Put on a pair of rubber gloves. Not only will donning the gloves help you to better remove the gum, but it keeps your manicure protected. It also does the dirty work and touches the gum for you, which is especially useful if the gum isn't yours -- ewww.
Pull whatever gum you can off with your fingers. Avoid tugging too hard on the sticky substance, as doing so can damage your sweater. Only remove what comes off easily.
Stick your sweater in a plastic bag. If you have a large resealable bag that will hold your sweater, use that. If not, use a trash bag or any other kind of plastic bag. Allow your sweater to get its chill on in the freezer for about an hour.
Pull the sweater out of the freezer and remove it from the bag, then use a dull knife to scrape the gunk off. Since the gum is frozen, its hard consistency should make removal easy.
Scrub the area with a toothbrush, if any gum remains. Scrub gently to remove the gum, but take care not to harm the material.
Apply warm white vinegar to the gum and its surrounding area, if it won't let go. Dip a toothbrush in the mixture and scrub away. The components in the vinegar should break down the gum, forcing it to take a hike. Immediately wash your sweater as you normally would once the gum is gone.