppose there's good news and bad news on the involving-girls-in-science front. These young women have good associations with science, mostly, and they believe they would be welcomed into the field if they were so inclined. Maria would be proud. Another piece of good news: science is about much more than paper airplanes. Now if we could only populate some of those stereotypically "scientific" white lab coats with women as well as with men...

Pentagon, Hexagon, Heptagon...

2010-03-29T14:53:00.003-04:00

So there I was in the kindergarten, and the children were showing me how well versed they had become in the shapes of the pattern blocks (thanks, Robbie and Bill!).

"This shape has three sides," said Robbie, holding a triangle so the kids couldn't see it, and the children chorused "It's a triangle!"

"This shape has four sides, and they are all equal," she continued, and "Square!" shouted the class.

"And this one has six sides..." "Hexagon!"

"Oh, oh!" called out a little guy upon seeing the shape displayed (and yes indeed, it WAS a hexagon--phew!). "I know another shape! It's LIKE the hexagon! It's a--a--" He screwed up his face, thinking hard... "It's an OXagon!"

All I can tell you is, I would dearly love to see an oxagon in the wild. Wouldn't you?

The Goldilocks Method

2010-03-21T22:38:00.004-04:00

We talk a lot about problemsolving strategies at PDS, especially in the 3rd-4th grades. One of the absolute favorites among the children is the one Icall the Goldilocks Method. Some texts refer to it as guess-and-check, or predict-and-test, but the name "Goldilocks Method" seems to have a greater "stickiness" quotient for kids.

The strategy is based, of course, on Goldilocks, Goldilocks of porridge fame, Goldilocks who could have been charged with breaking and entering, Goldilocks who encountered a trio of ursine forestdwellers...okay, okay, more to the point Goldilocks, who tasted the first bowl of porridge and found that it was TOO HOT, then tasted the second, which was TOO COLD, and finally tried the third, which was JUST RIGHT, and then repeated the process, replacing hot/cold with hard/soft and porridge with beds, but still coming out with JUST RIGHT at the end.

The students I'm working with in division right now used the Goldilocks Method the other day. They were playing a game to help them work with the connection between multiplication and division, and not so incidentally to practice mental math skills. I forget the name of the game (I usually do), but hey, grab a pencil, and you can play along with us at home:

First, choose a number between 600 and 800. No round numbers. (That is, no multiples of 10, like 790, 650, or 700. You will rarely hear me ban round numbers, but the fact is they're too easy to work with.)

Next, choose an odd number between 5 and 20.

Third, write a division expression with these numbers, such as "705 divided by 15."

The quotient will be--well, we don't know yet. But we can figure it out by using the Goldilocks Method. First, take the divisor (in this example, 15). Ask yourself: what do I have to multiply 15 by to get close to 705? We'll use a little mental math here: let's see, 10 x 15 is 150, so that's not close...20 x 15? Well, that would be 300. Okay, we're not getting there very quickly, so let's try 50 x 15. We'll write that down, calculate the product with either pencil-and-paper or a calculator, and discover that 50 x 15 = 750.

All right, what would Goldilocks say? She'd say TOO HOT. Or TOO HARD. Or TOO HIGH. Or something beginning with TOO. So, we need to try a smaller number. How about 45? Well, 45 x 15 = 675. TOO COLD/SOFT/LOW. Try something that's greater than 45. 48 x 15 = 720. Getting there! But still, TOO HIGH...

You see how this works. In this example, the original three-digit number was evenly divisible by 15, so it was possible to get something that was JUST RIGHT. Go, Goldilocks! Most of the time, it isn't possible in this game. That's okay too: we get as close as we can without going over, and then take the difference as the remainder. So for 696 divided by 9, we might say:

9 x 70 = 630 TOO LOW
9 x 80 = 720 TOO HIGH
9 x 75 = 675 TOO LOW
9 x 77 = 693 TOO LOW but oh-so-close...

And so our division sentence would be that 696 divided by 9 is 77, with a remainder of 3.

Goldilocks would be so proud...

From LOGO to Ladybugs

2010-03-20T12:24:00.015-04:00

Early in my teaching career, PDS decided to provide each lower school classroom with a computer. Well, I should probably say "so-called computer," as the machine that graced my own classroom bore practically no resemblance to the current PDS fleet of laptops.

The machine, IIRC (and I'm sure I do RC), consisted of a keyboard, a monitor with a black-and-green screen, a separate drive for floppy disks (and floppy they were), and a whole mess of unnecessary wires. No mouse, no trackpad. No internet connection, no CD drive. No color, no sound. No bells, no whistles.

Oh, there was a printer of some description, a noisy and unreliable machine that routinely shredded the paper you fed it and printed letters better suited to connect-the-dots than to actual, you know, legibility.

The machine, which looked something like the one pictured above (minus the mouse), could do two things. The first was word processing, or what passed for it during the early-to-mid-eighties. The word processing was courtesy of a program called Bank Street Writer, which had been developed specifically for use in "educational settings." From where I sat it was hard to see why anyone had bothered. Bank Street Writer was clunky. It was slow. It was inefficient. It was practically useless. You had to use the keyboard arrows to select text for editing, which took forever, and they kept throwing the mid-eighties version of dialogue boxes at you when you did ("Are you sure you want to select this block of text? Y/N" "Are you REALLY sure? Y/N" "Are you positive? Y/N" "Are you sure you want to move it to the indicated place? Y/N" "Do you have any idea why we are asking all these questions? Y/N" "Don't you wish you'd decided to write this out longhand instead? Y/N.") Saving was a slow and frustrating process, as was retrieving previously-saved files. There was one difficult-to-read font (though there may have been two sizes, I'm not sure), and formatting was just about nonexistent. After a number of ol' college tries to find any way in which this program represented an improvement over almost anything else, I washed my hands of it and went back to the trusty old typewriter.

The other program was better. It was called LOGO, which was always written in capital letters though I'm not sure it actually stood for anything. LOGO allowed kids to do simple programming in a geometric context. You had what they called a turtle, which actually looked like a triangle but what the hey, and it sat there on the screen waiting to be told what to do. Kids could then type in various commands to make the turtle move. Typing in "BK 20," for instance, got the turtle to go 20 units in reverse (BK=backward, clever huh? and you thought it stood for Burger King). "FD 5" made it go forward 5 units. As it moved, it drew a line behind it. You couldn't make it go directly up or down, but you COULD make the turtle turn. Typing "RT 90" instructed it to spin 90 degrees to the right; "LT 135" got it to...well, you can figure it out.

There were lots of things to like about Logo, scuse me, LOGO. Kids had to type the directions using a specific format: if they typed "FD85" instead of "FD 85," the program would give them an error message. That made the children focus on precision--and helped demystify the computer and its abilities ("yup, it can do amazing things--but it CAN'T figure out what to do when it sees 'LT50' because NO ONE TOLD IT WHAT TO DO when someone mistypes something"). Kids very much enjoyed pretending to be the turtle and giving each other directions: "Okay, forward six steps..." The spatial reasoning aspect of LOGO was excellent--which way do I have to turn if I want to go straight up? what number do I need to input? And the use of left and right and the intro to angle measures were both valuable.

LOGO did have an issue. One goal of the software was to have kids program the turtle to make certain figures--squares, houses, and so on. Can you make a triangle? The letter Z? How? A few kids did get into this. Many, however, quickly decided that the REAL point of the program was to get the turtle to make random lines. We got lots of "FD 400" "FD 40" "FD 400" "FD 989"-style programs in which the turtle made a line to the right, disappeared off the right edge of the screen, came back on the left, and continued to do this for as many commands as the children had told it while the onlookers giggled. Another popular activity was to ignore the FD and BK commands in favor of having the turtle spin endlessly in place: LT 900, RT 42, LT 656, RT 851. Somebody figured out that if you told the turtle to make a turn of 1 unit before doing the FD commands, you could eventually have the turtle criss-cross the entire screen, effectively whiting it all out.

These were cute, and they required some thought at first (especially the white-out one), but once that initial thinking was over the activities quickly became kind of useless educationally. Kids weren't learning anything by repeatedly typing in BK 77 BK 77 BK 77, and the more they did that the less willing they seemed to want to engage in the actual making of shapes. There was something highly motivating about watching the turtle spin this way and that, and in contrast the work of plotting how to make a square seemed considerably less compelling. How you gonna keep 'em down on the farm, as the WWI song went, after they've seen Paree? Under these circumstances LOGO rapidly became less a tool for learning than a diversion for entertainment, and after a couple of years that began to sour me on the whole program. When "real" computers came along LOGO and its derivatives were not high on my list.

This year, though, I returned to my LOGO-ish roots. For our ongoing geometry unit in the 1-2 classes, we decided that I would pull kids during some of their math times and do some computer work. I'd pull out the laptops and work with kids on one or more of the virtual manipulatives at the Utah Sate University website: http://nlvm.usu.edu/. A lot of these materials are really excellent. Rods don't fall on the floor. Pattern block designs don't get wrecked when someone accidentally shakes the table. Virtual rubber bands don't break when you stretch them across a virtual geoboard. While not all materials on the site are equally great, many are quite wonderful.

But after looking through the various manipulatives on the site, I decided to focus on the most LOGOlike one: a program called Ladybug Leaf. (If you click on Geometry on the home page, it will be about 6 or 7 buttons down in the list.) The object of Ladybug Leaf is to direct a ladybug, LOGOstyle, to hide under a leaf. The graphics are far clearer and more engaging than they were in the old LOGO program. The ladybug is a real ladybug; the bug moves in clearly demarcated units; the leaf is a real leaf. The order of commands remains on the screen as the ladybug moves, with each command flashing briefly as the ladybug carries out that particular action. It's easy to replace a command, too--much easier than it used to be! True, FD 25 and LT 90 are things of the past in this activity, and I do kind of miss them. Instead, there are buttons you can click that will move the bug one unit forward or backward, or spin it 45 or 90 degrees to the left or the right. On the other hand, angle measures aren't exactly a staple of first and second grade mathematics, and you can always introduce the terms 45 and 90 degrees yourself.

Anyhow, the kids have been very much enjoying their venture into LOGOlike technology--and, I would like to think, learning important stuff along the way. It helps that in the last twenty-plus years I have learned a few things myself. In particular, I made sure to focus this time around on specific tasks: hide the bug under the leaf, move the leaf and hide it again, make a square, make a triangle, make a house (a house! See below, courtesy of one very thoughtful and dogged second grader)

There's nothing wrong with entertainment for its own sake, but I want a little more from, you know, school.

And how can I complain when that first grade girl who is ordinarily so reserved and so serious, after successfully planning a route to the leaf for her ladybug, celebrated by standing up and chanting "Oh yeah, oh yeah" while doing some disco moves?

Now if only someone could make some improvements to Bank Street Writer...

Blaming the Teacher

2010-03-17T10:43:00.003-04:00

The world was better in the olden days. That's an article of faith among many Americans - and you will forgive me if I point out that it has been an article of faith for years and years and years. The "good ol' days" used to mean the period before World War I, or sometimes the 1920s. These days, though, the good ol' days have jumped forward to the fifties and the early sixties.

Ah, the Eisenhower/Kennedy years! A delightfully "innocent time," we read in Pete Hamill's review of the new book out about Willie Mays. A time when "anything and everything seemed possible," according to another book I recently ran across. A wonderful era when we had good old-fashioned values, when video games were nonexistent, when families ate dinner together every single night. Never mind the occasional problems: sexism, racism, McCarthyism, pollution, nuclear proliferation; it was the good ol' days, by golly, and everything was better back then.

As a teacher, I am especially tuned toward a particular mantra regarding the grandeur of the fifties/early sixties, which is that this era was the Golden Age of K-12 education. Everybody learned to read, quickly and easily. Everybody got really good at math. And in particular, the fifties-slash-early-sixties were a time when the education profession was respected, when parents and kids alike viewed teachers as professionals to be listened to and admired, not as lackeys to be walked all over and to be blamed for children's failures. Read the columns of child psychologist John Rosemond, just to name one strong proponent of this notion. Well, okay, I'll quote here from a typical Rosemond column, to save you the trouble of tracking them down yourself:

Back in the day, writes Rosemond, "when a child was reported to have made trouble in school, the child came home to even more trouble. Today, when a child is reported to have made trouble in school, the parents deny that the child is capable of making trouble, blame the teacher for having a 'personality conflict' with the child or failing to recognize the child's 'special needs' or 'boring' the child. In short, the school/teacher is in trouble."

Anyhow, I was reminded of this mantra while reading the Peanuts strip that appeared in the morning paper. I'm not sure how long this link will work, so I'll summarize the cartoon in addition to linking to it.

http://comics.com/peanuts/?DateAfter=2010-03-15&DateBefore=2010-03-15&Order=d.DateStrip+ASC&PerPage=1&x=7&y=8&Search=

Linus is distressed to find that he has failed to make the honor roll at school. Sweat pouring off his face and his wildish hair looking even more wild than usual, he tells Charlie Brown that he is "doomed," that his parents will be shocked and disappointed. (So far, so Rosemond.) Charlie Brown asks Linus what he thinks will happen, to which Linus replies, "Well, obviously, the first step will be to put in a complaint about the teacher."

The original publication date on the strip? March, 1963.

Oh. Okay. Perhaps things haven't changed as much as we thought.

Fathers, Sons, and Inequalities

2010-03-16T01:01:00.004-04:00

I did a couple of days of Professional Development last week for a nearby school district. Hard work, but fun in its own way, and the teachers were very thoughtful and responsive, which was great. A few former colleagues of mine are working over there now, too, and it was wonderful to see them.

To illustrate some of my points about how children think about mathematics, I told some of my favorite stories, a few of which have appeared on this blog. But I left out this one, which took place in a kindergarten class early in my teaching career:

*****************

The little girl is almost always late being picked up. Her mother works till 3, and pickup is at 3, and the mother hasn't figured out how to be in two places at once. Technically I am supposed to send the girl to the After program if she hasn't been retrieved by 3:15, but the reality is that the After costs money, which the mom doesn't have much of. And besides, the mom is almost always there by 3:25. And anyway I'm an old softy at heart, or something.

So we have worked out a silent understanding, the girl and I. I go about my business in the classroom from 3 to the time she is picked up, tidying up and organizing the next day's work, and she sits quietly in the big rocking chair just outside the meeting corner rocking slowly back and forth, her lunch box by her side. Sometimes she looks at a book while she rocks. Other times she just rocks. It seems to be a nice decompression time for her. Once in a while we talk briefly, but she's never been much of a talker under any circumstances; so more often this is simply parallel play of a sort: the day is over, and she is in her world and I am in mine. When her mom arrives at 3:20 or 3:25, she slides out of the chair and heads for the door. "See you tomorrow," I say, but she is the strong, silent type, and so she smiles and wiggles her fingers at me in a half-mast wave, and then she is gone.

