Abstract: An elementary construction of the moduli space of stable maps would be a projective variety X such that the moduli space of stable maps was a sequence of blow-ups of X.  Such constructions exist for Fulton-MacPherson spaces X[n] and moduli of curves \bar{M}_{0,n} (both which are special cases of moduli spaces of maps) and enable the calculations of the cohomology.  Towards this goal, we construct the moduli space of stable maps to projective space as a GIT quotient of the space of maps to the product P^r x P^1.  As a corollary to this GIT construction we find a projective variety X that is birational to the moduli space.   Time permitting, we will explain how this X fits into an "elementary construction".