That last post wasn’t really about the lattice algorithm. This one isn’t either. It’s about supporting claims with evidence. (The last one, in case you’re keeping score, was about crafting tasks that show instead of spiels that tell).
Another claim my future elementary teachers like to make is that the lattice is inefficient for problems with large numbers of digits. The idea is that you have to spend a lot of time making that lattice before you can multiply. This stands in contrast to the standard algorithm, which you can just get started with straight away.
They make the claim, but they don’t back it up with evidence. (This, after all, is part of what a college education is supposed to teach-how to build arguments-so I’m not complaining that they don’t. I understand that it is my job to teach them to do so.)
So I began to wonder whether the lattice-drawing really did set one back.
So I put it to the test. Ten digits by ten digits. On your mark, get set, GO!
By the way, I was going to do three algorithms head-to-head. I was going to do partial products in the middle of the board. But about a third of the way through, I got fed up and quit. That one really is inefficient for large numbers of digits.
Overthinking my teaching 
Very nice, Chris. Here’s something related to consider: speed vs. accuracy as components in evaluating “efficiency” of algorithms. If you can do it faster but make more errors with algorithm A than with algorithm B, which is more efficient? Worth testing with 10 digits numbers and your class split into groups, perhaps. Or, to avoid bias, perhaps get people to use the algorithm they believe is more “efficient,” as long as you can get a decent number for each algorithm you want to test.
Yeah, accuracy matters, doesn’t it? Notice that you can’t see accuracy in the video; only speed. You can’t tell whether I get the same answer with both algorithms, never mind whether either one is correct!
Very cool video. Don’t forget that if you’re a middle school student, the lattice example would have preliminary steps (at least for a problem of this magnitude) – i.e. “Mrs. Otto, do you have a ruler I can use to draw straight lines?” then drawing, drawing. . .”Mrs. Otto, do you have an eraser I could use? My lines aren’t straight. . ”
PS I’m trying to use more student made and teacher made media in my classroom. Students are using DM’s “Graphing Stories” as a project this week and a colleague and I made a scale factor vs. alternate method video this week. It’s very goofy but hopefully the kids will enjoy it. Very fun. 🙂
Yeah, I dig where you’re comin’ from with the preliminary lattice steps, Alex. But seriously…10 digits by 10 digits?
I get a couple of standard reactions to that video (besides mind-numbing boredom-that’s the third one, but probably most common). (1) Yours-that the set up time is ridiculous, and (2) A wondering why I am defending the lattice algorithm.
(1) Yes. I’ll say it now…the set up time for a 10-digit by 10-digit multiplication problem is ridiculous. But I would like to emphasize that NO ONE EVER MULTIPLIES 2 10 DIGIT NUMBERS TOGETHER BY HAND. When was the last time YOU did, for instance? And remember that you are a math teacher.
(2) The only sense in which I am defending the lattice algorithm is in its efficiency. Students tell me that the lattice is inefficient because it takes so long to draw the grid. True enough, but watch how quickly I catch up once the grid is drawn. And watch how hard it is for me to keep the columns straight in the standard algorithm. You’re telling me that a kid who struggles to draw the lattice is gonna keep those columns straight? No, I am not really defending the lattice algorithm in this video. I’m adding evidence to the conversation.
Finally Ms. Otto, you’ll be sharing this video you made somewhere, right?
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