The flight instructor reported that the student was practicing stop-and-go landings in the tail-wheel equipped airplane. The student had just completed two takeoffs, and the roll outs on landing were satisfactory with a little help from the flight instructor. On the third landing, the airplane was about halfway through the roll out to stop when the student suddenly pushed on the right rudder. The flight instructor immediately applied left rudder and brake, called for an addition of power, reached for the throttle, and pushed on it. The airplane straightened out, but was in the grass beside the runway. The airplane became airborne, and subsequently collided with a runway light in its path. After the airplane hit the runway light, the flight instructor flew over the runway, landed, and taxied back to the hangar. The bottom of the airframe sustained substantial damage. The flight instructor reported no preimpact mechanical malfunctions or failures with the airplane that would have precluded normal operation.