Here’s a way to visualize multiplication that reduces it to simple counting:

Express the digits in each factor with rows of parallel lines, as shown, and then count the intersections to derive the product. This is more cumbersome than the traditional method, but its visual nature is appealing, and it permits anyone who can count to reach the right answer even if he doesn’t know the multiplication table.

The example above uses small digits, so no “carrying” is required, but the method does accommodate more complex sums — it’s explained well in this video:

See Two by Two.

(Thanks, Dieter.)