The certificated flight instructor (CFI) reported that the student pilot was practicing short field procedures and had completed several uneventful landings. As they approached the runway, the CFI noted that the student had leveled the airplane too high. He instructed the student pilot to add power; however the student pilot reduced power and lowered the nose. Subsequently, the airplane landed hard. The CFI stated that he immediately took control of the airplane after touchdown, bringing it to a stop on the runway. The CFI reported no preimpact mechanical malfunctions or failures with the airframe or engine that would have precluded normal operation. The airplane sustained substantial damage to the firewall during the accident sequence.