First Week of School

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

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...

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

First grade, a lesson on estimation.

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

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

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

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

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

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

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

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

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

True, too!

You're Wrong HA HA HA

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!)

Pentagon, Hexagon, Heptagon...

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?

Fathers, Sons, and Inequalities

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.

A PalindromemordnilaP A

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

"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

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

"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?)

More Rabbit News

--> 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

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.

Consecutive Numbers Alert

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.

A Mathematical License Plate

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.

Practical Arithmetic?

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...]