Translation:The schoolgirls are learning with their friends.
I don't understand why this isn't "Ihnen" rather than Ihren? It's after mit so it is in the dative case. "Their" in the dative case should surely be the version of sie (pl.) and therefore Ihnen? Ihren would mean something like "her" friends, but don't see how this can be used given that "the students" is plural?
Actually, 'their' in the dative case isn't the dative of 'sie'. Those are distinct words (no different from in English, where 'their' and 'them' are not interchangeable), and the one you're looking for here is a dative case possessive adjective, not a dative case personal pronoun.
The dative of the third-person pronoun 'sie' is indeed 'ihnen', but if the sentence said "mit ihnen Freunden", that would be like "with them friends", not "with their friends". The word that goes here is the possessive adjective 'ihr' (cognate to the English 'her', which has the same ambiguity between object pronoun and possessive adjective). That means either 'their' or 'her', so it could indeed mean "with her friends", but you're supposed to figure that out from context; since there's no singular 'sie' antecedent, but there is a plural one, you know the possessive is plural as well.
'ihr' is declined like any other adjective, which is where the -en comes from: it's the normal ending for an adjective modifying a dative case plural noun not preceded by an article. http://en.wikipedia.org/wiki/German_declension#Attributive_adjectives
I got this one correct, basically, because (as you said), I figured by context "their" was the answer. HOWEVER, I'm not patting myself on the back, because I still had that question in my head of "her".
With... "That means either 'their' or 'her', so it could indeed mean 'with her friends', ..." Are you saing that ihren Freunden could be translated as "her friends"?
Sorry if this sounds repetitive. I'm just truly wanting to make sure I understand.
Yes, “ihren Freunden” can indeed be “her friends”, and it's only by context (because there's no possible feminine singular antecedent for that “ihr”, but there is a plausible plural one) that we can tell that “their friends” is meant.
If the sentence said „Die Schülerin lernt mit ihren Freunden”, the same phrase would mean “her friends”.
You may have had another typo. I've lost a point for a typo, even when it says the reason was for a missed umlaut. I've also discovered that typing e after the umlaut letter (fuer, for example, instead of für) will give you credit for the correct spelling, without having to mess around with clicking for umlauts.
A different language website translates a possessive pronoun or adjective or whatever "ihre" and the like are called followed by "Freunde" as boyfriends. To express the idea of friends-no possessive word can proceed it, but von ihnen could follow it. Where lies the truth?