  We analyze the relationship between two compactifications of the moduli space
of maps from curves to a Grassmannian: the Kontsevich moduli space of stable
maps and the Marian--Oprea--Pandharipande moduli space of stable quotients. We
construct a moduli space which dominates both the moduli space of stable maps
to a Grassmannian and the moduli space of stable quotients, and equip our
moduli space with a virtual fundamental class. We relate the virtual
fundamental classes of all three moduli spaces using the virtual push-forward
formula. This gives a new proof of a theorem of Marian-Oprea-Pandharipande:
that enumerative invariants defined as intersection numbers in the stable
quotient moduli space coincide with Gromov--Witten invariants.