One day, though, another teacher stopped by my room at 3:20 to consult with me about something. The room was empty, of course, except for me and my late pickup, the girl in the rocking chair. I was taking clothespins off a bulletin board, if I remember correctly (and astonishingly, I think I do), and she was rocking, of course, the chair creaking as she meditatively swung back and forth.

The consultation finished, the teacher noticed that I was wearing a sweater (this was in the days when I still occasionally wore long sleeves). "Nice sweater," she said approvingly. "It looks handmade. Did someone make it for you?"

"Um," I said. "Well, sort of. My sister made it, knitted it for my father. But it turned out to be too small for him, so he passed it along to me."

The teacher nodded. "It seems to fit you just fine," she said, "and it's certainly striking," and with that she ducked back out of the room, and I returned to my clothespins to the accompaniment of the familiar, faint creak of the rocking chair--

When, quite suddenly and unexpectedly, the girl spoke up. "Your daddy is older than you are," she said.

I had almost forgotten she was in the room. Turning, I saw that she had a satisfied smile on her face. "Your daddy is older than you," she repeated, just in case I hadn't heard it the first time.

"Yes," I agreed. "That's right." Well, of course it was right. But I couldn't resist finding out the details of her thinking process. "What makes you say so?" I asked.

"The sweater was too small for your dad," she said proudly, her chair busily creaking as always, "but it fit YOU. So you are smaller than your dad. And if you're smaller than he is, then you must be younger, because people who are young are small." Creak, creak went the chair as she rocked harder and more enthusiastically. "So that means your dad has to be older than you."

What could I do but congratulate her on her remarkable reasoning ability? And it WAS impressive, even if entirely unnecessary, and this tiny little girl, not yet even six years old and still unwise in the ways of the world, deserved all the praise she could get. "You're absolutely right," I said, nodding my head slowly. "My dad IS older than me. You did a great job of figuring it out."

"Thanks," she said, taking the compliment as her due, and just then her mother walked in the door, and the girl slid off the rocker, exactly as she had done a few dozen times before, and she wiggled her fingers at me with a larger-than-usual smile. And though it's been probably twenty-five years since that incident, and though I lost track of that little girl long ago, I can still hear the creak of the rocking chair and see the self-satisfied grin on her face as she explained her impeccable logic...

Ah, memory. It's a funny thing.

Teaching Teachers to Teach

2010-02-25T12:07:00.010-05:00

The "About Me" section in this blog states that I have worked with the education departments of a couple of local colleges. That's more true than ever this semester. With the usual instructor of Vassar's Math and Science Methods course on leave this spring, I have stepped into the breach and am now trying out being "Professor Currie" (that's ADJUNCT professor Currie to you--I'm not actually sure what my official title is).

I have a lovely class of what the education people like to call "preservice teachers," and so far the class is going well, and the students seem to be enjoying the activities and the discussions BUT there's a certain disconnection I'm noticing, which is that things seem pretty theoretical without, you know, actual children in the room. And teaching is all about children, right?

So for my most recent class I brought in a few children. Three, to be exact: three students who are currently fourth and fifth graders at PDS. The kids wandered in about 3:45 (class starts at 3:10) and sat down in too-big chairs in the seminar room. They looked a tad uncomfortable at first but soon warmed to the situation.

I started by asking them some questions about their enjoyment of math (they said they liked it, and a good thing too)--what they liked best about it ("it's challenging," a couple of them said), what was not so good ("when it's boring"--but I could've told you that in advance). I asked them to talk a little about how they learned multiplication as well. (Part of the focus of the class session was on multiplication and division.) Mathups, the kids agreed, had been very helpful, and using the array model had helped them understand the concepts too.

Other questions followed, both from me and from the students. What was your most memorable math project? (The corn kernels experiment, one student said. See the blog entries from October 2009 for more details on this one.) Do you prefer mental math or pencil and paper? Why? I had asked the college students to write an explanation of why the multiplication algorithm worked; one of the students was a bit unclear about one part and asked if the kids could explain it more clearly; one was happy to rise to the challenge. What manipulatives did they like? (NOT the pattern blocks, one child explained, adding "I like numbers best.") The kids were poised and articulate and knowledgeable, which was great to see. ("My, you know a lot!" I was tempted to say at one point. "Your math teacher must be HIGHLY skilled and no doubt deserves a medal and a large cash prize!" But I resisted.)

Finally, I'd arranged for the kids to teach the students a few multiplication games they had played in the past to help them learn and practice multiplication. So we divided into small groups, each with a child in charge, and off we went! The kids really enjoyed the chance to be in charge and the experts, and took well to the role.

We could've gone all afternoon, I think, but homework and other obligations loomed for the younger ones and us older ones had more discussion to do--so I paid the kids with some Freihofer's and thanked them much and off they went.

So the kids enjoyed it, and the students enjoyed it, and I enjoyed it, and it was an excellent reminder that teaching math involves teaching actual children. Fun stuff!

Obi-Wan, Mini Hot Dogs, and the Ninety-Dollar Jar of Olives

2010-02-15T12:47:00.005-05:00

Hungry? You’ve come to the right place. Some of our second graders have been spending a few of their math periods lately planning menus for a special dinner. If they have leftovers (and judging from the amount of food some of them are planning to buy they’ll have lots), I’m sure they’d love to have you drop by.

Each of the children was instructed first to draw up a guest list consisting of seven fictional characters they’d like to get to know better. Their lists skewed heavily toward fantasy fiction: Lord of the Rings, Harry Potter, the Star Wars crowd. Obi-Wan Kenobi, in particular, will be kept busy nearly every evening for a week with invitations to these foodfests.

Next, they sketched out seating arrangements. One child set up four tables of two people each. Another planned for two tables, each seating four. Someone else constructed an octagon. (Eight people? Yes--seven guests and a host.) Then they planned a menu. “Not just desserts,” I cautioned them when it seemed clear that this was the way the wind was blowing. “A main course, or even two, so people can have a choice. An appetizer, perhaps. Soup, salad. Fruits and vegetables. Drinks.” “Oh, okay,” they said, and proceeded to put together menus that were...well, idiosyncratic, or what else would you call it when red snapper and turkey are served side by side?

Now came the most mathematical part: shopping. We did this in the comfort of our own classroom, using grocery ads from newspaper inserts as well as the circulars that arrived in my mailbox over time. I gave the children record sheets as well. “Find the items you want to buy,” I told them. “Record what the item is and how much it costs to buy just one. Then decide how many of the item you need. What will the total cost be? Figure that out and record it here in this column; then round that to the nearest whole dollar to make estimating the grand total easier.”

The children fell eagerly to work. “Mini hot dogs, mini hot dogs,” one boy repeated over and over, scanning the ads in vain for his favorite entrée. “Pizza!” crowed a classmate, jabbing a pencil at a listing in the frozen foods section of one advertising supplement. “Pizza?” asked a classmate. “Hey, I need that too!”

The kids needed help deciphering some of the prices: supermarkets, we learned, don’t often write prices the way teachers say you should. Instead of $1.59, for instance, they often write a big 1 followed by a smaller 59 and no decimal point or dollar sign at all: “Looks like a hundred and fifty-nine dollars,” one child said disapprovingly. “5 for $2” was confusing enough, but “4/3” to mean “you can get four for three dollars” was enough to make strong second graders cry. (Well, not literally. They were good sports about it and chalked it up as yet another example of the weirdness of adults.)

Determining how many of each item they needed was again idiosyncratic. One child decided that each person would probably eat 5 cookies, no, make that 6, so he settled on 48 as the number to buy. Somebody else, less generous or perhaps expecting that everyone would be full when dessert came around, decided to purchase just one small apple pie for the table. Did you need one carton of orange juice, or were you better off with 10? Different kids answered similar questions in wildly different ways.

Determining the cost of n cartons of orange juice, when n was more than 1, was often tricky as well. Some of these hosts have a beginning knowledge of multiplication, which they used to good effect. “Each cookie costs ten cents,” mused our 48-cookie-buying friend from before, “so 48 of them would cost, um, $4.80.” One child bought two jugs of milk at $2.99 apiece, set up the costs in columns, and laboriously regrouped to arrive at the total, $5.98; another, trying to determine the cost of two $4.98 items, rounded up to $5 for each, added the estimates to get to $10, and finally subtracted the extra four cents for a total of $9.96. We shared strategies and tried out newer, more efficient ones.

What about items that appeared on menus but were nowhere to be found in the ads? I hear you ask. Surely there were some of those. There were indeed, the aforementioned red snapper and mini hot dogs were among them. “Not a problem,” I said. “You have an idea of what similar things cost. Estimate the price. Just put a circle around the price on your record sheet to indicate that it’s an estimate.” Some estimates were IMHO pretty good. Others were...well, one cent for salad dressing was perhaps overly optimistic, and over $100 for a few other items seemed a tad excessive, but hey, who am I to complain?

Once the estimates were done, it was time to find a grand total. Making an estimate (using the rounded-to-the-nearest-dollar figures on the shopping list) was a job for mental arithmetic. Finding the actual total was a job for a calculator. Children compared the answers to make sure their totals were reasonable, then looked again at the total bill.

At first each was flabbergasted by the high cost of groceries (especially given the $90 jar of olives one of them purchased--an estimate, natch), but after a moment flabbergast vanished to be replaced by pride. Yes, indeed, they seemed to be saying, a meal that costs over $500 must certainly be something special. Now if only they’d found more expensive broccoli and purchased twelve loaves of bread instead of just two--

So the menus are done, the shopping lists are ready to go, and all that remains is to check on the availability of the guests. Anyone know if Dumbledore is free the evening of the 23rd? The invitation says six pm sharp, but he can arrive at seven if he doesn’t mind missing the goldfish and gummiworms being served as appetizers...

The Pirates of Poughkeepsie

2010-02-14T12:12:00.003-05:00

It was Pirate Day in the 1-2s a little while back (arr!), and the buccaneers were out in force. Their captains, um, teachers, had a number of activities planned for them. One of these was locating a treasure chest that pirates of a previous era had buried somewhere in the Chapman Room. Luckily, these modern pirates had an appropriately antique map and faded directions.

“What’s it say next?” one pirate asked.

“Turn left,” another replied, squinting at the directions, and as one they all turned right.

“No, no, the other left,” their teacher explained helpfully.

They rotated 180° to the proper orientation. “Next,” said someone, “it says to take ten steps forward.” and they all took ten steps forward except for those who took twelve or thirteen because if ten steps are good then twelve or thirteen are even better, besides which if you’re the pirate in the lead you might get a bigger share of the treasure if and when it appears.

“Ten steps,” suggested the teacher....

So there they were, these young terrors-of-the-seas, thinking they were engaged in a search for hidden booty. But how wrong they were! Us Math Guy types knew the truth: They were actually engaged in Spatial Reasoning. Left, right, forward, backward, mapping, following directions, the whole nine yards. Proving that even pirates are mathematicians at heart--scuse me, at h(arr!)t.

Camera Shy

2010-02-13T20:37:00.003-05:00

For those of you loyal readers who don't know what I look like, well, wonder no more! Though photographs of me are hard to come by on the web, a student kindly sketched my picture on the whiteboard that lives in the Chapman Room. Thanks to the miracles of modern technology, I can get you that image faster than you can say "Jackie Robinson." Here you go:

Thank you, Joanna in the 3-4 for the extremely accurate likeness (well, not quite; I am not nearly that handsome, but never mind). In any case, if you ever run into someone who looks almost exactly like that picture, you will know it must be me.

Cheerful Charlie and the Cupcakes

2010-02-05T21:19:00.004-05:00

My good friend Cheerful Charlie, I told the third and fourth graders, was having a party, at which he planned to serve cupcakes. One hundred cupcakes, to be precise. I'm not sure who-all is on his guest list (other than me, of course), but he's either inviting a lot of people, or a few big eaters, or else he just wants a lot of leftovers.

Cheerful had (wisely) decided not to make these cupcakes from scratch (his measuring skills aren't what you would call real accurate), but was having some trouble determining which store he should go to. He had three choices, I explained, and he wanted to spend as little money as he could and get as good a deal as possible, and if the students could advise him that'd be great.

So, there was Cupcakes R Us, I told the kids, which sold baskets of 50 cupcakes at a shot, and each basket cost $40 but he also had to pay a fee of $5 to park.

And there was Cupcake Depot, where cupcakes were $9 per bag for a bag of 10, plus which Cheerful had a coupon for $5 off his total purchase.

And there was Cupcake's Discount Warehouse...

Look at all the information, I told them, and do some calculations if you need to, and decide if there are other considerations Cheerful should be thinking about, and then write Cheerful a letter suggesting what he ought to do. The students could turn in a handwritten letter which we'd forward on to Mr. Charlie, I explained, or they could email him directly at cheerful.charlie@yahoo.com, an account which he checks but not as regularly as he should because he often forgets or mistypes the password.

For some students, the assignment was just a relatively straightforward problem in arithmetic. You determine how many bags, boxes, or baskets of cupcakes Cheerful needs to buy at each store so he has 100 cupcakes in all; you multiply that number by the cost per box/bag/bucket/basket; you add the parking fee or membership fee, you subtract the coupon...and if you've done it right, you inform Cheerful that the cost at all three stores is the same, eighty-five bucks, and he can go wherever he likes and it doesn't make any difference. And this is a fine way of looking at it.

But what makes the problem interesting is the real-world nature of it. Sure, price is important. But is the lowest price always the best deal? I remember when my wife and I discovered WHY generic spaghetti sauce was so cheap (hint: it contained mostly water)... So the question becomes, what other things should Cheerful be taking into account?

Well, a lot of kids came up with lots of ideas.

"You should go to Cupcake's Discount Warehouse," one student wrote. The membership fee of $5 was annoying, she pointed out, "but if you have to go back again you will have your membership card so it'll be cheaper."

"Maybe you can walk to Cupcakes R Us," said somebody else. "Or ride a bike. Then you might not have to pay to park." (Note the "might." Hedging your bets, we call it.)

"It depends," wrote another student. "How far away are the stores?" A good question. If the nearest Cupcakes R Us outlet is in Albany, Scranton, or Paramus, it isn't worth the time and the gas to get there.

"You should ask them each for a free sample," suggested one optimist. "If you say you'll buy a lot they'll probably let you have one. Buy the one that tastes best." Not much point in saving five dollars if the cheaper cupcakes tasted like sawdust or carbon paper. --Or were made primarily of water, like the generic spaghetti sauce referred to above.

