The pilot reported that when he landed, he didn't pull the nose up, and the airplane landed on the nose wheel. The airplane bounced up. The pilot didn't get the nose up again, and the airplane landed on the nose wheel a second time and blew the tire. The pilot had difficulty steering the airplane, and it veered off the runway. The nose wheel went into soft dirt and the airplane subsequently nosed over. Later, an investigation revealed the 54 year old pilot was wearing monovision contact lenses to correct for distant vision in his right eye and near vision in his left eye. When correcting lenses were prescribed for the pilot (before the accident), his optometrist was not aware of FAA requirements concerning use of contact lenses. After the accident, the pilot flew with a flight instructor (CFI) to prepare for an FAA-directed evaluation. The CFI reported that while the pilot was using monovision (contact) lenses, he had a tendency to flare too late. During this approximate time, the pilot visited his aviation medical examiner (AME) for his annual flight physical. The AME found the pilot to be using monovision contact lenses and advised him that these were not approved for use in flight, and that he should discontinue using them for vision correction. After the pilot changed to bifocal glasses, the CFI noted a significant improvement of his landings.