The certified flight instructor (CFI) informed the student pilot to make a touch-and-go landing. During climb out, the CFI informed the student they would be conducting some mountain flying and to climb straight ahead over a ridge line. Upon clearing the ridge line, the CFI instructed the student to make a right turn into a canyon. After entering the canyon, the CFI realized he was in the wrong area and the airplane would be unable to clear the ridge line to their immediate front. He took control of the airplane and made a left turn towards the left side of the canyon. He then initiated a hard right turn and realized there was insufficient room to reverse course. He leveled the wings of the airplane and it collided with the trees. The airplane sustained structural damage and came to rest on the side of the ridge. The CFI stated there were no mechanical problems with the airplane and he was unfamiliar with the geographic area.