Being the basis of Gromov-Witten theory, Kontsevich's moduli spaces of stable maps are important in different areas of mathematics and theoretical physics. For this reason, it is interesting to study their geometry, for example analyzing their (co)homology. In fact, determining their Betti/Hodge numbers is already a nontrivial problem. In this talk, I will present a method for computing the Betti/Hodge numbers of moduli spaces of stable maps from genus 0 curves to a Grassmann variety G(r,V). First, by using the combinatorial properties of these spaces, I will show that the problem can be reduced to the computation of the Hodge numbers of the open locus parametrizing stable maps from smooth curves. Then, I will show how the latter can be explicitly calculated, by means of a suitable Quot scheme compactification of the space of morphism of fixed degree from the projective line to G(r,V).