For a compact semi-Fano toric manifold $X$, Givental's mirror theorem says that a generating function of $1$-point genus $0$ descendant Gromov-Witten invariants, the J-function of $X$, coincides up to a mirror map with a function $I_X$ which is written using the combinatorics of $X$. The procedure of obtaining the mirror map, which involves expanding $I_X$ as a suitable power series, is somewhat mysterious. In this talk we'll describe some attempts at understanding the mirror maps more geometrically.