The fingers are numbered in piano for the purpose of designating which finger to use to strike a key for best finger technique.
Finger numbers written above or below the staff are suggestions given by composers for best finger action. For example, this is an excerpt from Joseph Haydn, a brilliant composer and teacher. Notice the fingering suggestions he has given.
A finger number can sometimes be a clue as to which key to strike, but a finger number is never a label for a key because we only have 10 fingers to work with while there are 88 keys on a piano.
To move across the keyboard, fingers can crossover, hop, reach, and thumb-under. As students advance, they will gain finger agility that is unfathomable at the beginning of lessons.
Read Music Method teaches children to read notes on the staff on the first day of lessons. Students gain a deep comprehension of the musical staff on which to build their skills.