It is not as hard to play piano by ear as it seems!
Here is an example where a student (Eliana 5 y.o.) is being taught to listen attentively to the musical texture of the one-voice melody and to obtain not only an exact realization of the musical text and its correct phrasing, but also an intelligent reproduction of the content of the piece and its musical character. She is learning to play piano by ear.
Here are a few words on how it can be done in six steps:
Step 1 - learn the words of the song. If student is unable to remember the whole line of the words the song should be subdivided into short phrases and read in sections.
Step 2 - read the words of the song while clapping the rhythm (with help of a teacher).
Step 3 - find "long" words/sounds of the song.
Step 4 - read the words again while clapping the rhythm and making stops on the long words. Step 5 - try to sing song with tune (while clapping the rhythm). When singing it is vital to have clear intonation as well as musical expression and rhythmical accuracy (teacher plays song on the piano, helps signing it during the exercise, uses Kodaly hand signs method). Step 6 - sing and play piano by ear.
"In the shadow of the hill"
"Twinkle-Twinkle little star"