My school started this RPing club, but in my school right now only 2 people know how to DM and do it effectively, including me. As the teacher running the club knows nothing about Pathfinder, it seems that I will have to get some of my friends who play Pathfinder during lunch to DM for other groups. How can I teach 3 people how to be a DM in 15 15-minute sessions?
Assign homework and use a "flipped classroom" approach.
No, really. Have them use the various recorded play examples (some referenced here), and use the sessions to answer questions. (Technically speaking, this is called a "flipped classroom" and is an actual pedigogical approach.)
Once they have consumed the actual play examples, then spend time inculcating system mastery by assigning problems in system mastery. (Here's this situation, how would you adjudicate it? Why?) and spending the 15 minute sessions in discussion of the answers of the last homework.
It's also critical that they learn a rounded game. For a "midterm", have each of them run a game for the others. Each GM must make characters for the group and explain those characters and their mechanics to the group.
Once they've demonstrated system mastery and a familiarity with the literature. for Pathfinder, they can start learning how to DM. In many ways, I would teach them how to DM Pathfinder using different systems, both to break them of the "one system rules all" mentality that... is just a horrible habit, and to teach them the fundamental insights of these games. For this, the latter half of the course, I would explore Apocalypse World, Mouseguard, and Paranoia. (With them finding resources and listening to those resources on their own time. If anyone fails three weeks in a row to have done the prep, ask them to leave.) For Apocalypse World, the fundamental rules of storytelling are just super-super important, in terms of framing, pacing, and visualising scenes. For Mouseguard, the mechanic of yes/yes-complication is crucial: it will help your Pathfinder DMs not require umpteen hundred stealth checks for no-reason and will (hopefully) teach them that every roll must be important. And finally, Paranoia, because there is no better book to teach a DM how to handle Players. For their final exam, have them run the game of those three (or similar) that they like the most.
Get the 5 people together. You can either try to teach them over a single session of a few hours, or if you can only do 1- hour sections of lunch, several sessions.
Take an introductory adventure, and some pregens. Run scenes or parts of it to show off aspects of the rules - not the whole adventure. Explain what you're doing as you run it, in a metagame sense to the new GMs.
"So, I'm having this NPC come and talk to you to help move along the action. I can see you guys aren't sure which direction to go next, so I'm having this character help come give you some clues." Let them ask metagame questions about why you're pacing things the way you are, when you're choosing to use the rules or ignore them, etc.
Cover things generally in this order: - Basic Rules (D20, basic skill rolls, when to roll, when not to roll, how to pick DCs) - Scene setting (how to set up a scene, when to end it, how to feel the pacing, etc.) - Combat rules (To hit, damage, etc.) - Building good encounters - Magic (in a general sense, specific things to look out for, etc.)
By using stuff in an adventure to teach this, this will also teach them how to run it. Have them run short scenes or sections to do it in practice.
This, of course, assumes that you're only interested in teaching them Pathfinder. There's a lot of other games that are good training options that don't require as much rules knowledge - for example Lady Blackbird, FATE, etc. are options.
Before the session, print out and hand them copies of Greg Stolze's free PDF "How to Run Roleplaying Games". At the session, take questions and discuss the text, like a book club.
This will do two things: it will separate out those who aren't able or willing to do the kind of reading and prep required of a GM, and it will teach the rest the principles of GMing.
Then, they learn on the job, having occasional meetups to discuss how it's going and get feedback and advice.
Bolded for the tl;dr thesis in each paragraph
Brian Ballsun-Stanton's answer is excellent if you only have the lunch time to do it, however I have a two parter for a second idea:
Is there any core game actually taking place? If there is a weekly game, you only need one person to actually run the game, and you can even let the other prospective DMs be the party. As they throw challenges at you, make sure to write down specific points that hinged on semantics, made you whip out Rule 0, or that simply streamlined the process through methods. They too can take notes and then during your lunch sessions after the game you guys discuss everything that you feel needs illumination.
Of course depending on how many people are in this club you may want to let them take turns DMing on "training wheels". If they at least know the rolling system it's the biggest head start to running the game in my eyes, and you can relegate tasks to the trainee on the spot and discreetly explain why they are doing it. "Here's a notecard with a trap in the room. Wait until someone hits the trigger listed here, and follow the card down" or "Bob cast Spell X. Here's the page the spell is on, how does it affect the target(s)?" It may be a little messy at first but that's why you're there. It's not complete control as you're running the game, but now they can focus on little bits at a time instead of trying to surf a tsunami.