Joanne,
According to Fowler (H. W., that is, not Roger: _A Dictionary of Modern English Usage_), both rponouns are useable. One use is numeral ("One double-checks his quotations, but another does not.") -- the possessive is someone other than the person designated as 'one'; the other is impersonal ("One double-check's one's quotations as well as one's spelling.") -- the possessive is the same as the 'one'. I'm not sure I understand why Fowler calls it 'impersonal'. It seems to me that it's really 'presonal', but who am I to disagree with Fowler?
It seems by the examples you gave that your students were taught the opposite of what Fowler says. Their teacher seems to have gotten it backwards.
Paul D.