They may not be cognates, but they are equivalents in their respective languages, fulfilling the same functions, or at least overlapping functions, so don't get hung up on what you call them, the original point is still valid - you can say, "use '-ly' in English where you would use '-mente' (-amente/-emente) in Spanish." You can also warn them against confusing the English use of '-ment' with the Spanish '-mente.'
Another example of building different parts of speech from component words plus affixes is that comparatives are much more rarely made in English than in Spanish by adding the word 'more' (mas), except as in this sentence where it is modifying an adverb. So where in Spanish you might say 'mas alto' in English you don't say 'more tall' but rather 'taller', adding the comparitive suffix '-er.' And you don't say 'the more tall' or 'most tall,' but 'tall-est' The general rule you could teach there is that most times, 'mas' equates to the suffix "-er" and 'el mas' to the suffix 'est.'
However, this also highlights the whole problem with English in that there are so many exceptions that simply have to be learned by rote or usage and correction. For example, you don't say 'popularer' but 'more popular'! On the other hand, 'less' is pretty much the same as 'menos' and we don't have a suffix meaning less - in fact when we do suffix 'less' to a word it means 'without' (Spanish 'sin'). English is a stupid language. I love it!
