Mix water and sugar in a medium saucepan and bring just to a boil to dissolve the sugar. This is called a 1:1 sugar syrup, which means half water and half sugar, by weight. When it comes to a boil, remove from heat and let rest with a cover while you prep the kumquats.
Prep kumquats. Wash and slice into 3/16" thick crosswise slices, just a little thicker than a quarter inch. Remove seeds by picking them out with the tip of a paring knife. Discard seeds and end pieces.
Cook kumquats. Stir kumquats into the syrup and bring back to a boil. Turn down heat until simmering, then cover with a round of parchment paper the diameter of the pan. Cook until tender but still crisp with nice texture. Start tasting for doneness after 15 minutes. Taste every 5 minutes thereafter until done. The approximate cooking time is 25-35 minutes. Don't cook until mushy. Turn off heat and allow to cool in the syrup.
When cool, pack into plastic containers and refrigerate until needed.
Use for a garnish for desserts or as a topping for ice cream.