"You Must Be Smart at This"

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

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...

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)

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?

"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

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!