[DrSAR]

Not sure I agree with 'appearance [...] is much worse'.

Given the choice between a class room of first years who believe (in themselves and) an appearance of calculus knowledge or a room of scared undergrads that recoil from any calculus-inspired argument they 'have never learnt it properly', I'll take the former. I can work with that much more easily. Sure, some things might break - but what's the worst that can happen?

We'll sort out the rigour later while we patch the bruises of overextending some analogies.