I wouldn't agree with the advice above to push through it. In my experience, and those of other dancers that I have met, shin splints are not something that go away with repeated stresses. Rather, that will further aggravate the sheath surrounding the tibia, and cause further inflammation. If this continues to get worse, it can develop into stress fractures, and in extreme cases into compartment syndrome. Rest. Ice if inflamed. Elevate the legs (lying on the floor with legs up on wall) after any exercise, to help the blood drain from the legs. This will feel quite strange, especially if you're getting quite a lot of pain. Basically, the more the calves get tight, and the blood gets locked into them, then the more stress that places on the connections to the bone, as you run (repetitive impact). So these tactics, as well as massaging the blood out of the calves, will help.
Another aspect, is making sure that your legs and feet are tracking properly in alignment, so that strange stresses aren't being put through your shins. Check when you're extending your legs and feet, that this action happens in a straight line. When bending your knees while standing, make sure that your knees are tracking directly over your ankles and toes, and not twisting either way.
Lastly, when the pain lessens with some rest, start to do calf rises to strengthen your calves and ankles. This will help with the tracking, and also absorption of the impact. Use your feet when your run, and don't rely purely on padded running shoes.
Hope this helps.