Set aside a large bowl filled with water and the juice of half a lemon for the artichokes; the lemon will keep them from discoloring. You probably will need two knives: one long and thin for doing the rough cutting and a smaller one for finishing.
Slowly turn the artichoke against the sharp edge of the knife while making an abbreviated sawing motion with the knife. (It’s easier to control if you use the base of the knife rather than the tip.)
You will begin to cut through the tough outer leaves; when you can discern the natural cone shape of the artichoke, adjust the knife to follow it.
Keep trimming until you’ve cut away enough that you can see only light green at the bases. Cut off the top half-inch or so of the tip and dip the artichoke into the lemon water.
With the paring knife, trim away the very tip of the stem, then peel the stem and base going from the tip to the base. There should be no dark green tough spots left, only pale green and ivory. If you’re leaving the artichoke whole, put it in the lemon water and repeat the instructions for the remaining artichokes.
For most recipes, you’ll want to quarter the artichoke. Set the artichoke on the cutting board upside down, resting on the cut surface. Cut it in half vertically, then in half again. Check the choke. With babies, there probably will be just a little fuzz; if this is the case, put the quarters in the lemon water and go on to the next artichoke.
For medium and large artichokes, there will be what looks like very fine hair. Cut just below that to the very base of the leaves and it will pop off, leaving a clean heart.
Put the cleaned quarters in the lemon water and go on to the next artichoke.