I often hear people talking about living in the present moment as if it is a struggle, some cosmic game of attempting to grasp something that is fleeting, illusory. They say things like, "the moment I have it, it's gone." While this is true and can be frustrating, the last thing present moment awareness is about is grabbing serenity. I have always liked the metaphor of the river (borrowed from my Christian contemplative practice of centering prayer) in talking of the flow of thoughts. Imagine the stream of consciousness as a river, with boats and debris representing thoughts. You're sitting on the bank of the river watching it. Normal awareness has you looking at each individual boat-thought, following it down the river with your eyes -- and to strain the metaphor, getting on it and opening hatches -- then suddenly shifting your awareness to another boat and so on. If your mind is particularly cluttered, you can feel overwhelmed by all the boats you have to look at and it can feel like that classic I Love Lucy skit with the conveyor belt at the chocolate factory, like you're falling behind and they start slipping by. There can be a sense of panic that a thought that's getting past you without attention is important and you're missing it.
Present moment awareness is simply sitting on the bank and watching the river, not the boats. Boats cross your field of vision and you do see them, but you don't follow them with your eyes or get on them. They're not out of focus, but you don't focus on them.