"Buy them in bags of 10," someone else advised. "The prices are all the same but if you need more then you only have to buy 10 more, not 20 or 25 or 50, and that will save you money."

And another student went right to the heart of the matter: Measure the cupcakes. "Buy the ones that are biggest," he suggested.

Unfortunately, Cheerful (who likes things simple) is still uncertain what to do: he's definitely bummed at the prospect of needing to find more information. We'll keep you posted. In the meantime, it was nice to see how many students recognized that there might be other considerations besides the cost of the cupcakes. We like to remind kids that math is about real-world situations, and in particular we like to point out that answers may not be as cut and dried as the textbooks sometimes suggest they are. This was a good example of both--and a good example of how a little knowledge can be a dangerous thing.

(Though regarding "other considerations," you CAN have too much of a good thing: see one of my favorite cartoons of all time, http://xkcd.com/309/. --Cut and paste the URL into your browser window if the link doesn't work for you. The two folks on the extreme right? That would be me and my wife...)

Sleepovers

2010-01-09T16:43:00.004-05:00

(Warning: This is not about math.)

Back when I was a classroom teacher, I once taught a girl we will call Claudia. Claudia was new to the school and was a member of my 1-2 class. She was shy and quiet, but she took school seriously and was quite eager to please.

Writing skills are of course a major part of the 1-2 curriculum. Most children try out different subjects and themes as they are learning to write. Not Claudia. Her first three pieces of writing were all essentially the same. Allowing for some variation in wording and the name of her host, they all read like this:

"I slept over my friend Kari's house. First we played. Then we ate dinner. After that we watched TV. Then we played some more. We went to bed. In the morning we had breakfast. Then we watched TV. We played. Then I went home."

At first, I simply worked with Claudia to improve her writing given this basic framework. Could she add some details? What show did she watch, what did she have for dinner? How about putting in some describing words? Was the bed hard or soft, was the TV show funny or scary? I offered some ideas for varying the sentence structure and helped her explore other ways to organize the information. "Conferencing," we call it (though I'm not sure we called it "conferencing" back in the day), and Claudia dutifully added the information I told her to include, and each of the first three stories wound up a little bit different in final form, though not, I must admit, a lot different.

Perhaps, for deep psychological reasons, it was very necessary for Claudia to tell this same story again and again, but where her writing was concerned the constant repetition seemed to be getting in her way. Ideas, after all, were important too. I decided that the "Claudia and the Sleepover" series, Vols 1-3, was borne less of emotional turmoil and more of being in a rut. So when it came time for her to start her next piece of writing, I called her aside.

"You've been writing a lot about your sleepovers," I told her. "I know you like writing those stories, but I think you're ready to write about something new. Let's try this. What's the weirdest, strangest, most unbelievable thing you can think of?"

Claudia thought for a moment. "You mean," she said slowly, "like, my stuffed animals came to life?"

This was a greater leap than I had expected--a highly imaginative response, and given almost immediately. YES!! I thought. "Sounds great," I said, smiling. "So, here's your first line: 'Last night my stuffed animals came to life!' Got it?"

Claudia nodded.

"Then go get 'em," I said, steering her back to her seat. She picked up her pencil and started writing. Excellent, I thought as I moved off to consult with the next child. I'd said the right thing at the right time. I'd handled Claudia well; I'd started her down the road toward more creativity and a greater interest in writing. This was why the school paid me the big bucks. (So to speak).

That warm glow lasted only ten minutes, however, because when I returned to check in with Claudia this is what she had written on her paper:

"Last night my stuffed animals came to life. And they had a sleepover. First, they played..."

A PalindromemordnilaP A

2010-01-02T21:46:00.002-05:00

John McAdam, professor of education at Marist College and a fellow Math Guy in spirit if not in title, reminds me that today is a palindrome: when written in month/day/year format, the digits are the same read forward as read backward:

01/02/2010

Okay, okay, you have to put a couple of leading zeroes in there, but pretty darn good, wouldn't you say?

And FYI, the next one will be 11/02/2011. A loooong way off. So as Paul Simon (the singer not the senator) never put it,

enjoy it while you can, Nan
it ain't all that bad, Dad
there's no need to sob, Bob
just sit down and chill, Lil
and eat a banana, Hannah
(or some other food, Dood)
Just leave it to me... .

On Homework

2009-12-19T18:28:00.004-05:00

"Okay," says the teacher, relentlessly cheerful as teachers of youngish children are expected to be, "does everyone understand this chapter?"

"YES!" bellow the assembled students.

"Have you all finished the sample problems?" the teacher continues.

"YES!" the kids chorus again.

"Are there any questions about the homew--"

"NO!"

You probably can guess the punch line. But if not, check it out (just copy and paste):

http://www.babyblues.com/archive/index.php?formname=getstrip&GoToDay=11/29/2009

The Week with Less Pizza

2009-12-12T14:30:00.007-05:00

As you may know, the 3-4 students have been keeping track of pizza sales thus far this year. Yes, we have records stretching back as far as, let me see, September 10 or so!

For quite some time, as you'll see on the graph pictured below (in two parts), the total pizza order was a rather dull oscillation between 144 slices (18 whole pizza pies) and 152 slices (19 pies). Week after week, 144 or 152, 152 or 144. You could set your watch by it. It was like, I don't know, jazz music or Blue's Clues or driving on Interstate 65 in northern Indiana or something. As the graph shows, the median (the middle value when the data points are ordered) stayed within a very constrained band of numbers, and the range (the difference between the lowest and highest values) remained absolutely, boringly, even mindnumbingly consistent.

[Note that the number of slices actually ordered by lower school students doesn't match the number of slices we actually buy. Why is that, I wonder? Hmmm...]

Then, all of a sudden one Friday, the number of slices ordered took a nosedive. Fell off a cliff, or at least rolled down a slope, as the graph makes clear. Woke us all up, I tell you that. Boom, all the way down from the 150 region to...104. 104! Think of it! The median didn't change (why it didn't was food for thought for some of the students), but the range changed, oh boy did it ever.

Why would things be so different this week? I asked the gathered third and fourth grade children (after swearing to secrecy Ellen's class, which had handled the order and therefore knew the answer). What possibilities do you think there are?

They came up with four:

A) There were a LOT of kids out with swine flu.
B) Some of the classes were on a field trip.
C) The pizza place ran out of pizza partway through.
And
D) Not very many people were hungry for pizza that day.

I wonb't tbell ybou thbe rbeal ansbwer. But wbith anby lbuck, yobu cban gbuess.

What to Wear in Winter

2009-12-09T14:54:00.003-05:00

As a high school student, I came up with a foolproof (and very mathematical) way to determine what winter clothes I needed.

I attended a PK-12 school, and the estimate depended on the behavior of two very different student groups: kindergarteners and sixth/seventh graders.

The K students were sent in (by parents) with masses of winter protection--coats, boots, hats, mittens, earmuffs, scarves, alpenstocks, beeveils, etc--and sent out (by teachers, into the elements) the same way.

The 6th/7th graders, regardless of what they were sent in wearing or sent out wearing, very quickly removed as much outer clothing as possible. There was something truly cool about wearing short sleeves as the mercury dipped down to the single digits Fahrenheit. (Also something truly frostbitten about it, but when you're in middle school you don't care.)

My formula was simple: to determine what level of clothing I needed, I found the halfway point between the overdressed kindergarteners and the underdressed middle school students. That was what I put on before leaving school.

Worked every time. And to judge by what I see out on our playground this winter, the formula continues to work today!

On Family Size

2009-12-01T22:58:00.003-05:00

It's important to connect numbers with real-life situations. Which is why I had 4th graders tell "stories" about the multiplication expression 4 x 6 as a warmup for a lesson this week. By "stories," I hasten to say, I don't mean great literary efforts, with foreshadowing and metaphor and plot twists and poetic license and all those great things. No, I mean simple situations like these:

"There were 4 glasses and each glass had 6 ice cubes in it."

"There were 4 people and each one ate 6 hot dogs."

"I saw 4 flowers. Each flower had 6 petals."

You'll note that in each case the 4 [the first number in the expression] represents the number of groups, and the 6 [the second number in the expression] represents the number in each group. Of course, 4 x 6 is equal to 6 x 4, which all the children I was working with that day knew perfectly well; but it's useful to think of the first and second numbers each playing a slightly different role in the expression.

And we were progressing swimmingly until one boy said, "There were 4 families and each family had 6..." Then his voice trailed off, and he thought, and then he said, "I mean, there were SIX families, and each family had 4 people in it."

Real-life situations indeed. No prizes for guessing how many people there were in his family!

"You Must Be Smart at This"

2009-11-22T18:52:00.002-05:00

I've been reading a fascinating book called "How We Decide," by a guy named Jonah Lehrer. The book contains many odd and interesting (and useful) tidbits of information relating to psychology, probability, and more. I'd mark it up with lots of underlining and margin notes, only I won't, because it's a library book.

One of the more intriguing stories in the book details an experiment done by a researcher named Carol Dweck. I've read about this study before, but not in such detail as it appears in this book. Here's what Dweck did:

1. She gathered a bunch of fifth graders and had researchers give them some simple nonverbal puzzles .

2. Then she had her researchers offer the children a one-sentence statement of praise--EITHER "You must be smart at this" OR "You must have worked really hard."

3. Then the researchers offered the kids a choice of two followup puzzles.

--Option A: A "harder" puzzle, "but you'll learn a lot just from trying it," or

--Option B: A puzzle that's "about as easy as the one you just tried."

The results? 90% of the "worked really hard" group opted for choice A. Less than 50% of the "must be smart" group did.

Dweck wasn't done. She gave the kids a REALLY hard puzzle. The "must be smart" group worked at it for a little while and got discouraged and frustrated. They gave up pretty quickly, on the whole. The "worked really hard" group--well, they worked really hard. "This is my favorite test," many of them claimed, even some of those who never actually solved it.

And when Dweck told the kids that they could see the work of students who'd done better than them or the work of kids who'd done worse, the "must be smart" kids typically chose to see the work of kids who'd done worse. The "worked really hard" kids, in contrast, tended to look at the work of kids who'd done better than they had. The "must be smart" group, Lehrer sums up, "chose to bolster their self-esteem" by looking at the work of students who hadn't done as well--who weren't as smart. The "worked really hard" group "wanted to understand their mistakes, to learn from their errors, to figure out how to do better."

The reasons for the split, to Dweck, were clear. "When we praise children for their intelligence," she writes, "we tell them that this is the name of the game: Look smart, don't risk making mistakes." The "smart" kids acted in ways that avoided putting their supposed level of intelligence to the test. In particular, they did their best to put themselves in situations where they'd be unlikely to make mistakes. "Mistakes," reports Lehrer, "were seen as signs of failure; perhaps [the children] really weren't smart after all." The "worked really hard" group, on the other hand, acted in ways that tended to reinforce the notion that they really WERE hard workers. The results were telling: they showed more curiosity, enjoyed themselves more, and in the end LEARNED more. Which is, after all, the point of school.

This has implications for all subjects, but perhaps especially for math. People tend to believe that math is something that you either CAN do or you CAN'T: you're "smart" at math or you're not. "I was never any good at math," parents (and teachers!) sometimes tell me. "I just don't have the knack for it....It's like other people have a math brain and I don't." I don't usually hear those kinds of things about social studies or even about reading.

For the record, there are lots of good reasons to reject the notion that some people have a "math brain" and others don't. But EVEN IF IT WERE TRUE, it isn't something I'd ever want to hear, because it simply isn't helpful. Dweck's research strongly suggests that if we changed the question "Which kids are smart when it comes to math?" to "Which kids work hard when it comes to math?", we'd all be better off--that kids who find math a little alarming might develop a more resourceful and positive attitude toward it; that kids who are already quick with numbers but accustomed to coasting might find themselves motivated to delve a little deeper and think a little harder; that kids of all ability and interest levels might be inclined to take more risks, show more persistence, and in the end, like the students in Dweck's study, learn more.

So. Two conclusions (for now, anyway).

One: when we teacher types say, "Mistakes are a natural part of learning," we really MEAN it.

And two: Yes, we know your kids are smart. Of course they're smart; they've got good genes, they've grown up in wonderful homes, they're verbal, they're curious, and they're as bright and funny as all-get-out. But do us (and yourselves, and your children) a favor:

Don't tell them.

How to Measure: An Illustrated Manual

2009-11-18T12:00:00.014-05:00

The Definitive Treatise, by PDS First Graders.

1. “There can’t be gaps when you measure. You have to push the measuring things together, like this.”

2. “And you can’t make the rods all zigzag. You have to put them together in a straight line, like this. Don’t get off track.”

3. “The first thing is you have to estimate how many rods will fit.”

4. “You should look at it carefully. Then you can use your fingers to help you estimate.”

5. “You start by putting the first rod down so its end is right at the edge of the thing you’re measuring.”

6. “Then put more of them along the side, like this.”

7. “Go until you get to the other end of what you’re measuring! Then count how many rods you used.”

And now you know how to measure!

A Dot Can Be...

2009-11-16T17:30:00.008-05:00

This morning, Bill and Robbie's kindergarten read Donald Crews's picture book Ten Black Dots. It's a lovely book showing imaginative ways of transforming static black dots into familiar objects...

...as in the children's responses pictured below.

One dot can make a Cyclops...

...or a squirrel hole...

...or a window in a house...

As for 2 dots, they make great eyes...

Stop by the kindergarten to see the whole series!

(Why I Am Not a) Gamblin' Man (Mostly)

2009-11-13T17:37:00.012-05:00

I have purchased one lottery ticket in my life. It was a loser. I have visited two casinos, one on a "riverboat" in Mississippi and the other on drier land in Louisiana. I put about $1.85 in slot machines, total. I lost it all. In 8th grade I was invited to a "Las Vegas" party, where I played roulette all night long (well, till 9 pm anyway) using poker chips. I bet on 36 time and again. It never came up. I lost, and lost, and lost some more. I eventually needed two loans from the bank. So, I don't gamble (mostly), because I lose (mostly, or maybe always).

Still, there are times when you just have to place that bet...

Yesterday I was in Jan's third and fourth grade class, where students are working on logic and attributes. The focus for the lesson was on Venn diagrams. You know, overlapping circle thingies, like this:

(My college roommate, Bernie, was a double major in science and philosophy. After taking a class on Tibetan Buddhism he once accidentally referred to these things as "Zen diagrams." I'm still tempted to call 'em that sometimes.)

In addition to the Zen, I mean Venn, diagram, we also had a bunch of blocks of various sizes, shapes, and colors, and labels with categories that described the blocks: "triangles," "small," "red," "yellow" and so on. I had chosen two labels at random ("What does that mean, 'at random?'" I'd asked earlier in the day, and the response was "Randomly," which was accurate if not perhaps revealing) and placed one in each circle of the Zen, I mean Venn, diagram. I got to look at the labels. The kids didn't.

