Analyze That Algorithm!!

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

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!)