A Good Piano Teacher Looks for More Ways in Which to Improve His/Her Student
For a proper piano education the piano teacher would need to teach improvisation methods along with note training. Countless students with solely a note training are known to get frozen when left without notes. However many piano pieces you may have memorised, you are likely to forget some of them over time. When music theory and improvisation ability join forces with knowledge of notes, possibilities are endless.
When learning classical piano, it is best to approach the matter by learning about the classical composers. Any additional background information like the composer's father's name etc is just general information. You should however know the composer's period and style so that you can interpret his work appropriately. If the piano teacher is knowledgable, his/her suggestions regarding composers, periods and styles would highly increase the efficiency if you listen to them.