Kontsevich and Manin have introduced the moduli spaces of stable maps as background spaces on which to count curves with fixed properties in projective manifolds. In this talk I will describe a new construction of the Kontsevich-Manin spaces for rational curves, starting from the simpler example of the Grassmannian of lines in the projective space. This new construction allows us to understand the structure of their cohomology rings in a natural, geometric way. Time permitting, I will explain how this construction can be formalized by setting a family of different stability conditions on maps from curves to the projective space. This talk is based on joint work with Andrei Mustata.