Boys, Girls, and Paper Airplanes

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...

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

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

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

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

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.