I Want To Learn That Song That Goes....
When a kid says, "I want to learn that song that goes..." and then hums some melody, it has to be a moment of triumph for the intelligent piano teacher. It signals that a child has taken the piano on as their own, and are ready to explore music that they like.
Once a child makes this move, you should stand out of the way. Throw more fuel on the fire by finding every song in the child's head, and figuring out ways to arrange it so the child can play a bit of it. Every film, cartoon, nursery rhyme, TV show and computer game tune is eligible. Think of all the songs running around in your head, and then find what's going on in the child's.
Difference Between Work And Play
The opposite of this situation is when a child is bored with the piano, and doesn't care about any of it. The songs bore them, the piano bores them, even you, the teacher, are supremely boring to them.
But find that favorite song, and all of a sudden you are a much-needed partner in the search for a song. When a child crosses that barrier from "work" to "play" you will find the most fertile ground for teaching.
Like Kids Playing Together
Think about it. A child's experience of education is largely force-fed. When child plays with friends or by themselves, their entire imagination, inspiration and intelligence is stimulated. It is this same "play" state that we seek to awaken in the child's experience of the piano.
Without this sense of play, piano lessons are exactly what most kids claim they are. Drudgery. Don't expect enthusiasm for drudgery from a child.