On the quantum cohomology of the plane, old and new

We describe a method for counting maps of curves of given genus (and variable moduli) to $\Bbb P^2$, essentially by splitting the $\Bbb P^2$ in two; then specialising to the case of genus 0 we show that the method of quantum cohomology may be viewed as the 'mirror' of the former method where one splits the $\Bbb P^1$ rather than the $\Bbb P^2$, and we indicate a proof of the associativity of quantum multiplication based on this idea.