- Gridding is how I casually multiply things in my head already. I read left-to-right, so I usually start multiplying with the leftmost digits and track the zeros, then add things up until I've reached the desired accuracy. Really, it's just the traditional method in reverse -- this 10-slide explanation is much easier for teaching the basic concept of multiplication, though.
- Chunking is division explained as "how many portions does this make" rather than the traditional problem of "how big would each of these equal-sized portions?" -- which is great for two reasons: (1) physically performing the alternative -- measuring out a fluid into N equal-sized portions -- is hard! (2) it's more like how computers work, and makes the modulo operation trivial to explain; this kind of quotient clearly ignores the issue of remainders in the initial problem, and then later it's clear what you can do with the remainder to make a compound fraction.
But it's unfortunate that memorization has become taboo in Western education. Sometimes you actually need to memorize a table of facts and be able to recall them quickly, as with single-digit multiplication -- if you have to add up seven eights each time you encounter 7x8, you'll just be too slow to keep up. Know how to confirm what you've memorized, but also know that 7x8=56 as a basic fact.
(It's not that hard if you take advantage of spaced-repetition learning, as in: http://www.mnemosyne-proj.org/ )