My brother sent me a link to this video that teaches “Japanese” multiplication (thanks Tim!):
I learned about this technique in my History of Math class, and Vi Hart talked about it in a video back in 2011:
She does a nice job showing why there’s nothing particularly special about this Japanese “visual” multiplication. Here are a few reasons why it’s not better, as far as I’m concerned:
- It’s not faster (sometimes it is, but most of the time not). As Vi points out, counting the number of dots in a rectangle by hand is ridiculous.
- It’s painful when the numbers are bigger than 1, 2, or 3 and when there are more than 2 digits in the numbers (just try multiplying 976 x 8937 for example).
- Zeros make things difficult (use dashed lines?)
- Carrying is still required.
- It’s perhaps more error prone, since it relies on your counting all the intersections.
In the end, to multiply two numbers you still have to multiply all their digits by each other and deal with carries, no matter which method you choose. I think it’s still worth teaching various methods of multiplication to students in an effort to make the abstract more concrete.
Random Walks 
The siginificance here is that you don’t need to memorize the table of multiplication.