It's easy. They're alligator mouths, that always go for the larger number (greedy alligators!)...

I used to mix it up in elementary school because the school teacher explained that "<" is "less" and ">" is "more". Even worse, she suggested to memorize it like this: |<leiner = kleiner (german, meaning less)

At some point I realized it's possible to memorize it visually, since then I never mixed it up again

Yeah people frequently mix up left and right also. And my brother always wrote his numbers backwards or rather left to right (units on the left).

Was it possible to give up? In my place, they wouldn't be let on to the next grade if they'd fail basic math - i.e., they'd spend the summer repeating it + an extra year in the same grade if that wasn't enough; so it would have to be a real exception to see a 10 year old who can't somehow manage that.

Anyways, the classic mnemonic for that was simple - for < and > the wide part contains more 'thingies' than the narrow part. They're actually taught before numbers or together with numbers, like [pic of four strawberries] > [pic of three strawberries].