The notion of "game" is intuitive, but tricky to pin down, especially with the advent of videogames, MMO communities, etc. But here's a vague description that's not too far off the mark: a game provides challenges, and somehow quantifies the player's progress through (or victory over) those challenges. D for instance, and all RPGs, are "about" facing trial after trial after ambush after trap after trial, coping with each in one way or another, and getting rewarded with levels and booty, the better to face the next hurdle.
A class is also supposed to provide challenges, and quantify the student's progress through (and ideally, victory over) those challenges. It's harder to find distinctions between games and classes than similarities.
What are the traits of a good teacher? Some are strict, some are friendly, some are innovative, some stick to tried and true methods. But a good teacher makes classes compelling, makes the subject interesting, keeps students interested.
Thus, the post title. A dungeon master (or any other game designer) starts with a bare structure, and builds a compelling environment, and a dramatic storyline that doesn't just make you follow it, it makes you play a part in it.
If anyone's got a good campaign idea for College Algebra class, pleeeeeze let me in on it. Seriously.
Of course, every good DM uses some of the fine resources already out there. I'm going to start out with Module Q1, Queen of the Demonweb Pits.
"Okay, Stephen, you're the thief, hiding under the drow altar. Peering out from under the black velvet, you count 46 legs. Meanwhile, from her vantage point peering down from the balcony, Vanessa the barbarian sees 16 drowish heads. After your companions report what they've seen, it's up to Ashley the mage (who has the highest Int): how many driders and how many drow are out there?"
Do you want a module that also leads people to reason out proofs of theorems? Getting people to think through an algebra problem is one thing, but in calculus you also need them to discover and understand (e.g.) the Fundamental Theorem of Calculus. Computation is one thing, but theoretical results would require more.
Proving theorems is harder, but consider the genius of Raymond Smullyan. He's concocted puzzles and stories (and stories with puzzles are not far off from RPGs) that lead to Godel's Incompleteness Theorems (see his "The Lady and the Tiger").
Are you arguing that it's impossible, claiming that teaching calculus is harder than algebra, or asking for such a module yourself?
I disagree with the first two claims, and agree with the last one.
I noticed many years ago that I've solved problems in video RPGs, high-level puzzle kinds of things, that I'd never have even attempted outside of gaming, so I am certain Bob's idea is a sound one.
For some time I've had an idea for a geometry proof game that would ultimately work as a video blackboard, assigning names to points automatically and, to the best of its ability, taking the theorems that had been worked out in the game up to that point and presenting a relevant them from a list when points or lines are selected in the UL.
If there is something like that out there already I don't know, but it doesn't exactly sound like Mathematica's style.
I somewhat dislike the growing impression that it is the instructor's job to keep the content interesting for the students. Germaine, sure. Entertaining? Well, only to a point.
When choosing an undergrad institution, I remember a great-uncle looking at me and asking, "They all got libraries, don't they?" His is, on the whole, the proper mindset for learning.
Don't worry, I usually try keeping class fairly entertaining, both for the students and my own selfish motivations of, well, enjoying myself, but I've found the best people to ask how to make the class more engaging -- the people to ask what game should be Game Mastered -- is likely them schmoes registered for the class in the first place.
Though I'll admit it's very good to see mention of the driders. For me, that's the first time in about 18 years. ;^)