Will NASA ever recruit the average guy?

Kind of a bad title for the question, but my query is concerning those whom are recruited for astronaut training. I am a senior and applying to the Naval Academy at the moment. Despite my apparent competitiveness for this school I am lacking in Math skills. I want to go to space one day but I understand that the selection process is rigorous. Physically I have it down, but academically I could be lacking when the time comes around. (I hope to go into Naval Special Warfare and be a Commando and down the road apply to be a mission specialist.) As our technology develops will we ever begin to recruit more average people from society to go to space? I mean sort of like Star trek (lame i know but you get the point). When will we begin to have average people allowed to explore along with the intellectuals like it was back in the Age of exploration in the 15th century? If ever.