The object is for the students to identify the two labels. They name blocks one by one, and I place each piece where it belongs: in the overlapping section of the diagram if it fits both labels; in one of the circles but not the other if it matches just one label; or outside both circles if it matches neither one. For instance, a small green triangle goes inside a circle marked "small," "green," or "triangle." Students use logic and the position of blocks in the diagram to determine what the labels CAN and CANNOT be.

Yesterday, after just three blocks, we had the following situation:

In the left circle, but NOT in the overlap, was a small blue rhombus. (A rhombus, for those not in the know, is not a method of transporting rhoms; it is a four-sided figure in which all sides are equal.)

In the overlap between the circles was a large blue rhombus. And outside both circles looking in was a small blue triangle.

"Talk to each other," I said. "Tell your partner what the labels COULD be and what they COULDN'T be. Then share your ideas with the rest of us."

{WARNING: SPOILERS AHEAD. You may wish to see if you can solve the problem on your own based only on this information. Remember, labels name only colors, shapes, and sizes, and we only choose two labels. Read down when you're ready...}

After the partner conversation, one of the third graders raised her hand. "I think I know what it is," she said. "This circle"--and she pointed to the one on the left--"is the circle for rhombuses. And this circle"--and she pointed to the one on the right--"has to be for big blocks."

"Why couldn't they be for blue blocks?" I asked.

"Because we tried blue with the blue triangle," she said, "and the blue triangle didn't go in either of the circles. So it can't be blue."

I asked a couple more questions like that, inquired if anyone had other ideas, and then turned back to the girl who had spoken first. "How sure are you?" I asked.

"I'm pretty sure," she told me. "Maybe 80% sure. No, 90%." (I like having kids this age express "sureness" in percentages. They seem to like it too.)

I drew a quarter out of my pocket and examined it closely. "90% sure is pretty sure," I said, "but it isn't certain. We only have three blocks so far. It's kind of early to be naming both labels, don't you think? I'm thinking it COULD be something else. I'm thinking it probably IS something else." Pause. "What do you think?"

"Umm." The girl frowned and looked back at the diagram. A classmate next to her whispered something. The girl nodded. "I still think I'm right," she informed me.

I tossed the quarter into the air and caught it nonchalantly. "I have a quarter here that says you're wrong," I said. If the labels were "rhombus" and "large," I explained, the quarter would be hers. (That got everybody's attention.) On the other hand, I added oh-so-casually, if she was wrong she would owe ME a quarter.

"Don't do it!" somebody hissed at my, ah, victim, just as someone else leaned in close to her and said "Go for it!"

"All right," she said, rolling her eyes, "you can have my allowance..."

As it turned out, of course, it wasn't necessary. We went through her reasoning, failed to find any holes (bummer, man), and revealed the labels in the Venn, I mean Zen, diagram. The girl's reasoning had been one hundred percent correct, and she had stuck to her guns despite my best attempt to rattle her. I handed over the quarter as the class cheered and surrounded her to offer their congratulations to the kid who had, if not broken the bank at Monte Carlo, at the very least outwitted the Math Guy.

I'd lost (again). But though my pocket was lighter, I was convinced that the reasoning and confidence the child had demonstrated during the lesson had been worth the very real financial hit to me...

And at any rate, now you know why I am not (generally) a gamblin' man!

If Left and Right Are Opposites, What About Remaining and Wrong?

2009-11-09T12:53:00.006-05:00

"Look at the four cards I just gave you," I instructed the first graders I was working with today. We were warming up for some measurement work by reviewing some concepts from last month. "Look at the numbers on the cards. Show me an odd number...good job. Show me an even number...excellent! Which number is the least? Show me a number that's between 5 and 9." And on it went like that, culminating in the following exchange:

Me: "Okay, now find the greatest number. Put that card here in the middle of the table."

Children: [follow directions]

Me: "Now, look at the cards that you still have. Which is the greatest of the numbers that are left?"

One Child [looking back and forth at the three remaining cards, face up on the table]: "Which way is left, again?"

Oh, to have directional words that have just one meaning. That'd be great, right? (Wait...which way is right, again?)

Corn, Revisited

2009-11-01T22:29:00.007-05:00

I promised to write more about the corn project (see entry of October 13). Picking up the story from there:

Once all the students had the complete and accurate number of kernels, we assembled in the Chapman Room. "Who thinks they have the MOST kernels of anyone in all three classes?" I asked. Several people were pretty sure the honor was theirs, but one young man from Jan's class took the prize: he had 644 kernels on his ear of corn, a full 43 more than the next runner-up.

"Okay, how about the LEAST?" We had a few who coulda been contendahs, but again one student won out--another of Jan's students, down at 289.

All right. We had the greatest and the least values. One way of describing a set of numbers, I explained, is to find the range: the distance between the least and the greatest. (This tells you roughly what kind of a spread you have in the data: are the numbers generally pretty far apart, or are they mostly close together?) As a group, we estimated the difference, then subtracted to find out. "Close together, or far apart?" I asked when we had our result.

"FAR APART," chorused 48 voices.

How right they were. The range was--quite large. Taken together, the two lowest figures were less than the highest. There's plenty of variation among ears of corn, evidently.

Next we turned our attention to the median, or the center value when the numbers were all ordered. We had the students sit in a line--well, technically a curve--arranged from 289 up to 644. When everyone was in order I had them all stand and look around. "Where do you think the median value is?" I asked. "Point to the person who you think had the median amount of corn kernels."

Fingers waved toward the middle of the line. Most people in the middle of the line pointed to themselves. To find out the real answer, we started at the outside of the line and had students sit down two by two: 644 matched with 289, 601 matched with 293, and so on. Like a very slow row of falling dominoes, or perhaps like spectators doing the wave at a baseball stadium, they sat down, or fell down, depending on their level of coordination and their penchant for dramatics. Little by little, the number of children standing diminished. The 500s disappeared altogether, so did the 300s. The upper 400s took their seats. People began revising their predictions.

Before long, we were down to two students. One had amassed a total of 408 kernels. The other had--412. There was an even number of people. The answer, someone realized, was to split the difference, and that's exactly what we did. The median was 410. If you wanted to choose one number to stand for all the numbers in the group, you could do a lot worse than choose 410.

(The picture below shows the Final Two. Everyone else has been eliminated from contention as the Merry Median of the Corn Kernels. Thanks to Jan for the photo.)

One more project remained. You've heard of the Living Flag? Well, this was to be a Living Histogram. (A histogram has nothing to do with allergies--it's a bar graph in which the bars stand for a range of numbers rather than a single figure or response.) We had the students divide themselves into groups, according to the number of kernels: up to 299 over here, 300-349 over there, 350 to 400 in that corner. Then we called the groups over one by one and had group members sit in a line, creating eight lines of varying lengths in all. "What do you notice?" I asked, and they noticed quite a lot. The longest line was in the middle, they explained, the shortest lines on the outside. It was like stairs, someone said; it was like a roller coaster, said someone else. They were quite right, too. It was about the normal-est curve I'd encountered in the last few months--the nice bell shape you read about.

(Here are the lines. You might recognize the two almost-median-winners, smack dab in the center of the longest line there in the middle of the photo. See how neatly all these things work out?)

So, a nice way to spend a misty, mathy morning. The kids enjoyed getting their minds around the concept of range and median--and did it very well, I might add. They were surprised to see how big the range actually was, and they very much liked using their own bodies to locate the median. And while some of the players were beginning to get a bit restless toward the end, they kept their sense of curiosity about the graph and loved the idea of constructing it themselves. We'll continue to explore range and median--and who knows, we may get back out to the Chapman Room with a different set of data someday!

Mathtubs

2009-10-31T15:28:00.005-04:00

Give me a mathtub each morning,
Give me a mathtub at noo-oo-oon,
Give me a mathtub each evening,
But give me a mathtub soon.

One of the little perks of my job is distributing the mathtubs each Friday. {See the link below for more info about these tubs--they're boxes filled with math games, math materials, suggestions for math-related projects, and picture/storybooks with math connections; kids take them home for a few days at some point during the year.}

I walk into a classroom, around about the time pizza is delivered, carrying one or two tubs, and deliver the tubs to the children (CHOSEN AT RANDOM--NOTHING UP MY SLEEVES) who will get them for the next few days.

Children vary, of course, in how they express their excitement over getting the tub (from a small, self-satisfied smile up to an enthusiastic fist-pump and a chanted "Oh yeah, oh yeah, oh yeah"), but about 97% are very pleased to have their turn. It's quite gratifying.

The other children, meanwhile, are full of helpful comments such as "Is it my turn yet?" "When is it my turn?" "She's so LUCKY," and so on. For the moment, at least, the arrival of the Math Guy and the Math Materials outranks everything--even pizza. No small feat.

I thought I had invented the mathtub idea, or at the very least, um, repurposed it from a similar idea I'd read about somewhere in which teachers sent books home in backpacks for kids and families to enjoy. A couple of years into the mathtub project, though I was cleaning out some old papers and discovered to my surprise that in 1997, while at a conference in Rochester (NY), I had actually attended a workshop in which the presenter was describing how teachers could package up some math materials for use at home. I still take credit for the name "mathtubs"--I think that teacher used shopping bags or something similar--but as for the concept, well, I should know by now that there are few truly original ideas in education. Hey, it works, and the kids enjoy it, and that's what counts--right?

You can read more about mathtubs here, in an article I published in a teacher magazine a few years back:

http://www.highlightsteachers.com/archives/articles/the_mathtubs_are_coming_by_stephen_currie.html

People v. Tables

2009-10-30T11:45:00.003-04:00

"I am going to have a party," read the question given to a number of our 1-2 students the other day. "I want to invite ___ people." (The blank is standard: everybody gets a different number, which a) cuts down on the Problem of Roving Eyes and b) allows us to give somewhat harder numbers to kids who are ready for a challenge while keeping the same problem frame for everyone.)

"I have ____ tables where my guests can sit," the problem continues. "Each table has room for _____ people. Do I have enough tables, or do I need to get more?"

Different kids had different ways of attacking the problem, as usual. Some sketched the tables, drew chairs around them, and counted by ones. Others dispensed with the chairs and simply wrote the number at each table, then counted by that number if they knew how. A couple didn't bother with a sketch at all. One or two made groups with checkers or other materials--7 groups of 6 checkers, for instance, to represent 7 tables with 6 people at each--and then checked the number of people to see if they'd gone over or not. The strategies were generally quite accurate, if not consistently efficient: the next step will be to move kids away from the pictures and toward more abstract skip-counting and other strategies.

At any rate, children needed to show or describe their work and then answer the question (which, if you recall, had something to do with whether there were enough tables or whether we needed to get more). Several children didn't recall--they needed a reminder to do this part--but eventually we had the answers we sought.

"I have enough tables," one child wrote confidently and accurately. (Actually, she wrote "enuff," but let that pass...)

"You have enough tables. Am I rite?" wrote another child, perhaps a little less confident than the first. (Yup, I told him, you're rite. Um, right.)

"You need to get more tables," wrote a third responder, "because seven tables is going to be a smaller number of people. You need 8 tables." She included a careful sketch with the correct number of heads jowl by jowl at each table: an arrow then pointed to the last table, with the helpful label "extra."

"I have to sell one more chair," wrote still another girl. A somewhat convoluted way of saying that she not only had enough tables--she had an extra seat. I'm not entirely clear whether the sale would be an auction for the right to attend the party, or simply an attempt to convert an unwanted and unnecessary item into cold hard cash. Either way, this is a girl who knows the value of a buck.

And perhaps my favorite: the boy who discovered that he had space for 52 when he only needed to seat 49. After showing his method, he concluded: "You need more people."

Virtual Manipulatives

2009-10-26T22:14:00.006-04:00

We sometimes haul out the laptops during lower school math times and have kids work with Utah State University's National Library of Virtual Manipulatives website. We've made good use of this site for projects with both the 3-4 and the 1-2 classes, but my personal favorite is the subtraction.

See, you get these rods and cubes, just like base blocks only they're on the screen and exist only in pixel form, so they don't fall off the table and get lost and they can't be used as hockey sticks and pucks, or as drumsticks or grenade launchers or whatever else creative minds have in store for them.

--Oh, and then when you model regrouping (which I prefer not to call "borrowing," as I've said before, because you don't ever give it back--I prefer to use the phrase "stealing") you actually grab one of the virtual tens rods and bring it over to the ones column and let go and watch as it separates itself into ten little ones cubes.

Then you hear the kids saying WHOA! and COOL! and NEATO TORPEDO! (well, not that one, maybe) and the like.

Then you get to separate hundreds into tens the same way and thousands into hundreds and the whole thing is utterly charming and truly awesome and the best thing next to...

[Down, boy.]

[The picture below isn't actually from the virtual manipulatives website--it's from a powerpoint presentation I made dramatizing the process. What you see here is the ones stealing a ten, in the dead of the night of course, dragging it back to Ones Street, and breaking it into ten little ones cubes so there'll be enough ones to carry out the subtraction.]

Anyhow, children sometimes ask how they can get to the site at home. Unfortunately, the address isn't straightforward. If you google "virtual manipulatives," it's the first site that comes up (as of today, anyway).

The whole site's URL is http://nlvm.usu.edu/en/nav/vLibrary.html. If you're interested, take a spin around the site with your child(ren). It may not be the equivalent of a medieval European cathedral, but as the Michelin guide would put it, it's quite definitely worth a visit.

Corn

2009-10-13T22:46:00.007-04:00

How many kernels on an ear of corn? we asked the third and fourth graders the other day. They've been studying the Mayan people, who called themselves "People of the Corn," so it was a worthwhile question.

We started by having students find approximations; as you should know by now if you've been reading this blog, us Math Guys consider this a very important step. We asked students to choose a round number (a number that is a multiple of 10); the point, after all, wasn't to guess the exact number, but to use a number that makes some sense and is relatively easy to work with. You can always revise your estimate later, we assured them.

What is the estimate based on? Well, we gave them each an ear of dried corn to eyeball. Some did some quick-n-dirty calculations, fourth graders in particular. (Yes, we asked them to justify their reasoning. Some of them HATE this, but it's oh-so-good for them.)

"About 20 in each row," wrote one student. "Maybe 10 rows. 10 x 20 = 200. I estimate 200 kernels in all."

"I think there are 20 rows and 30 in each row," reported someone else, "but that might not be enough so I added a few more. I say 640."

"I think 260," wrote a third grader, who would have been happy to leave it at that, but who added, under duress from a teacher, "because it looks right. And because it's a good number." We might call this strategy "Pick-a-large-number, any-large-number, and-assign-it-great-virtue-so-critics-will-be-cowed."

