  These notes are intended as an easy-to-read supplement to part of the
background material presented in my talks on enumerative geometry. In
particular, the numbers $n_3$ and $n_4$ of plane rational cubics through eight
points and of plane rational quartics through eleven points are determined via
the classical approach of counting curves. The computation of the latter number
also illustrates my topological approach to counting the zeros of a fixed
vector bundle section that lie in the main stratum of a compact space. The
arguments used in the computation of the number $n_4$ extend easily to counting
plane curves with two or three nodes, for example. Finally, an inductive
formula for the number $n_d$ of plane degree-$d$ rational curves passing
through $3d-1$ points is derived via the modern approach of counting stable
maps.
