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."