The student pilot, who was practicing touch-and-go landings in calm wind conditions, inadvertently landed near the left edge of the runway. During the landing roll, he did not correct back to the runway centerline prior to adding power for takeoff. Soon after adding power for takeoff, the airplane departed the left side of the runway. By the time the pilot was able to bring the airplane to a stop, it had traveled about 900 feet over uneven terrain. Prior to the time the airplane came to a stop, its nose landing gear strut collapsed, resulting in substantial damage to the firewall. There was no evidence of any anomaly or malfunction in the airplane's nose wheel steering or rudder system.