This is a quote from Chess Master and author Bruce Pandolfini:
People often ask me, "How can you tell if a kid has talent?" By observing how a child looks at the board, I can tell if the child has a sensitivity to the game. When most children look at the board, they stare at a single point. But chess is a game of spatial relations. If I see a lot of eye movement, that's usually a reflection of real thinking. The thinking may be incorrect -- kids are kids -- but that eye movement tells me that the child may have something.
Its an interesting point but I don't think I entirely agree. For instance, if there is one part of the board where all the action is currently taking place, then you usually wouldn't need to look over the entire board.