The flight instructor was demonstrating a soft field takeoff to his student. During the takeoff roll, he applied full back pressure on the yoke; he heard the stall horn and he lowered the nose. As the aircraft left the ground, the flight instructor heard the stall horn a second time. The airplane's right wing began to rise as the aircraft reached an estimated altitude of 20 feet above ground level. The flight instructor checked that the throttle was full forward, but the airplane would not accelerate. As the airplane continued to turn left, it departed the left side of the runway, and the left wing impacted a runway information sign. The airplane came to a stop in the grass off the left side of the runway. The left wing's forward spar was damaged and the left fuel tank was ruptured. The right wing's outer 4 feet (estimated) was bent up and the engine's fire wall was wrinkled.