If you ever want to pass some time on a long flight, just turn to a fellow passenger and announce “the correct way” to cook an artichoke. A disagreement will ensue because there are about as many ways to cook an artichoke as there are connections through O'Hare.
Most newcomers to the world of artichoke cuisine can't believe we go to so much effort for such little payoff. After all, at least 85 percent of an artichoke is inedible — you're only after that tender morsel at the tip of each leaf and, oh yeah, that succulent heart.
Anyway, to eat an artichoke, my approach is to simply pull leaves off the main globe one at a time, beginning with the smallest ones around the stem, and work my way toward the center. When you pull off a leaf, you'll notice a plump little portion of artichoke meat at the base of the leaf. If this is your genuine first artichoke, then I highly recommend tasting it au naturel. No mayonnaise or butter or other type of sauce. Just you and a pure artichoke experience. To do this, just scrape off that plump tip by gently biting down on the leaf slightly ahead of the edible portion and scraping it through your front teeth. The fibrous leaf comes out; the tender pulp stays behind.
For the remaining leaves, you certainly have the option of continuing to eat them in their unsauced state. (Why pick up any bad habits if you don't have to?) Or you can do what most artichoke lovers do, and that is justify the consumption of vast amounts of mayonnaise and butter by dipping the pulpy tip of your artichoke leaves into one of those offerings before eating.
By the way, you'll want to experiment with the leaf orientation as it's inserted into your mouth. That is, try some leaves with the pulpy portion facing up, and some with the pulpy portion facing down. One way will feel more enjoyable than the other, and that's how you'll inevitably continue to eat them for the rest of your life.
Anyway, once you reach the fuzzy center, you're ready to enjoy the ultimate reward, the heart. Scrape the “choke” from the meaty bottom by using a spoon or knife. The big, thick, disk of artichoke you're left with is the heart. Using fingers or fork, dip portions of it into your dipping sauce (unless you're being pure), and enjoy!