Snap or cut the stems off the artichokes and trim off any tough outer leaves from the base leaving some still attached if the artichokes are young and tender. Cut the top 1/3 off with a sharp knife, then use a pair of scissors to trim any other leaves flat at the top so you are left with the fleshy edible parts still attached. Put the artichokes in a large pot, and fill it with water until they are covered. If they bob about, put a saucer on top to hold them down. Add the lemon halves and bring the water to a boil, then turn down to a simmer and cook for 20-40 minutes, depending on how big the artichokes are. You will be able to easily pull out a leaf when they are cooked. Drain the artichokes, thoroughly, upside-down.