[alankay]

This is easy to derive if you have a little algebra, but it is shoved at kids before they can derive it, so they don't understand it (which is anti-math). The dull ones go along with it, but the brighter ones often think they are stupid. I (and independently others) have worked out visual proofs of both fraction multiplication and division for younger kids.

But I mostly think that "fractions as ratios" can be left until the child is older and has more chops for derivations.

The Greeks thought about these as different ways to measure intervals, and they worked out the idea of "common measure" as a good solution to dealing with these problems.