Dave Marain of MathNotations is running a poll about how to teach multiplication, but the question has broader application:
How should we teach the arithmetic algorithms
— or should we teach them at all?
Algorithms are step-by-step methods for doing something. In arithmetic, we have standard algorithms for addition, subtraction, multiplication, and long division. Once the student masters the steps for any particular algorithm, he can follow the steps to a correct answer without ever thinking about what the numbers mean.
An efficient algorithm can free a student to concentrate mental effort on learning more difficult concepts (such as multiplication or division of polynomials) or thinking through multi-step word problems. But reliance on memorized algorithms can be a “follow the recipe” approach to math that disguises lack of understanding, leading to a student’s eventual crash-and-burn (usually in high school algebra).
What is the solution? Do you have an opinion? Head over to MathNotations and make your voice heard!
Denise Gaskins' Let's Play Math 
There is a (rather discursive) commentary on this very question to be found at Ethan Akin‘s faculty website.
Thank you for the link. I haven’t had time to read Akin’s entire article, but I do like the quote he begins with:
I also downloaded Akin’s Theses on Johnson, which looks (on first skim) to be the most sensible analysis I have seen of the “Intelligent Design” movement.
Denise,
Thank you for letting others know about this poll. I deeply appreciate your support.
There truly are bigger questions than the one I am asking but we cannot be afraid to start somewhere.
Can children develop an understanding of WHY a procedure works yet not be successful in implementing the procedure? Can children perform an algorithm mechanically and successfully and yet not really grasp why it works? These are rhetorical questions!
The central issue for me is what we, as professionals, should expect. Most teachers are expected to follow a prescribed curriculum, which is most often embodied in a particular textbook. The veteran teacher knows how to work around this, the less experienced teacher needs direction and support. BUT teachers must know EXACTLY what it is their children are expected to learn. Ill-defined expectations are tantamount to “garbage in, garbage out.” For myself, I want my own children to have understanding of what they are doing and why they are doing it, but, in the end, I want them to be able to ‘put the ball in the basket’! I will not apologize for that…
I suppose you probably know what I will say, but let’s get understanding, and then link it to the most efficient algorithm.
And before understanding should come exposure to kid-scale numbers. Lots of play with 2’s and 3’s and 5’s and 10’s.
How do you decide what’s the right time? Now that’s a tough question.
Jonathan
Eeks. Akin should have stopped after 2 or 3 pages. I really believe that most of us who are proficient at arithmetic don’t really know how we got that way.