I learned this technique for peeling a tangerine in Japan, and use it all the time now.
It's such an elegant way to peel the fruit, and creates a small bowl at the end, useful to hold the tangerine seeds.
Instead of a bunch of messy bits of peel, you have a pretty, flower like design at the end. You can easily teach children this technique and it's fun to see the different designs everyone makes with their tangerine peel! An everyday task like peeling a tangerine becomes an intentional act resulting in a small work of art.
Step 1: Dig Thumb Into Top Center of Tangerine and Start Peeling
Dig your thumb into the top center of the tangerine, getting under the skin, and peel outwards, being careful not to break the skin. Visualize this as a flower opening. After you have unfurled the first "petal" part way (no more than halfway down the tangerine), put your thumb in the top center, and peel the second petal. Repeat, until the top half of the tangerine has been opened completely, as in the third photo.
Step 2: Once Tangerine Has Been Opened All the Way Around, Remove Peel
One you have peeled away all your petals down from the top half of the tangerine, run your thumb around the bottom half of the tangerine to loosen the fruit from the skin, and pull away the peel.
Step 3: Admire Your Peel!
Some people consistently make 5 petal peels, others have 6 petals, and others a more random arrangement.
You can compare your peel with your friend's, or try to peel it in a certain pattern such as a 6 petaled lily. Afterwards, the peels can be neatly stacked and composted or used in a craft project.