What is the role of kōans in contemporary Buddhism?
The way my Zen Master used them (as much as I can pretend to have understood my Zen Master at all!) is to essentially give the person a hint about a particular pathological pattern the person was stuck on. So the koan would always match the elephant in the room that the student was working hard to ignore. The student could then resort to analysis, or simply mull over the koan while going along with the daily activities. In any case, solving the koan required getting an insight into oneself and one's fundamental relationship to the world, as opposed to figuring out some new external fact or gaining an analytical understanding.