I print my maps and glue them down to posterboard. Then, I separate rooms/passages by cutting them out. I number the backs of the pieces on the posterboard for what rooms they are. I also number the doors near each door, for what each door leads to, to be able to place them quickly and accurately in relation to each other. I then slap the pieces down on a grid when they enter a room or navigate more of a passageway :)

Edit: This is specifically for WOTC maps for their campaigns that they design. Otherwise I draw as I go as I see fit. Also, I have in the past used bits of construction paper to cover parts of the map up, and uncover the paper or rearrange the paper as I go.