Goals for teacher knowledge of multidigit arithmetic:

(Definition: an algorithm is a set of steps you can follow to perform a computation or to do something else similarly particular and well defined.)

Multidigit arithmetic is one of the important goals for elementary mathematics.  In order to effectively teach these arithmetic algorithms, though, it helps as a teacher to understand not only the standard algorithms, but a lot of the strategies that are similar and related. 

Some newer curricula have abandoned teaching the standard algorithms, or at least delayed them until later.  Instead teachers guide students to discovering alternate algorithms.  You'll see a video this week of a teacher doing that with a class.  I think the verdict is still out on whether that is overall more successful than a traditional approach, but even with a traditional approach it helps you as a teacher make better pedagogical decisions if you have seen several common invented algorithms, so that you are able to more quickly evaluate new things that your students do.

There are also some algorithms that are taught as the standard algorithm in other countries, and I think it's instructive to see a few of these.  I don't know if or which ones you might see again, but it gives on a sense that the way you think of as simplest might not be simplest to someone else, and the different technique a student brings in that their aunt showed them might be correct, even though it's unfamiliar to you.  I want you to be confident in your ability to figure out other ways of computing.

I think, as we do the mental math exercises, that you'll find that there are even some alternate ways that you use for computing some things.  Most of us are more flexible than we realize.

I do give the most time and attention to the standard algorithm, however.  I think it's likely you'll spend a significant amount of your time somewhere along the line, teaching one of the algorithms that are standard in this country.  I want you to be able to show how to perform the calculation using manipulatives, and explain the reasoning.  I want you to be able to step-by-step show how the manipulative work corresponds to the written work, and I want you to be able to put the algorithm steps in an appropriate (word-problem-type) context for the manipulative work. You'll also be looking at some of the common computation errors that children make in the standard algorithms.