The student pilot stated he returned from a solo cross country flight and entered the traffic pattern for landing. During touchdown, the airplane's nose pitched up and the pilot subsequently pulled back on the flight controls. The airplane became airborne, then settled and bounced a second time on the runway. After a third, and more pronounced bounce, the nose gear collapsed, and the airplane slid to a stop toward the left side of the runway. When the pilot was asked how the accident could have been prevented, he stated he should have performed an aborted landing after the first bounce. Examination of the airplane revealed structural damage to the engine firewall, engine mount, and the rudder. The student pilot stated that he did not experience any mechanical problems with the airplane before the accident.