Wash the kumquats and half them using a sharp chef knife. Discard all the seeds.
Place the halved kumquats, water and sugar in a pot. Let the sugar macerate for at least 3 hours in a cool area. Bring to a boil, then lower the heat to a medium low and cook until the fruit becomes transparent and the syrup thick (about 10 minutes). Remove from the heat. Remove and discard the star anise seeds. Cover, set aside and let it sit overnight (at least 6 hours) at room temperature.
The following day, bring the kumquat syrup to a gentle boil. Stir every now and then using a wooden spoon. Skim any foam that develops on the top before pouring into the Mason jars. Add the anise extract.
Fill a big pot of water and bring to just under a boil. Place the Mason jars, lid, heat-proof funnel and tongs in the pot and let it boil for 10 minutes. Be sure to read the instructions that come with your Mason jars. They'll give you exact time and techniques.
Remove the jars from the water and fill with kumquat preserves leaving about a 1/4-inch head space or whatever your canning directions say. Carefully place the lid on the jar. Tighten the collar around each jar. Bring your big pot of water to a boil and place all the sealed jars in it for 15 minutes.
Remove the jars from the water. If the top of the lid still pops, the vacuum didn't form and you'll need to consume the product in the next couple of weeks, storing it in the refrigerator. Otherwise you can store it in your pantry for up to a year.
To insure a good sterilization of the product, I stack a dozen magnets together to create a stick and use it a gripper to place the lids on the jars without touching them. Just make sure the magnets are clean!
The next day, when you bring the preserves back to a boil, your candy thermometer should read 220°F. Remove from the heat and start canning.
The preserves will taste better if you let the sealed jar sit for at least a week before opening. Patience is a virtue.