"Paper Prototyping" involves building a close approximation of game systems using physical materials rather than digital tools. It often saves us time, though admittedly at the expense of a few trees.
Paper prototyping for a game can look like a giant arts and crafts party. Game pieces are made out of wire ties, scribbled number lines track scores, and post-it notes turn into beautiful maps if you squint at them really hard while mentally picturing a beautiful map. Math is simplified to basic addition and subtraction—we can always model complex math later with the help of computers. One of the rules of prototyping is to know what you're looking for, in part so that you don't get bogged down with irrelevant details.
While paper prototypes are fun to put together, a decent amount of thought and planning goes into their construction. We have to make sure that the prototype models something similar enough to the proposed digital systems so that we can learn something by playing with the prototype, but we also have to keep the prototype simple enough to handle physically. Also, these materials take up space, which is a point that seems obvious until your grand prototype ends up running off the edges of any table you try to build it on. Still, a lot of creativity can go a long way, and in the long run few problems are insurmountable when duct tape is easily accessible and imagination is liberally applied.