Scrape as much of the gum from the polar fleece as you can with a craft stick or spoon.
Wrap the clothing or blanket in a light plastic bag, gum side up. Place the bag in the freezer. The bag will help prevent the fabric from sticking to other items in the freezer and ruining the fleece.
Keep the bag in the freezer for at least two hours, or until the gum is frozen solid.
Remove the bag from the freezer and remove the gum with a craft stick or spoon. It should pop or crack right off.
Rub either an egg white, mayonnaise, lemon juice or white vinegar on the gum spot. The vinegar will work even faster if you heat it before applying it to the gum. Saturate the gum thoroughly.
Give the gum time to dissolve, or at least 30 minutes.
Check to see that the gum is extremely pliable, then remove it with a craft stick or spoon. Be gentle, as fleece is prone to pill and/or fray.
Apply a generous amount of peanut butter, baby oil, Goo Gone or WD-40 to the gum with a clean rag or paper towel.
Let the oily substance absorb the gum for 15 minutes.
Lift off the gum with a craft stick or spoon. You may have to repeat the process again to completely remove the gum. And be prepared to spray a stain lifter or other pre-laundry treatment to the fleece in case the peanut butter leaves behind a faint stain.