Neither one is actually correct. Something can't hurt good (except in annoying 70s songs), so it can't hurt bad or badly. It hurts a lot, it hurts intensely, or as Nirerin said, it aches. Bad is an adjective of quality, and the verb "hurt" needs a quantifier or an adverb of intensity.
Or you could take it as an opportunity for vocabulary building. "Hurt + adverb" is pretty vague. A word like sting, ache, throb, etc gives a lot more information; it even lets the other person start guessing at why.
Also, if the teacher corrects your child on something minor like this, it's going to be better for the child in the long run if you support the teacher. Help the child learn to choose his or her battles and not sweat the small stuff. If she says 1/3 of 15 is 7, stick up for the kid. Bad versus badly... that's when you say, "I'm not sure myself, honey, but I think that since Ms. Teachly has probably studied grammar more than I have, we should listen to her unless you find a really good source that says otherwise."
Edited by: ANARIE at: 12/1/2012 (17:28)