Satie Gymnopedie - the famous one was recently an ABRSM grade 6 piano piece: seems fair enough. A few years ago another exam board set it at grace 3!!! I pity the souls who went through that.....
Exams or not? I find that pupils' progress definitely improves when they are preparing for an exam. I am well aware that some teachers believe that exams can impede a growing love of learning an instrument; Fair point. But the progress of pro-exam pupils definitely outstrips anti-exam pupils. If pupils dont want anything to do with exams then that is fine: I will proceed without that option; but rarely do anti-exam pupils make anything like the same progress. I operate as a private teacher: perhaps things are different in a school setting.
To improve cross rhythms - try C major contrary motion scales: triplet quavers in left and duplet quavers in the right. Take it slowly at first and use a metronome ONLY on the crotchet pulse.
Singing - I'm thinking about it again, even though I am a piano teacher primarily. This is for teachers and parents: if a pupil is preparing for an aural section of an exam and they are shy of singing, make sure that you hum or sing with them simultaneously. It helps them to feel less self-conscious about the sound they make.
Are you a piano teacher? a snippet of wisdom: if a piano pupil loves singing, why not utilise that by giving them piano pieces with words so that they can sing along? This has brought fresh life to lessons with a 9 yr old pupil I have.
Increasingly I encourage pupils to play piano blind - don't look down at your hands! Keep your eyes on the music! It helps with sight reading as well a pieces with a massive compass. Any comments?
Piano and keyboard teacher based in Eccleston, south Lancashire, England.