After reading to my three year old son his third and final bedtime story for the night last night, we started talking about going to sleep and dreams. He asked me, "Do people still dream when they're awake?"
And I thought, that's a good question. And then I thought, it could be, people could dream all through the day (not day-dreaming) and not even know it, and then when they go to sleep they gradually get in touch with their dreams.
There is no direct observation that proves one begins to dream once one reaches deep REM sleep. So I said to my son, "That's a good question. Maybe people do dream when they're still awake, but they don't actually see their dreams until they fall asleep."