The fact the poppies are "spreading like mad" goes to show that you don't have to cut the first years flowers off.

Jordan, I do rather doubt the verity of that advice. I suspect it may be what we often refer to as as "urban legend" or "old wives' tale". True annuals are just that, and there is no way to make them perennials. The same goes for perennials -- unless one is growing them in unfavorable conditons. (For example, tuberous begonias are perennials. However, in my climate, they are annuals unless one digs the tubers up in the fall and stores them appropriately. They simply cannot survive our winters.) Now that said there are also short-lived perennials like columbine that often die out after a few years. For that reason it is a good idea with such to allow some reseeding.