I've always agreed with nyte's point that, if a person is too focused on falling asleep, they put pressure on themselves to do so and therefore can't. It's similar to the idea that, if you spend your life looking for love, you might never find it; simply because you were looking too hard. The solution is to relax and let it flow.

I disagree, however, with the concept of trying to stay awake in order to fall asleep; you're not going to trick your body with reverse psychology. You're also not going to tire yourself out, because you'll be focusing on staying awake, or keeping your eyes open. The point is that, if you focus on anything at all (whether it be falling asleep or staying awake), your brain is too active to slip into unconsciousness.

This is where the idea of counting sheep comes from; the idea is not to try and stay awake, but to forget about sleeping and waking. You busy your mind with something trivial and repetitive, and soon you will fall asleep. This also works with daydreams; if you construct a fantasy in your mind, the creation of ideas is enough to tire out the brain. Remembering things doesn't work, however; you're accessing a separate part of your brain and this only serves to keep you awake. Counting sheep is a creation of ideas, not a recalling of memory; dreaming is also the generation of new thought.

In conclusion, try them both. Try staying awake to fall asleep, and try involving yourself in a fantasy as you close your eyes: different things work for different people. One thing is certain, however: if you put pressure on yourself to go to sleep, you will never be able to do so. If you start to panic because it's late and you need to get up early, you will not get to sleep any faster. Take your mind off it, however, and you'll soon be sleeping soundly.

Oolong has just informed me that recent studies show that counting sheep doesn't work for many people, but visualisation does (e.g. of a tranquil scene, oceans, waterfalls, lush meadows etc). This is perhaps a corollary to the section about creation of ideas; the point is to involve yourself in imagination to the extent that you fall asleep. Thanks to Oolong for the information about the importance of visualisation. See also falling asleep visuals.