The next step: Count the kernels! The classroom teachers had prepared egg cartons with ten cuplets (better them than me). Kids used their fingernails to push the kernels off the cob (great fun). Then they distributed the kernels 5 or 10 at a time into the cups, making groups of 50 or 100. Record the number, dump out the kernels, lather, rinse, repeat.

At some point along the way several students noticed that their estimates weren't looking as accurate as they had back before counting had begun. This was especially true for those whose initial strategy had been "Pick-a-large-number, any-large-number &c," but other more careful estimators ran into this difficulty too. No problem! we said. Just revise your estimate, record it--oh, and explain why you wanted to change your original prediction. (My favorite: "Because I passed my first estimate a long time ago.") You will no doubt be shocked to learn that the second set of estimates were considerably closer than the first.

Eventually, all corn kernels were off the cobs and had traveled through the eggcups and into plastic bowls or paper bags (except for a few strays which had found their way onto the floor), and everyone had an exact answer. Some were surprised to see how many there were. Others found the results unsurprising in the extreme, or claimed they did: "I knew it," crowed one boy whose answer was not, perhaps, as close as he thought.

As students finished, they compared their totals with friends and thought about questions such as Why aren't all the totals the same?, What could you do to get a better estimate next time? ("Nothing," said the young man quoted above), and About how many kernels do you think there might be in the whole class?

So, three-digit numbers, ordering, estimating, grouping by tens, fives, 50s, and 100s, and explaining reasoning. Plus, a fun project (there's something truly satisfying about flicking those kernels off the cob, and something even more satisfying about running your fingers through a nice big tub full of everyone's kernels), and one that relates to science and social studies. A worthwhile math period indeed. Next up: data analysis with these results. On Thursday we'll be in the Chapman Room calculating the median and range of the data and forming a Living Histogram. Pictures to follow, assuming my camera behaves itself...

p-a-t-t-e-r-n-s

2009-10-09T16:24:00.006-04:00

What do you call it
When things repeat?

We call it...
A pattern.

Head, shoulders, knees, and feet,
Head, shoulders, knees, and feet,
That
Is a pattern.

A, B, C, A, B, C, A, B, C, A, B, C
That
Is a pattern,

1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3
That
Is a pattern.

What do you call it
When things repeat?

We call it...
A pattern.

Pizza!

2009-09-30T22:53:00.010-04:00

As many of you know, the third and fourth grade classes order pizza each Friday. Children throughout the lower school put in their order; runners from the 3-4s pick up the orders and the money, determine the number of pizzas to buy, and hand-deliver it when it arrives.

Pizza is a major undertaking. There are times when we teachers wonder whether it is all worthwhile, especially when we discover that one class is short $15 or that a dozen or so children neglected to sign up until the pizza, you know, arrived... BUT we continue to do it because pizza a) tastes good, b) is convenient for parents, and c) IS A GREAT TOOL FOR PRACTICING MATH SKILLS. Of the three, c) is by far the most important in my book, though your mileage may vary.

How does pizza relate to mathematics? Glad you asked. Let us count the ways...

1. Pizza order takers get good practice in counting money and determining if it matches the number of slices ordered (hint: it does only about half the time).
2. Students get practice in giving and making change.
3. Students round the number of slices ordered per class to the nearest multiple of 10 to make estimation easier.
4. Kids practice addition skills by calculating the total number of slices ordered.
5. We look at number patterns. Hmmm: when a class orders 14 slices at $1.50 per slice, we get $21. Interestingly enough, 14 plus half-of-14 equals 21--the same number, only in regular numbers rather than in money. Now why would that be?
6. Especially later in the year, we use pizza as a real-life example of multiplication--if there are 8 slices per pizza, how many slices in 5 pizzas? In 7 pizzas? In 13 pizzas?
7. Kids calculate the profit for each week's worth of pizza: if we charge a dollar-fifty per slice after buying it for [sorry, trade secrets removed--suffice it to say "less"] per slice, how much money is left over? What operation can we use to calculate it?

And there are many other ways we mathicize pizza, especially this year, but I've been typing all day and my fingers are about to fall off. So you'll have to wait for another post. Sorry! In the meantime, how about some pictures? ...Yes, yes, the very thing!

Some of the proceeds, up close and personal:

This young man is clearly enjoying himself.
Think Scrooge McDuck.

Doublechecking that the amount of money from one of the 1-2 classes actually matches the number of slices ordered:

One of the "Grand Totalers," making bundles of ten for easier counting:

And at last, the fruits of our labors--or seven eighths of them at least (did I mention that pizza and FRACTIONS go well together? No? Consider it mentioned...):

Getting to School

2009-09-30T22:34:00.008-04:00

If you're like most of us, you've always wanted to know how kindergarteners get to school. Lucky you! Now's your chance to find out--because the kindergarten recently put together a graph showing that information. Voila--!

(You can click on the picture to enlarge it.)

The kids enjoy the drawing, of course (some of them enjoy it quite a lot--I'm very fond of the multicolored, creatively shaped cars at the top of the cars column, along with the 3-wheeled truck on the far right). But it's also a good learning experience for these newly minted K students.

There are the reading-the-graph questions, of course:

*Which way of getting to school was the most common?
*Which was second most common?
*How many children came to school by truck?
*Which way of getting to school was used by 7 children?

And the interpreting-the-graph questions too:

*Suppose we asked the same question tomorrow and made a graph about that. Would the graph look exactly the same? Mostly the same? Not at all the same? Why?
*What if we made a graph showing how people got home? What do you think that would look like? Why?
*Why do you think no one got here on a skateboard? A surfboard? A motorboat?
*Do you think the school needs a bigger parking lot? Why?

But mostly, there's the notion that you can take information and show it in a way that makes it available to everybody who comes along. You can tell, just by looking, that more people in your class come to school by bus than come by van, and that a LOT more come by car than come by truck. You can locate your own name (or your own truck) on the graph and show a friend how you got to school that day. You can find out how a friend arrived. The information will be there today--and tomorrow--and the next day--and it will remain available forever, or at least as long as the teachers choose to hang it in the hall. Knowledge is power, we like to say; and graphs, I believe, are a really good example of that adage.

n (or maybe n+1) Flies on the Wall

2009-09-22T08:35:00.009-04:00

The 3-4 classes generally begin the year with work on number sense, number patterns, place value, and number in general. This year we're starting off with some projects involving functions and some simple algebraic ideas.

Here's a fly-on-the-wall view of an introductory lesson (shh; don't let them know you're in the room):

Teacher: Suppose we choose a number from 1 to 100. We'll call that number n. We often use the letter n in math to stand for any number. Someone pick a number for n--

Student: 38!

Good enough. So if n is 38, what's n + 10? 38 + 10, right? Which is--

Students: 48.

That's right. Okay, let's make a table and try it using some other numbers for n:

n n + 10 Result
--- ------- ----
38 38 + 10 48
17 17 + 10 27
90 90 + 10 100
45 45 + 10 55

Looks good. Okay, what patterns do you see? How does n change when you add 10?

Students: The ones digit stays the same.

Yeah? Always, or only most of the time?

Students, a bit hesitantly, because you always have to watch out for trick questions: Always...always so far, anyway.

That's right. Can you think of a number n where the ones digit would change after you add 10?

Students: several suggestions, all of them withdrawn upon further reflection.

...Why doesn't it change?

Student: The number 10 has 0 in the ones column, so it doesn't change the ones.

Another student: Oh, and when you add ten on the hundreds board you just go down to the next row, so if you're in the threes column you stay in the threes column...[We use the hundred board a lot; one is pictured here.]

What happens with the tens? The tens go up? Good; by how much? By one? Always, or only sometimes?

Students, less hesitantly than before: Always.

How do you know? So, okay, let's put the rule into words: When you add 10 to a number n, the ones digit stays the same but the tens digit goes up by one.

Nice job! Okay, let's try it again, only this time we'll look at what happens when you add 11 to n.

n n + 11 Result
--- ------- ----
12 12 + 11 23
28 28 + 11 39
77 77 + 11 88

Student, bursting to be the first: I know, I know! I know the rule! It's the tens digit goes up and the ones digit goes up too!

Student, bursting to be the second: Yeah! It's the tens digit goes up and the ones digit goes up too!

By how much? Let's say it as a rule.

Students, cautiously: It goes up by one in the tens column and one in the ones column.

Always, or just sometimes?

n-2 or n-3 students, where n is the total population of the class: Always.

Two or three students: Sometimes.

Why sometimes?

2 or 3 students: Because what happens when the number is in the nines? Say you add 11 to a number like 59...

2 or 3 more students: Ohhh!

Let's extend the table--

59 59 + 11 70
69 69 + 11 80

n/2 students: It goes up two in the tens!

The other n/2 students: And it goes down to 0 in the ones.

Okay, let;s make the rule. Help me out here:

[And so we develop the rule: When you add 11 to a number n, the tens digit usually goes up one and the ones digit goes up one as well, EXCEPT that when the ones digit is 9, the tens digit goes up by 2 and the ones digit goes back to 0. We talk about why this might be the case--and then out go the students to work on developing rules for n+1, or n+19, or n-2, or perhaps even n x 5...]

Okay, class is over for the day. You can come down from the wall now! Aren't you glad none of the kids brought flyswatters today??

[Edited to add: I should note that this lesson is adapted from a set of activities in a new book by math education guru Marilyn Burns. In 2008, I spoke at a national conference of math teachers. I was disappointed to discover that I was scheduled at the same time as Marilyn, which was disappointing for two reasons...first, I didn't get to hear her, and second, hardly anybody was left to come to my workshop...]

Learning from the DVD...Player

2009-09-18T11:27:00.003-04:00

Teachers of today can choose from a wide array of technologies to spice up their lessons and increase students' understanding. There's Powerpoint, of course, and calculators, smart boards and video cameras, wikis and spellcheckers, voice-to-text programs and DVDs, Excel spreadsheets and Activote systems, GPSes and, um, electric pencil sharpeners; the list goes on.

Most of these educational technologies get plenty of respect within the educational world. (Well, maybe not the pencil sharpeners.) Whole conferences are organized around these technologies and how they can help teachers do a better job of preparing students for the 21st century [Q: At what point will we start saying "preparing students for the 22nd century"?]. BUT there is one technology that is sadly overlooked. It is the Rodney Dangerfield of the educational technology world. I refer, of course, to the lowly DVD player. Not the DVD; the player.

"How did you know so quickly that 8 + 8 was 16?" I asked a first grade girl earlier this week. (If this question sounds familiar, it's probably because you read the previous entry in this blog.)

"Well," she said, "we have this DVD player at home and it has arrows. And if you want to speed through the movie it says 2, 4, 8, 16, 32, and then it goes back to 2 again. And I know that 2+2 is 4 and that 4+4 is 8, so 8+8 must be 16, and I guess that 16+16 would be 32. But then the pattern stops because it goes back to 2 and 32+32 is...something, but it isn't 2."

What can I say? Clearly, we should as a nation reduce our spending on old-boring-and-ineffective technologies such as computers, projectors, smart boards, and digital cameras, and load up classrooms instead with DVD players. Who's with me?

--Actually, this is a really good example of a child not only noticing but using math in everyday life. No one taught her that 8 + 8 was 16. She was struck by a sequence of numbers that appeared in her environment, and spent time and energy deciphering the pattern--learning, and evidently mastering, the fact that 8+8=16 along the way. This is the kind of thinking we always want to see in our students. As our report form puts it, one of our goals for children is that they "recognize and construct mathematics in daily life." It's lovely to see such a clear example.

That's All She (w)Rote

2009-09-14T14:31:00.004-04:00

You have 8 cubes, I say.

The child, a first grader, nods happily. He's just counted them, accurately, and showed me how you could split them up so that we each had the same number (4 apiece, if you were curious), and answered several other questions about them as well.

What if you had 8 cubes and I had 8 cubes too? I ask. How many would we have in all?

This isn't necessarily an easy question for six-year-olds, and they vary in their approaches--also in the speed with which they answer. 28, says the boy, just as automatically as you please. There's no lack of confidence here. (Not a lot of accuracy, either, but hey, it's still early in the year.)

28? I ask, just to make sure.

28, he says. No. I mean, um, 34. Yeah, 34.

34, I repeat, resisting the temptation to ask, Regis-style, whether this is his final answer. And how did you know?

Oh, I didn't know, he says with a grin. I guessed.

Okay, I say, and go on to do a few more activities with him. I wrap up with a nice open-ended question: What else do you know about math that you'd like to tell me?

Well, he says eagerly, one thing I know is that 8 plus 8 is 16...

SMACK! goes my hand (metaphorically at least) against the side of my head.

This little anecdote nicely illustrates the difference between knowing a fact and KNOWING it. This boy knew that 8+8 was 16, but he didn't KNOW it--that is, while he could repeat it, he couldn't use that information in a real-life context. His verbal knowledge isn't yet supported by his conceptual understanding.

There's nothing wrong with learning some kinds of things by rote. Indeed, sometimes it's necessary. It's just that you have to be careful with kids and not automatically assume they KNOW everything they know....if you know (KNOW?) what I mean!

Spiiiiiiiiders

2009-09-13T14:37:00.005-04:00

So there I am in the Pre-K, "just visiting" as they say on the Monopoly board, and the children are doing watercolors, and one child brings over her picture to show me.

"I painted a tabantula," she explains, her eyes as wide as a four-year-old's can get. Wider, if possible.

"A tabantula, huh?" I say. "Sounds mighty scary."

"It has eight legs," she says, and proceeds to count them, which she does very well (us Math Guys notice these kinds of things), and lo and behold, guess what, there ARE eight.

"Well, that's a good thing," I say, "because taban, I mean, taRANtulas are spiders, and spiders are supposed to have eight legs. Good for you for knowing that. I guess you're an expert on spiders."

She ignores this comment as the typical babbling of the Adult and points instead at a swirl of red paint. "That's the tabantula's head," she explains. "Do you want to know what that red is for?"

"Tell me," I say.

She leans in very close, stretches up, finds my ear, and stage whispers "IT'S BLOOD."

Can't wait for Halloween!

The whole nine yards; or, Dressed to the nines

2009-09-09T09:10:00.002-04:00

If I'm timing this one correctly (and I'm probably not as the margin for error is not exactly huge), this post will be time-stamped

09 [month]
09 [day]
09 [year]
09 [hour]
09 [minute]

or

09/09/09 at 09:09

The nines have it!

[Edited to add: Ooh! Missed by ONE MINUTE. Oh well...I tried.]

