I have a five year old male and a ten week old female. They follow me around and stare at me. If I am doing something, I look down and they are sitting there watching me. What is that about? The puppy, in particular, follows me everywhere. I have to be careful not to step on her. If I am cooking at the stove, she is sitting between my feet.
These are "velcro" dogs. Mine follow me everywhere also and stare at me when they want attention. Mine go as far as to paw me to get my attention and occasionally try to move my laptop mouse to get to my lap Mine also love to play, so the "stare" may be that they want me to play with them.