Keen on (British English) the students' taking part: the teachers like the idea that the students participate.
Keen on the students who take part: the teachers like the students.
Sentence one can be understood as meaning the same thing without the apostrophe, but its ambiguous. You should avoid this construction because it isn't clear. I'd say The teachers are happy that the students are participating in the activities.
Thanks for the reply!
I would like to make some points for discussion and clarification: (1) the difference between
"N's + Ving" and
"N + Ving"; (2) the difference between
"be + adjective +prep + N/N's + Ving" and
"transitive verb or phrasal verb + N/N's +Ving".
(1) To me, "... the students' taking part..." and "... the students taking part..." have different meanings. The focus of "... the students' taking part..." is on WHAT the students have been doing (It sounds like, for instance, a habit or something the students do for many times), while "... the students taking part..." emphasizes "the action done ACTIVELY" (Which is the reason why V-ing is considered an active voice).
(2) "keen" here is an adjective, indicating "the state" rather than "the action" (More specifically, it's different from "I don't
mind you/your smoking here" or "He
insisted on his daughter('s) returning with him"). I mean...Could the part of speech in this case ("
be keen on someone" and "
mind/ insist on someone") determine what "... take part in..." turns out to be, as shown in the following options?
(A) ... the students' taking part in ...
(B) ... the students taking part in ...
(C) ... the students who take part in...
(D) ... the students who are taking part in...
If not, I suppose that each sentence has its own meaning that is different from the other three. And the reduction mentioned above is to shift one "focus" to another.
Context really matters. A single sentence may have different interpretations, or even worse, lead to ambiguity.