SummerMath, Part 4: On the Road Again

2009-08-04T18:05:00.004-04:00

If you're like most Americans, you and your children spend a lot of time on the road. There are plenty of ways for kids to pass the time on car trips, whether of 5 minutes or 500 miles. Several of them I'm quite sure you know:

Screaming "She's on MYYYY side!!!! Mom, make her get off of MYYYY side!"

Asking as often as possible, "Are we there yet?"

Threatening to throw up.

Singing vaguely risque songs like "The boys and girls are kissing in the D-R-A-K, D-R-A-K, D-R-A-K, Dark!"

Singing "George Washington Bridge." OH MY EARS QUICK RINSE THEM OUT WITH RUBBING ALCOHOL (George Washington Bridge, for those of you not in the know, repeats the lyrics "George Washington Bridge" again and again and again to an essentially unmelodic melody. My sister pulled this one on a 5-hour car ride between Chicago and La Crosse, Wisconsin, many years ago, and when she was finally told she could NO LONGER sing "George Washington Bridge" she said "Okay, then I will sing 'Adobe Bricks.' 'Adobe Bricks,' as it turns out, is 10 times worse. We eventually threw her out of the car)

But those days are over, for here is a list of WONDERFUL MATH ACTIVITIES that you can do in the car. (Assuming that your children can actually see out the windows.)

1. LICENSE PLATES. In New York State, most (not all) plates are of the form AAA 1111--three letters, then four numbers. Have kids hunt for license plates withe certain characteristics. Who can find one that has four even digits? Four odd digits? Four digits that are ascending (like 4689)? Descending (say, 8521)? Two digits that are the same? Three digits that are the same? If you're stuck at a stoplight, ask your child to add the four digits on the license plate of the vehicle in front of you--which two are best to add first? What shortcuts can your child find? (To add 6364, kids might start with 6+6=12, a "doubles fact," or with the "ten-friends" fact that 6+4=10.) Estimate by looking if the sum will be greater than or less than 20. Then check. Whoops, green light--better move on...

2. STOPLIGHTS. How many stoplights do you think there will be between here and the mall/camp/Grandma's house? Let's keep track. Will they mostly be green when we get to them, or red? You count the red ones, I'll count the green ones. Do you think it'll be about the same on the way back, or will it be different? Let's find out. For a route you drive frequently, choose a couple of lights and keep track of whether they're red or green over a period of 8-10 days. These are exercises in counting; they also ask kids to gather, use, and interpret data. More than half of the lights are green? Why do you think that might be?...That light where we cross Route 55 is almost always red when we get there--how come?

3. VROOM, VROOM. On a 4-lane highway, have kids count the cars you pass and the cars that pass you. Make this an exercise in counting forwards and backwards: start with a score of 10, add one for every car you pass, subtract one for every car that passes you. Or, start with 50 or even 100. Try not to give in to your children's pleadings to do whatever it takes to avoid being passed by that in-your-face Oldsmobile or to overtake that weirdly painted appliance truck. Does it matter who's driving?...why yes, yes, it might. (And can we correlate that with speeding tickets received? Why yes, yes, we can...)

4. NUMBERS ON SIGNS. There are lots of these running around the roads: speed limits, mileage markers, route numbers, distances to upcoming cities. We call Route 376 "route three-seventy-six," but what's its "proper" name? (Three hundred seventy-six. Yes, I know it isn't *really* a number, because it doesn't indicate three hundred seventy-six of anything...) Who can find a two-digit number on a road sign? A three-digit number? An odd number? The sign tells how many miles to Montreal and how many to Buffalo. Which is further away from us right now? How do you know?

5. MENTAL ARITHMETIC. We're at milepost 27--look, there's the sign. What milepost will we pass in five miles? Ten miles? (Careful--are we driving towards milepost 0 or away from it?) How many more miles till milepost 40? (See above.) The sign says Albany is 65 miles away. Our speed is, guess what, 65 miles per hour. About how long till we're in Albany? (Only use easy numbers for this kind of question!) The car can be a good place to go over basic facts as well: "7 + 2." "9!" "Tell me two ways to make 10." "Um--"

6. MAPS and DIRECTIONS. Have children direct you to a location they've visited many times before. Ask them to tell you where to go straight and where to turn, and whether you should go left or right when you turn. Obviously, don't break any traffic laws--you are still the captain of the ship! As for maps: Print out a map showing your route to a (relatively) nearby place. Go over the map with your child before you leave. Have your child hold the map and try to track your position along the route as you drive. Try using a road map for longer distances: Find a long thick blue line with a shield and the number 84. That's the road we're on now. We're heading west...which way is west? Can you find a city called Middletown? Excellent--we're just a little bit west of that right now. What's the next town you see? Some third and fourth graders can become quite good at navigating. Just be sure that the "road" they're having you follow isn't just a marmalade stain on the map (this happened once to Paddington Brown and his family, I believe).

As always, these are ideas, nothing more; you can come up with others yourself. Be sure not to push too hard. REPEAT: BE SURE NOT TO PUSH TOO HARD. If you find you're suddenly more invested in these activities than the kids are, cut the games short and do something else: talk, sing, tell jokes, look at scenery. But if people start yelling about siblings being on theirrrrr side, it helps to have games like find-a-license-plate-with-4-odd-digits in your back pocket; and if you can find the right combination of math activities for the car ride on any given day, you will NEVER EVER EVER have to put up with anybody singing George Washington Bridge, and that will make any hardship worthwhile.

My Heart's in the Heartland

2009-08-04T10:23:00.009-04:00

A cool map for you geography geeks out there:

http://strangemaps.wordpress.com/2009/07/27/402-homeland-is-where-the-heartland-is/#comments

Note that some of the sizes are a bit off. Also, something seems to have happened to Dutchess County. STILL!

More Rabbit News

2009-08-04T10:13:00.003-04:00

--> NEWS FLASH <--

THE MATH GUY'S RABBIT CAN MULTIPLY!!!!

(Technically, this is NOT the Math Guy's rabbit. It belongs to his daughter, who purchased it from a pet store a couple of years ago and then hid it in the garage for three days while she worked up the nerve to tell us what she'd done. However, now that she's away at school, guess who does the bulk of caretaking for Miss Rabbit? --That's right.)

Anyway. The other day I was communing with the rabbit and asked it a question, offhandedly:

"Rabbit!" I said. "What is 3 x 5?"

And darned if she didn't woffle her nose 15 times.

"Remarkable!" I said. "And, Rabbit, what is 2 x 4?"

I counted eight woffles.

"Such a mathematical rabbit!" I crowed, and took her through several more examples: 6 x 2, 4 x 4, and 2 x 10. Woffle, woffle, woffle, and she was right on the money EVERY TIME.

My family seems suspicious of her awesome abilities. They seem to believe that I am cheating or perhaps misinterpreting her responses, but I swear she woffles the correct number of times. Anyhow, who cares what they say. I'm considering taking her on the talk show circuit--your thoughts?

Of Rabbits and Math Guys

2009-07-21T18:16:00.007-04:00

Several years ago, early in my incarnation as Math Guy, I walked into Sue's third and fourth grade classroom ready to present a lesson. I was surprised to see that a bunch of rabbits had replaced the children that day.

The class had been reading a novel about rabbits or rabbitlike creatures, Sue explained, and several children had come up with the idea of dressing like rabbits one day, and the idea had met with approval from basically everybody.

Some had done just the basics--a few face-paint whiskers, a kush ball for a tail. Others had added a carefully-stapled set of ears made from construction paper. A few had gone whole hog (whole bunny?) and dressed all in white or brown or black with socks and slippers and even gloves. They looked...different. They looked...creative.

"Greetings, rabbits," I said, and asked them to take their seats so we could begin the math instruction for the day. For rabbits, they did reasonably well sitting still, and they did an even better job of listening (must've been the big ears).

My planned lesson was on what kids often like to call "timesing." We began by reviewing some basic multiplication facts and then moved on to multiplication strategies and the link between multiplication and addition, and just before I sent them to the tables to do some independent work, it suddenly occurred to me that I was--

--that's right--

--teaching rabbits to multiply.

Bada-bing!

True story, too.

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

2009-07-16T15:31:00.015-04:00

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...

A Games-in-Education Site

2009-07-14T10:01:00.004-04:00

This one is intended mainly for teachers who want to make better use of games of all kinds in their classrooms (and I'll be passing on the link to lower school teachers, of course), but parents should be able to benefit from it too. I recommend it!

http://www.gamesforeducators.com

Happy July 14, by the way. In, let's see, 12 years this will be a special date of its own: 7/14/21. No prizes for guessing the pattern, but you might try it out on your third grader.

SummerMath, Part 2: Card Games

2009-07-07T16:39:00.007-04:00

I wrote a few days ago about board games. Now it's the card games' turn. What can I say? They whined and groaned until I HAD to include them...

Card games are if anything even more math-y than board games. In fact, cards themselves are pretty solidly mathematical. Consider: There are 4 suits with 13 cards in each; there are 4 seasons in the year, with 13 weeks (give or take a day here and there) in each. Coincidence? Nah!

Many of you know the game Crazy Eights, a version of which is marketed under the trade name UNO. This is a great game for getting kids to think about attributes--the different categories that cards fit into. On a seven of diamonds, for instance, you can play any diamond or any seven. Young children often scan their hands and then say with disappointment "I don't have any cards that will work." "You don't have any diamonds?" I'll ask. "No," they'll say. "And no sevens either?" "No--oh, wait!" The ability to keep two attributes (such as suit AND rank) in mind at the same time is extremely useful in math. In geometry, for instance, kids will need to know that a square is a kind of a rhombus and a kind of a quadrilateral (and on and on); in numbers, kids should recognize that 44, say, is divisible by both 2 and 11 (not to mention 4 and 22). So a hand of Crazy Eights before dinner is a nice painless way to encourage mathematical thinking--and knock off a few of those prerequisites for geometry, division, and more.

Okay, okay, the game War is exceedingly dull and involves no strategy whatever. I get that (boy, do I ever). But your 4-7-year-old is busy practicing concepts of greater than and less than while playing, which ALMOST makes up for the boredom issue. Ask questions as the game goes on, too. (And it DOES go on...okay, enough carping.) "Your 9 beats my 2...by a little, or by a lot?" "I'm going to put my card down first--oh, a 3. Do you think I will probably win with a 3? Let's check your prediction."

There are any number of variations on rummy. These games are especially good for third grade and up. Basically, players try to get groups of three (or more) cards that are all the same rank (as in three queens) or same suit and in a run (as in 4, 5, 6, 7 of spades). You pick up and discard various cards in an attempt to make these groups. We're talking strategic thinking and probability in addition to attributes and sequencing. Scoring requires adding the values of cards, too.

Then there are the approximately one zillion forms of solitaire. Many of these games deal with attributes, or with addition, or with sequencing; all of them are good for strategic thinking. The game Spit was extremely popular as a snacktime/rainyday activity for third and fourth graders last year; despite its unsavory name it helps develop sequencing skills, both backwards and forwards, and encourages kids to know what's one less or one more than a given number automatically. Concentration isn't much of a math game, but you can make it one by playing only with cards A-9 and having the object be to draw 2 cards that have a sum of 10. (Instead of matching two 8s, say, you match an 8 and a 2.) The same principle applies to Go Fish, another not-very-mathy-game, which becomes "Tens Go Fish" when you ask for a card that goes with one of your own to make 10.

A little more purely mathematical, but still fun: For younger kids you can try Close to 10 or Close to 20. For Close to 10, deal out 3 cards after removing the face cards from the deck. Focus only on the rank (ace = 1). Choose two cards with a sum that is as close to 10 as possible. How close are you? That's your score. Record it. Play 5 rounds. High score loses. You can play this cooperatively or competitively, which each player having a different set of cards. For Close to 20, use five cards and choose three, or try some other variation. This game is great for estimation, for practicing addition strategies, and again for strategic thinking.

Then there are various betting games. "Can we play Cash Cab poker?" one of Ellen's fourth graders used to ask me almost every day last year, and though the answer was usually "Not today," kids ages 7 and up very much enjoy the mixture of skill and luck in --> HIGH STAKES <-- card games. I don't advise using actual money, but counters work just fine. Here's a basic template, which permits a whole mess of variations:

*Remove the face cards (and sometimes the tens). Remind players that ace counts as 1.
*Deal each player a card face up. High card bets (or folds). Other players follow (or fold). (I generally don't do raises, but you can if you like.)
*Next, deal a second card face down. High card showing bets again; others follow.
*Finally, deal a third card face down. High card showing bets again; others follow.

Who gets the dough? Here are some possible ways to do it.

*Multiplication practice. Choose two of your three cards. Find the product (what you get when you multiply them). Greatest product wins the pot. Alternatively, play high/low in which players who have low cards still can win. Before revealing their cards, players announce whether they're going for high or going for low. Those who announce they're going for high reveal their products; highest product gets half the pot. Those who announce they're going for low do the same; lowest product gets the other half the pot. Sneaky, huh?

*Greatest 3-digit number. Or greatest 2-digit number chosen from the 3 cards. Or high/low. Which way should you order 4, 7, and 2 if you're going for high? Which way for low? Which gives you a better chance of winning?

*Greatest sum. Make a 2-digit number and a 1-digit number (so if your cards are 4, 7, 2 you can do 47 and 2, or 24 and 7, or...). Add them, mentally or with paper and pencil. Greatest sum wins; or do high/low...

*Make it 5 cards instead of 3. Your goal is to have the 5 cards that add to a total nearer 25 than anyone else. This one's especially interesting because what looks like a "good" hand early on may prove to be a "bad" hand as those nines and tens don't stop coming

Or other variations that you and your children come up with.

As before, these games should be considered an opportunity for some fun rather than a chore. They're games, after all. Be aware of when your child starts to squirm, or when the brain begins to turn off, or when the beautiful day outside is becoming more appealing than the king of hearts. But if you don't overdo it and play your cards right (hardy-har-har), these games can be great ways to help your child have fun--and practice a little math in the bargain.

Consecutive Numbers Alert

2009-07-06T13:21:00.005-04:00

My cousin Susannah, an elementary school teacher in Iowa, sent me this link:

http://news.yahoo.com/s/afp/20090705/od_afp/britainoffbeat

Since I use the standard US convention of month/day/year, July 5 '09 didn't seem so very impressive to me (07/05/09? Okay, whatever...), though of course I'm glad to know the scoop. However, I was awaiting the consecutive-number extravaganza that arrives Wednesday, the 8th of July:

07/08/09

And especially shortly after 4 that morning, precisely six seconds past the changing of the minute, when the time will be

04:05:06
on
07/08/09.

