I let my piano beginners learn these pieces by ear.
So they can control their arms, hands and fingers. Afterwards they are really ready to start with playing sheet music.
 
The first four pieces are children's songs, the Flea Waltz is something like a traditional, the other three pieces I made myself.

MP3's

• Ist ein Mann in' Brunn' gefallen
• Kuckuck
• Hänsel und Gretel
• Summ, summ, summ
• Little Waltz
• Flea Waltz*
• Etude
• Blues In A-Minor

PDF Downloads

• Children's Songs
• Little Waltz
• Flea Waltz*
• Etude
• Blues In A Minor

* >> Flea Waltz Blog

Learning and playing by ear and thus by heart has another advantage. Our brain has two halves. When we play by heart our right brain-half is working. Piano students which start without playing by heart in the beginning are nearly only working with their left brain-half: they may play correctly but it sounds mechanical. Also J.S. Bach pointed to the importance of playing by heart in the beginning.