Here is a bit more on how I run mapping in my own game. This is mostly new to me - none of my previous groups mapped, I don't ever remember us mapping back in the "good" old days, and we didn't really need maps in my previous game. Not a lot of dungeons in that game, fantasy though it was.
As I've discussed before, I don't give specific sizes and cardinal directions when describing places.
I did rule that if the party has someone with Absolute Direction, they get specific measurements - I figure that makes up for the lack of utility of knowing North in a dungeon. This makes having the ranger-type who took that along very useful!
But I generally lay out what they see on the hex map, using props, building blocks, minis, etc. If they want to write it down, they can, if some character is mapping. Same for any other props - if they want to keep the handout picture of the six-fingered hand, the map of the surface, the picture of the critter - the characters need to spend in-game time and in-game resources making that picture. Then they can keep it. I've found this adds a bit of fun to the game - PCs paying for extra paper to make copies, PCs handing the picture or map to NPCs and then trying to explain to illiterates what the symbols mean, and PCs debating selling their inaccurate maps they re-did as real treasure maps.
So moment to moment, the players can see what the environment is like. But I'm not promising to leave it out there or show it to them again anytime they want - no way. You have to map for that.
We also use the DF rule (also in DF2) that lets the mapping character roll Cartography skill to get to ask me if what's down is correct. That's pretty amusing, when they're sure something is wrong but equally sure they've mapped their current location correctly. Warped dungeons, oddly shaped rooms, tricks and twists - all serve to make this tricky. I can at least tell them if they drew it correctly.
I also force the group to move a bit more slowly when mapping. Or at least charge them more passing time to move while mapping than when they move without. This is partly why they hit so few wandering monsters on the way out of the dungeon - they move back at a much greater speed. And any time spent in the real world arguing about the graph paper orientation, fiddling with the maps, erasing and re-drawing - it's minute for minute real time in the dungeon, which means more wandering monsters. There is a real cost for doing the map.
Finally, I did tell my players the scale of my graph paper, just so they know that they aren't "getting close to the edge" or any other meta-gamey type problem like that. I'm using a tiny 8-to-the-inch map, and some maps are portrait, some are landscape, some fill the page, some don't. They still use a much easier to read 4-to-the-inch map.
All in all, I've found that insisting on a mapping character (rarely the player of that character, though) and these ways of running the mapping/props/battlemap have really made the dungeon come alive. It adds another dimension of interest to our games in a way I didn't expect.
Plus it's funny to hear them say, "No, no, it should T out here. What the hell?" when they made some silly boo-boo six rooms back and can't figure out where they went wrong. Then, the slugbeasts come . . .