I would plan a celebration to mark such a momentous occasion. However, mathematical import or no, I admit that I intend to be asleep. Maybe we can schedule something for another year, when the consecutive numbers will appear at a more civilized hour. What are you doing at 09:10:11 on 12/13/14? What, your calendar's clear? Good! Save the second.

SummerMath, Part 1: Board Games

2009-07-04T09:33:00.013-04:00

Yes, I know it's summer, and very few of you are actually reading this, but those of you who are most likely saying to yourselves, "Selves, it's summer, and there's no school, and WHATEVER are we to do to keep little Ashley and little Braden from forgetting all they ever knew about that most important of disciplines, Mathematics??"

Fear not! says the Math Guy. Today I begin a survey of Ways to Keep Your Children Thinking Mathematically During the Summer Without Screaming or Excessive Boredom. Since this is a bit of a mouthful, I call it SummerMath for short. You can tell it's trendy and cool because of the way I combined two words with Capital Letters and NoSpaces, like StubHub and ExxonMobil.

Part 1 is Board Games.

Board games are great mathematical tools. They are wonderful ways of teaching and reinforcing math concepts from number sense to probability and from addition and subtraction to estimation. They're fun, too, and the math isn't always obvious, so if little Ashley and little Braden have already decided they don't much like math (which would be EXTREMELY DEPRESSING) they may not rebel.

A few examples:

MONEY. Kids love money. The like it best when it's real, but they enjoy it when it's pretend, too. Games that involve buying and selling help reinforce money skills. When they play Monopoly, for instance, kids have to count out specific dollar amounts (and they usually need to do so as efficiently as possible--$20 + $5 + $1 rather that 26 one-dollar bills, for example). They have to think about giving change. They have to focus on the difference between $200 and $2000--oh, those zeroes. They need to have some idea of how much ca$h they have on hand--enough to buy six houses, three houses, one house? And they need to compare: Who's really ahead? Is my stack of $10s more or less impressive than your three puny $50s?

ADDITION and SUBTRACTION. Money involves adding and subtracting, of course. Games with dice are particularly good for this, too. Rolling two dice and moving that total is basic to lots of board games and provides great practice for sums to 12. The game Sorry has a feature where if you draw a 7 card, you can split the move between two pieces--a nice way of developing fluency with these facts. Chutes and Ladders: "Ooh, I hope I get a 5; then I'll land on the ladder that will take me up to the top row!" And many games require players to add in order to figure out their score: think Scrabble, for example.

MULTIPLICATION and DIVISION. Scrabble, again, with its double and triple letter and double/triple word squares. Monopoly occasionally calls on players to multiply. I couldn't quite figure out my nephew Davey's "Lord of the Rings" version of Risk--his mom bought it at a yard sale for $1 ("best purchase I ever made," she says), but I'm not clear if the rules were included--but the original Risk game required players to divide the number of territories they owned by three to determine the number of reinforcements they got at the beginning of each turn.

PROBABILITY. This is a big one in any board game that includes chance. Kids may not be able to articulate precisely WHY they "probably" won't land on Free Parking this roll, but the more they play the more they notice that some dice rolls come up more often than others. Attacking 3 on 1 in Risk gives you better odds than attacking 2 on 1 or 3 on 2. My Stratego piece is powerful and is probably more powerful than yours, so attacking makes sense; this other piece is of very low rank, so I'll run away because your piece is probably stronger.

GRIDS, GRAPHS, and SPATIAL THINKING. Battleship (which can be played easily enough with pencil and paper) uses a coordinate grid system. Games like chess and Stratego require players to think in more than one dimension. So does Connect Four. I once had a three-dimensional tick-tack-toe board (I was TERRIBLE at it; fortunately, so were most of my friends). Othello, also known as Reversi, is another good example. There are lots more too.

ATTRIBUTES, LOGIC,and STRATEGIC THINKING. Mastermind (another one that you don't really need a board for). Checkers. Chess. Thinking ahead: if I do this, then she will probably do that. I should move this piece in parcheesi instead of THAT piece because then I might be able to catch that other piece on my next move...I'll test your unknown Stratego piece by using my lowest-ranking piece because it might be a bomb; I can afford to lose a scout but I can't afford to lose a colonel. And of course logic is at the heart of Clue: [drumroll] Miss Scarlett [drumroll] in the kitchen [louder drumroll] with the WRENCH!

NUMBER SENSE and ESTIMATION. You're on Boardwalk in Monopoly. You roll a 5 and a 2. Where are you now? Oriental Avenue--1 space to Go, then 6 more to Oriental. You're on Kentucky Avenue. You roll a 9. There are 10 spaces along each side of the board, and Kentucky is one space after Free Parking--so go to the end of the row--oops, "Go to Jail." You've got a bunch of beans in a space on the mancala board. Where will the last one land if you distribute them, and will that be advantageous to you? You drew an 8 in Sorry--can you tell by looking whether that will get you to the space where your opponent's man is right now?

I hope you get the idea! If you play these games with your kids, it can help to "think aloud" some of your moves and to ask leading questions: "Let's see, I just spelled the word BOX; I get double value for that X, so that's double 8, which is 16; the B is worth 3 points, and 6+3=9, so we're up to 19, and one more point for the O makes 20." "I owe you $36. Here's two twenties. What do you need to give me back for change?" "You told me that E-6 was a hit, but E-5 was a miss, and I know that E-8 was a part of your cruiser; I'll try other spaces in the 6 column." But that's not necessary, and I wouldn't advise pushing it. Unless they're playing absolutely mindlessly, they will pick up some of these concepts even if the game is played in complete and utter silence!

And yes, I DID play lots and lots of board games as a child--why do you ask? In fact, my family's old Monopoly board has a faint track worn into it by the pieces moving steadily around the spaces... (Hint: The railroads are a really good deal in the early stages of Monopoly, a nice steady income, but trade them in mid-game.)

A Mathematical License Plate

2009-06-20T13:50:00.003-04:00

This SUV was parked at my motel at a convention I attended in Salt Lake City last year. I expect it belonged to a fellow Math Teacher. It would be a major shock if it didn't.

Perhaps someday I shall have my own mathematical license plate. What about:

E 2 THE X

or

3 X 3 EQ 9

or

SQRT NEG1, for those days when I feel imaginary....

Other ideas? Send 'em along. I'm open to suggestions.

37 Cities, 32 States, and Lord Knows How Many Unnecessary Miles

2009-06-08T19:35:00.003-04:00

Cheerful Charlie's Tour of the USA is at an end. Between early October and early June he visited, as the title says, 32 states plus the District of Columbia and a total of 37 cities--large ones like San Diego and Seattle, Minneapolis and Houston; smaller ones like Cedar Rapids, Iowa and Springfield, Massachusetts; and small ones indeed such as Wall, South Dakota and Virginia City, Nevada. The students got used to enormous cross-country jumps and routes without rhyme or reason--just excess mileage and wasted gas. Still, they dutifully marked in the origin of each postcard he sent, used concepts of scale and ratio to estimate the distance from one city to the next, and made helpful suggestions about ways to improve his efficiency.

We are still calculating the total mileage. But it ain't gonna be pretty.

Here is the Official T-Shirt of Cheerful's travels. Please note that the list of cities should be quite accurate, as Cheerful sent me his list for proofreading (and boy oh boy did it need it). He did NOT tell me, however, that he was going to include the three lines at the top, so I had no opportunity to do the proofing. The mistakes--OF COURSE--are his and his alone.

You can click on the image for a closer look...

Fifteen Tires Were on Top of a Hill...

2009-06-08T19:14:00.009-04:00

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!

How to Annoy a First Grader

2009-06-05T16:31:00.006-04:00

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

2009-06-01T22:07:00.010-04:00

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...

Things Your Children Probably Shouldn't Be Telling Us, Part 1

2009-05-29T22:20:00.004-04:00

So we were working on time in the 1-2s this week. Today, the kids named and ordered various units of time, from milliseconds and seconds up to centuries and millenniums, and explained what they knew of the relationships between them, using nice NUMBER SENTENCES (60 sec = 1 min, 1 day = 24 hours, etc). On the whole, they did quite well, though why "half an hour" and "5 minutes" don't count as separate units of time was a bit mysterious for a few of our first graders. Next year--

Anyhow, after this lead-in, I asked children to fill out a sheet about time units. The structure was simple enough. "It takes about one SECOND to..." was the first one, and kids were supposed to think of an activity that takes about one second. Then they followed it with one minute, one hour, and one day.

The responses were fun and revealing of children's understanding: one second to "squash a bug," "pick up a feather," or "say four letters of the alphabet," one hour to "clean my room" or "draw a perfect picture" (quick, tell Picasso!), one day to "make a really good sculpture."

My personal favorite, though? "It takes about one minute to do my homework." Given that mathups, reading, and spelling alone are supposed to take at least 20-25 minutes each night, this is the sort of statement that is perhaps better left unsaid. Ah well--by high school I'm sure this child will have figured that out!

Finding the Center

2009-05-28T19:54:00.010-04:00

One of the perks about being a member of NCTM (http://nctm.org, the National Council of Teachers of Mathematics) is that you get a subscription to a journal called, what else, Teaching Children Mathematics. This journal has a monthly feature called "Problem Solvers," which presents an open-ended problem and encourages teachers to try it with their classes. Teachers are then invited to write up their experiences and send 'em in. From time to time I've tried these problems out, and once I even got around to sending in my reflections.

Anyway, a recent Problem Solvers challenge caught my eye: How would you go about finding the geographic center of the United States (minus Alaska and Hawaii)? O-ho! I thought. This will be an interesting problem to do with all the grade levels I work with! But then field trips and special events got in the way, and so did division and fractions and 3-d geometry and other such valuable topics--so in the end I managed to do the problem only with a few 4th graders and a few 1st graders.

At some point I'll talk more about the 4th graders, who generally did quite well--they showed some sophisticated thinking about the assignment, and made use of a number of different mathematical skills to come up with an answer. This post, though, will be about the 1st graders, whose work was...um...

*************************************************

For the children , this was one of the easiest questions I'd asked all year. “It’s right here,” said one girl, touching the middle of the border between Kansas and Nebraska. The others nodded agreement and, not to be outdone, put their fingers on roughly the same spot themselves. That part of the Great Plains has never been so crowded.

This was a good estimate—a very good estimate, in fact, but I was looking for an explanation of how they'd figured it out too. When no clear explanation was forthcoming--in fact, when there was no explanation of any kind--I asked whether there were any tools they could use to show me what they were thinking. When I said tools I had in mind, oh, rulers, or some other kind of measuring device. They did not.

“A jackhammer?” suggested one boy.

"You could use a compass," said the girl who had made the initial estimate. "You would walk with the compass. You can start anywhere, like in California. Then you walk this way.” She put her finger near San Francisco and slid it eastward on the map. “When you get there, you stop.”

“How do you know when you’re there?” I asked.

She shrugged. “Because you’ll get to that place, and then you’ll be there.” She was too polite to say Duh!, but you could hear it all the same.

“What do the rest of you think?” I inquired. A chorus of “I agree”s and “Uh-huhs” rose from the other children. I believe this is called proof by intimidation.

I decided we'd better back up. “How about this table?” I asked. “Where’s its center? And how do you know?” Several hands slapped down in a place reasonably close to the center, if not the exact spot. The center, they explained, had to be in the middle of the lines that divided the table in half. Duh! Again, politeness reigned, but I knew what was what.

“So now we know about the center of the table,” I said. “I wonder if that might help us find the center of the country.” I opened up the map again. “What do you think?”

There was brief discussion. One child pointed out that the United States wasn’t a nice regular shape, such as a circle or a square, so it didn’t really have a center. Another argued that the whole world would have a center, “because that would be a sphere and then you could find the middle of it.” But they all deferred to a girl who cut to the chase. "The center would be right here," she said, stabbing a forefinger at a spot in the middle of Kansas, just south of the original place chosen. “That’s the center.”

Back we’d come to our starting point. “How do you know?” I asked once more, feeling like the twenty-first century version of a broken record and hoping she'd say something about lines that divided the country in half...

Nope. The child looked at me with something resembling pity. “You go up in a plane,” she said, “and then you can see where the center is and you go there.” Duh!

From which I conclude one or more of the following:

*First graders seriously underestimate the size of the country.
*First graders see no reason to calculate the exact position of a center when eyeballing it will do.
*Sometimes it’s really hard to explain your thinking, especially just before lunch on a Monday morning.

Oh well--onward!

P. S. If you'd like to know more about the geographic center, here's a rundown: http://en.wikipedia.org/wiki/Geographic_Center_of_the_Contiguous_United_States.

Psst! Rules!

2009-05-20T22:01:00.008-04:00

PDS is a progressive school. right? Right. So it doesn't bother with silly things like rules, right?

Well--no. We do bother with rules. As well we should. The first rule of the school is, or used to be, "No one may interfere with the learning of another." A good rule, and a sensible one, which recognizes the important distinction between rights and responsibilities, even if it lacks the pizazz of what used to be the school's second rule, "Windows are not doors."

We have plenty of other rules regarding behavior, too, many of them developed by the children themselves (most of these begin with the word No, as in No hitting/No kicking/No biting/No poking people with sharp sticks/No knocking over other people's block buildings) or by the children in conjunction with a teacher (these are stated much more positively--Walk in the halls/Be nice to people/Share classroom materials/Treat other people's block buildings with respect). My favorite of these from my classroom-teacher days was Eat regularly, which had nothing to do with having frequent snacks but was one second grader's attempt to condense No burping at lunch and No opening your mouth while you're eating to show people what's inside into one relatively positive statement...

But the rules I want to discuss in this post are math rules. Yes! Math rules, as I use the term, are statements about math that are always true. We encourage children to come up with these rules as they work on problems and as they talk about math. Here are 3 examples, named for the children who first suggested them:
These may seem basic to you, and of course they are. None of these will make the pages of American Mathematical Monthly. They've all been discovered before. On the other hand, let's keep in mind that these are first and second grade children who are just beginning to make sense of the number system. The idea that numbers will always behave in a certain way is by no means obvious. Most things about school aren't that predictable. Think about reading: the letter o can stand for several different sounds, some words add an s in the plural while others add es, a word like set or right can have multiple meanings. Math is different. Math is a little more--reliable.

So we ask children to look for these always situations in math. We ask them to think carefully about how numbers work, about whether a particular result will happen no matter what or whether it will happen only sometimes. The benefits, I think, are clear. A child who recognizes that an odd number plus an even number will always be an odd number is thinking hard about numbers and their properties. She's doing some basic number theory; she's acting like a scientist, making and then testing a conjecture. A child who sees that adding or subtracting zero never changes the original number is finding a pattern, generalizing from special cases, and boldly going where (to the best of his knowledge, anyway) no man has gone before.

