A precis of our lecture on the topic:
Mathematical division (over, say, the reals) is not a function, because it violates the property that every element of the domain (say, pairs of reals) maps to exactly one unambiguous value in the codomain.
To make division into a function, we can either exclude (x,0) from the domain, or make the codomain a union of the reals with a singleton of some sentinel value indicating failure (e.g. { ⊥ } ). The corresponding pattern in Haskell would be either to restrict the input (via a newtype) or add a sentinel value to the return value. The latter is more common, and Maybe is the standard vehicle.
I’m sure you know all of the above! But this is more or less the exact formulation that is used to explain it to students who have (at that point) less than 3 weeks of functional programming knowledge, and they are able to take this in their stride as well. This is off topic though, so I’d recommend we don’t go on about it 