[og1]

I was taught with the older techniques, but I like the new chunking and gridding system better. I think it is a lot closer to how you would break the problem down if you couldn't use pen and paper and had to do everything in your head.

[sesqu]

This is exactly how I do it in my head. I was never taught to, but found myself multiplying multifigure numbers in my head often enough to come up with the gridding system. It becomes laborious after five figures or so. I was never that fond of the long forms, and didn't use them after they stopped appearing in the exams.

Incidentally, I didn't bother with a calculator until 12th grade. Since then, I've started making more mistakes, and now have to do any mental calculations several times over.

[microtherion]

Yes, the advantage of gridding seems to be that it decomposes the multiplication into simpler steps.

I would suspect, however, that if you habitually had to do long multiplication, the old method would be faster (precisely because you do a bit more in your head and less on paper). Multiplying two 5 digit numbers involves adding up 5 numbers instead of 25.

But what has changed over the last 40 years is that NOBODY does habitual long multiplication anymore, because calculators are ubiquitous. Therefore, the technique is purely a building step for further understanding, and picking an algorithm optimized for casual users of multiplication is appropriate.