Yes, we could tell children that odd + even is always odd, and very often we do just that. But it's more powerful, and the effects are longer-lasting, if children come to discover these rules themselves. Now if you'll excuse me, I need to go determine whether every even number greater than 2 can be written as the sum of two primes...

Unringable Numbers

2009-05-14T14:23:00.005-04:00

To succeed in doing division, especially with large numbers, there are two prerequisites. The first is that you have to know a lot of multiplication facts like THAT . Our 3-4 students--fourth graders in particular--have done a great job with this. The children, as a rule, seem to recognize the progress they've made in committing these facts to memory, and also in understanding the broader relationships between numbers (if you know that 2x8=16, how does that help you find 4x8?). It's been gratifying to get to the point in the year where I say "7x5" to a group and hear a grand chorus of "35" shooting back at me.

The second prerequisite is to think flexibly. Now that children have gotten good at telling me the product for a given expression, we need to switch the task (isn't that always the way??). So these days I give them the product and have them tell me the factors (the numbers to be multiplied to make my number). When I say "35," then, the students need to respond "7x5" (or 5x7); when I say "36," they should tell me "6x6" or "9x4." The language I use with children is that numbers such as 35 should "ring a bell"--that is, they should spark an immediate connection in children's minds with 5x7.

This ability is critical for doing division. One thing that makes it hard, though, is that not all numbers ring bells. An example is 17--the only multiplication expression that goes with 17 is the dull-and-boring 17x1/1x17. 43 won't ring any bells, either. Children not only have to be able to shoot back "3x9" when I say 27; they have to be able to recognize which numbers don't go with any expressions. That's new and different, and it takes children a while to figure out what's going on. Fortunately, young brains are quite malleable, and when they've gotten the idea, they're usually quite good at it.

So we were practicing some of these relationships this afternoon in one of the classes. 49, I said to the fourth graders I was working with, and they quickly responded with 7x7. How about 30? I asked, and got, variously, 5x6, 2x15, and 3x10. Twenty-five? That was easy, they scoffed--5x5. Then up came 37. Ring any bells? I asked.

There was silence for a moment as the students considered. At last, one child raised her hand. "It's unringable," she said confidently.

The usual mathematical term for this concept is prime, a term that this child knew but had temporarily forgotten. I must admit, though, that I have a certain fondness for unringable. Perhaps we can get the mathematicians' union to recognize it as an alternative. In a discipline that features the Pigeonhole Principle and the Generalized Ham Sandwich Theorem, I'd say it isn't entirely out of the question...

POW!

2009-05-12T20:42:00.004-04:00

Three fifth graders came by the office the other day to visit at lunchtime. Two of them were there mainly to show off the shirts they had recently picked up at a museum. (The third, as you'll see from the picture, had something else in mind altogether.)

"POW!" I said cheerfully, hoping they'd congratulate me on my ability to read ALL CAPS. "Great shirts. Are they Batman-related?" I was not much of a Batman fan as a kid, but I do dimly remember the TV show, with the fight sequences that would pause so the POW! WHAM! OUCH! BOP! EEK! sound effects could appear on the screen. The style of the writing on the shirts reminded me of the series. (Right??)

"No," they said in that exasperated tone that only early middle school girls can achieve. "Look! This is what it stands for!" And they pointed one by one to the letters. "Problem Of the Week!"

POW! Problem of the Week! For two whole years these girls dealt every, well, week with the POW, which their teachers and I sometimes called the Problem of the Week but sometimes simply called the POW. The Problems of the Week, stored in my computer under the name of (what else) "POW stuff."

"Oh," I said, embarrassed. "Problem of the Week! Of course it stands for Problem of the Week. I knew that!"

And if I say it often enough, maybe it'll become true.

Practical Arithmetic?

2009-05-06T20:01:00.004-04:00

While doing research on something completely unrelated to math, I ran into a cute (and tongue-in-cheek) article in an Indiana newspaper of 100 years ago, give or take.

Having trouble with his homework (sound familiar?), young John, a seventh grader, seeks out his father for help (sound familiar again?). It's a word problem, of course. "Asked how much money he has in the bank," the problem reads, "[a man] replies, 'If I had $10 more I would have $1,000 more than half what I now have." John is supposed to find out how much the man has in his account.

Dad is no help. That's because he's a card-carrying curmudgeon. Anyone who wants to know the actual amount, he snarls, should simply ask the teller. "In my day," he announces, unable to resist a dig at "modern" methods of teaching math, "we had practical problems in our arithmetic." [Kids, get off my lawn.]

Curious, John now repairs to the library to examine arithmetic books from the previous century. Each of them, it turns out, is chock-full of the sorts of problems his father is enthusiastically disparaging. John eventually tracks down an 1805 text, where he finds the following gem:

"A good man driving his [geese] to market was met by another, who said, 'Good morrow, master, with your 100 geese.' Says he in reply, 'I have not 100 geese, but if I had half as many as I now have and two and a half geese besides the number I now have already I should have 100.' How many geese had the man?"

At which point the writer of the article temporarily abandons both John and his father and addresses the reader directly. "How long," he demands, "would you permit a man to live if he made such an answer to you?"

[Well, let me think. What's the best way to express the answer, I wonder? Ah--got it. If the time interval were increased by seventeen seconds, it would be precisely four times one third the actual number of seconds diminished by the cube root of...]

George's Excellent Adventure

2009-05-03T15:19:00.005-04:00

Sometimes the best lessons are the ones you don't plan.

Friday morning, Ellen poked her head into the office as I was preparing for a fraction lesson with the 1-2s. "Elizabeth found a Where's George dollar in her lunch money," she said. "Okay if we take a few minutes to enter it at the beginning of math time today?"

Where's George, I should explain, is a lovely internet project that tracks paper money as it moves across the country (www.wheresgeorge.com). Since the 3-4 classes handle lots of money in their capacity as Pizza People, they occasionally run into Where's George bills, which are recognizable by special markings. We log onto the site, enter the bill's serial number, note our location, and press Enter. If the sound on my laptop is turned on, we'll then hear a cash register noise and the bill's previous location(s) will appear. Most of the bills we've found thus far have come from nearby places such as Pennsylvania and Massachusetts, Brooklyn and Kingston, but we've had bills from Missouri, Tennessee, and Texas as well. It's fun, and suspenseful, and teaches a bit about geography--and you never know when someone will find "our" bill and put it in again.

When we entered Elizabeth's bill, the screen showed that the bill was now 1128 miles from its original location. I had a sudden brainstorm. Instead of scrolling down and telling the class where the bill had come from, I'd have them narrow the possibilities by using math--specifically, their measuring and estimation skills. They'd been working on maps all year long, after all, filling in states that Cheerful Charlie had visited in his round-the-US tour. Ellen got one of the students' maps, and we hung it up. We determined that 1128 was very close to 1100, in double-round numbers, and at 200 miles to the inch, the class quickly calculated that the starting point was about 5.5 inches away from us.

It was clear to most of the students that the possibilities would form the arc of a circle, and so we did a little measuring. We ended up with a curve beginning at the western end of Michigan's Upper Peninsula and then zagging through Wisconsin, Minnesota, Iowa, Missouri, Arkansas, and Mississippi--all of them marked on the students' maps--before catching a piece of south-central Florida and disappearing over the Atlantic Ocean. "Why can't the bill have started here?" I asked, indicating where the arc crossed the Gulf of Mexico. That was obvious. "It's too wet for money in the ocean," a third grader answered (unless, he added, there were islands he "didn't know about"). As for why we didn't go north of Michigan, that was obvious too: Canada has its own money.

I scrolled down on the webpage and revealed the answer: the bill had originated in Florida. ("I knew it!" half the class exclaimed.) I named the town, which I'd never heard of. But Ellen had: her brother lived there. She asked if there was any way to find out who started the bill on its travels. Well, yes, there was; I clicked on the profile button and found a first name, Bob.

It wasn't Ellen's brother. But that was all right. Bob had provided us with a nice map of the US, each state filled in with one of six colors. Now I had my second brainstorm. We'd done a little real-life estimating and measuring with scale; it was time for some real-world data analysis.

"What do you think this map shows?" I asked, turning the computer so the students could see. Temperatures, guessed one boy. Good thought, but no. How many people live in each state? asked a girl. Close, I said. Think about what website this is, Ellen suggested, and suddenly hands were flying up all over the meeting area. Bob, they realized, had marked dozens and dozens of bills and sent them into the wild. The colors showed how many of those bills had turned up in each state.

[Here is Bob's Hit Map, by the way:]

Right on the money! (So to speak.) The only question now was which colors stood for the most bills and which for the fewest. To help, I had them identify a few key states on Bob's map, and then I gave them a little extra information. California, I explained, had the most people of any state. Texas, New York, and Florida were next. Wyoming took up a lot of space, but it had fewer people than any other state.

Working as a group, the class swiftly came up with a sensible schematic for the colors. Red, the color of Florida, California, and New York, would be the most. Bright green, it seemed apparent, would be next, judging from what the children knew of population and distance, and so on, down to lowly Wyoming, the only state that was colored gray.

The guesses were in. It was now time for the Great Unveiling. I had everyone's full attention: they were deeply invested in the outcome by now. And the results were entirely satisfactory. The class had four out of six colors right; the only error had been reversing the orderof the fourth- and fifth-most colors.

Not bad, not bad at all, I told them, and we moved on to the regularly scheduled lesson on division.

Shrubbery Ice Cream?

2009-04-30T19:00:00.004-04:00

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

2009-04-30T14:39:00.005-04:00

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...

Ozzes and Libs

2009-04-27T09:36:00.006-04:00

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.

Fractions + Transformations = ?

2009-04-21T23:29:00.005-04:00

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!

AWARDS!!!

2009-04-20T14:56:00.004-04:00

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!

Seventy-four

2009-04-19T16:47:00.005-04:00

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.

Ways to 100

2009-04-15T13:08:00.010-04:00

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

2009-04-14T11:34:00.013-04:00

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

2009-04-10T22:00:00.004-04:00

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

2009-04-07T22:19:00.004-04:00

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

2009-04-06T17:05:00.011-04:00

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

2009-04-05T22:05:00.001-04:00

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

2009-04-02T14:51:00.003-04:00

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.

Math and Poetry

2009-03-31T19:18:00.001-04:00

The Mostly-Annual PDS Math Contest (TM) is drawing to a close. This year, the challenge is to submit a poem about your favorite number, or your favorite number sentence. Entries are pouring in. 7 and 9 seem to be popular choices, as are the ol' reliable 1 + 1 = 2 and 2 x 2 = 4, but we have some, um, more creative favorite numbers and sentences as well...All will be revealed shortly.

In the meantime, another thought about poetry and math. Some people claim that the two disciplines are unrelated. How very wrong they are! Here's an example from just last week of how an understanding of poetry can help children improve their math skills. Yes! Srsly!

Background: We asked first graders to solve a word problem that involved finding the sum of three addends--5, 7, and 5, in this case--and to EXPLAIN how they got the answer (a big step for children that age). Most of the children wrote something like "I added 5+5=10 because it's a doubles fact and I know that 10+7 is 17." Short and sweet. Some used their fingers to help, or drew sketches as evidence. BUT one young lady had a very different approach:

"I remember when we were studying haikus. Haikus go 5, 7, 5 [that's the number of syllables per line--SC]. I counted how many altogether in a haiku and it was 17. So I know 17 is right."

Another example of the interdisciplinary learning PDS does so well...:)

Pi Day

2009-03-27T11:55:00.000-04:00

One of the great mathematical holidays of the year is Pi Day, which occurs every March 14. (It may be the only mathematical holiday of the year, in fact, but never mind.) The third and fourth grade classes celebrated Pi Day this year by deriving this number--or as close as we could get.

We began by distinguishing, of course, between pi (the number) and pie (the food). Several students said they'd heard of pi (the number). All students said they'd heard of pie (the food). Once we had that out of the way, we introduced the concept of DIAMETER (distance across a circle, through the center) and CIRCUMFERENCE (the distance around a circle). "Okay, here's the question," I said. "Are these distances related? If you know the diameter of a circle, can you use that to calculate the circumference without measuring?"

To explore, we had pairs of students use tape measures to determine the diameter and circumference of various circular objects in their classrooms--clocks, round tables, woven mats, stools, and more, rounded to the nearest whole centimeter. Here's a sampling of their results:

Bowl d = 9 cm C = 28 cm

Stool d = 21 cm C = 66 cm

Wheel d = 13 cm C = 45 cm

Garbage can d = 49 cm C = 150 cm

"Any patterns in the data?" I asked. Ye-es, the students said tentatively; the circumference is always more than the diameter. "By a lot, or by a little?" I queried. By a lot, they agreed. Usually, one cautious soul hastened to add. "So maybe if I add the same number to each diameter, I'll get the circumference," I suggested. "What number would I have to add to get the circumference? Talk with a partner and see what you come up with."

They considered the question and then they shook their heads. The numbers didn't work. You need to add a small number to the diameter of something small, like a magnifying glass, they pointed out, but you have to add 100 or even more to the diameter of something big. Clearly, addition was NOT the way to go.

If addition didn't work, the obvious answer was multiplication, and the students quickly moved in that direction. (Multiplicative thinking at work! Hurray!) "The circumference is, like, double the diameter," one student suggested. "More than double," a classmate countered. "Triple the diameter," said someone else. We tripled a few diameters using mental arithmetic. "When you triple it," someone concluded, "you get just a little bit less than the circumference." "Usually," piped up our cautious friend from before. "Oh, I know!" shouted an excited third grader. "You triple the diameter and then you add one! Oh, wait a minute--"

This, as it turned out, was a job for a calculator. We keyed in circumferences and divided by the diameter, then recorded the ratio (a new word for most students) on the board, cutting the endless stream of decimals to two places. 3.33, 3.00, 2.97, 3.24... "They're mostly around 3," students noted. I nodded. "We can't measure exactly with our tape measures," I explained, "but if we could, we'd discover that the ratio is always the same--a little over 3. This number is pi. The digits of pi go on forever, but are there any guesses about what pi would be if we just used two decimal places?" There were plenty of guesses, of course, there always are, but one exceptionally observant fourth grader had a reason for her answer. 3.14, she said, for why else would Pi day be March 14?

Why else, indeed?!

A footnote: This year, Congress passed a resolution officially declaring March 14 to be Pi Day; the resolution was also an attempt to highlight the importance of math education. Incredibly enough, ten representatives voted against the resolution. It's fun to speculate why. Perhaps they have had bad experiences in the past with irrational numbers, or maybe they think the value of pi should be determined by the open market, not the government--oh, wait... More information about the bill can be found here.

(Photo credits to Jan Campbell, and to Ellen DeLong's camera.)