|Your mediator at times|
There is common method for problem solving in software engineering which is commonly called rubber ducking. The story from which this technique derives its name comes from the book The Pragmatic Programmer:
A very simple but particularity useful technique for finding the cause of a problem is simply to explain it to someone else. The other person should look over your shoulder at the screen, and nod his or her head constantly (like a rubber duck bobbing up and down in a bathtub). They do not need to say a word; the simple act of explaining, step by step, what the code is supposed to do often causes the problem to leap off the screen and announce itself.
This technique works equally well outlining your problem to a rubber duck, bear, or an interested ear. A problem having to be explained clearly, verbally and thoroughly, has a tendency to open up its own solutions space. The change of perspective from internalizing to externalizing the problem description yields insight into the source of the issue and opportunities for answers that were not previously present.
That this method was described in The Pragmatic Programmer is no coincidence. As the rubber duck method is a pragmatic way to problem solve technical impasses, mediation is a pragmatic solution to interpersonal conflict. In the mediation, the mediator acts initially as the rubber duck.
A mediation would typically begin with each party to the issue describing their understanding of the issue. This informs the mediator and grants all some perspective as to the understandings of each party. In this way the mediator does an initial session of rubber ducking requiring each party move their internal understanding of an issue into external verbalization. The simple act of explaining the perceived issue often allows parties to self-discover the underlying problems. And through that process already begin to open the issue's solution space.
As the goal of a mediation is to have the parties resolve their own conflict, mediators are trained to be that rubber duck.