I’m going to assume you know the lattice algorithm for multidigit multiplication. If you do not, and if you would like a primer, here is one.
This post isn’t really about the lattice algorithm, but it’s the context for what I’m really trying to say, which is this: It is worth the time to craft classroom tasks carefully.
I have used the lattice algorithm for years with my future elementary teachers. We learn the steps in class, they go off and practice it. And then they write about it, using the ideas of the course to analyze the algorithm.
After a number of semesters of this, I became tired of reading in their work some variant of the following claim,
The lattice algorithm is very good for teaching place value because you have to pay attention to the places as you work with it.
I could not disagree with this claim more strongly. As I work the lattice, I am going digit-by-digit. I am absolutely NOT thinking about the values of those digits. And I suspect most children are not either. This makes it an efficient algorithm.
Last semester I decided to put that claim to the test. If these future teachers thought the lattice algorithm exposes important ideas of place value, then what task could I give them to demonstrate that it does not?
Well, they have been analyzing the algorithm; they have written papers about it. So if it teaches place value, they should be able to ace any place value task involving the lattice, right?
So here’s the task: Perform the lattice algorithm to multiply 7,343 by 1,568. When you are done, use a marker to highlight each and every tens digit in the lattice.
No follow-up or clarification questions allowed. If the premise is that the lattice helps us to learn place value, then we should know enough about place value to make a commitment to the meaning of a tens digit.
Can you guess which of the answers below is the more popular in my classroom?
When both are presented, a really useful discussion of the algorithm and its position with respect to place value ensues. And that discussion helps to explain the really clever “slide trick” for placing the decimal point (as seen about 2:30 into this video).
But back to my point. I can tell my students that the lattice doesn’t bring place value understanding along for free. Or I can show them. Showing requires carefully crafted tasks. But I find it’s worth the time.
When I have the choice between telling and showing, I nearly always choose to show.
Which is why I’m always running behind on content coverage.
I made my peace with that years ago.