C172 Flight Instructor reports bounced landing by his student that results in aircraft damage.

Narrative

I had my student practice landings on Runway 22. The wind was shifting between straight down the runway and nearly a direct crosswind. There were also moderate gusts present at low altitude. About halfway through the lesson I suggested that we use Runway 31; which the winds were favoring by that point. The first pattern and second pattern to short final were uneventful. During the second landing the aircraft touched down initially then bounced back into the air. The aircraft climbed to about 30 FT with a slowly increasing pitch attitude and decreasing airspeed. I called for power to be added two times; followed by a call to go around. As I reached to push the throttle forward I felt the sink rate increase; and the impact occurred as I was pushing the throttle forward. The impact was hard enough to knock my headset and sunglasses off my head. As we skidded to a stop I pulled the mixture to idle cutoff and turned the master and ignition off. My student and I then exited the aircraft and surveyed the damage.