Here is a little idea I had to prime the pump in terms of playing by ear. I've made a video of me playing some very simple two-note phrases that you can try to learn to play by ear. The notes are from the four notes that are used in the Main Melody of the New World Symphony.
How to use this video:
1. I play each little phrase twice
2. While looking away, play the video and listen to the first phrase
3. Stop the video and try to sing what you have just heard.
4. Then try to play it.
5. Then watch me play the phrase for the second time to check if you were right.
