Learning is not a one-way street.
Teachers teach students, but students also teach teachers.
I was reminded of that recently while working on one of Haydn's last sonatas, No. 52 in Eb. I was noticing how he was using more of the piano than his earlier works, fuller chords, lots of 32nd notes, and greater contrast between dramatic and lighter sections. As I played, I thought, "this reminds me of Beethoven!"
I always teach my students that Mozart and Haydn were friends and influenced each other's compositional style. And I also teach my students that Haydn was Beethoven's teacher
As I worked on the Haydn Sonata, I realized that it was composed after Beethoven had been studying with Haydn, and had begun having success in his own musical career. Of course Beethoven influenced Haydn! Beethoven' s strong temperament and musical personality influenced everyone with whom he came into contact. When you have been influenced by someone, then you have learned from them.
I do not know why the idea surprised me that the student Beethoven influenced (taught) his teacher. I cannot count the number of times I have learned from my students, just by their asking me a question that made me think about something differently. A fingering, a chord, a phrase. Sometimes they may play at staccato or accent differently than I would, but it still works! Maybe I should try that for a change! I have more knowledge and experience than my students, but their questions and observations make me look more closely and think more deeply.
If a teacher is observant, the student is also teaching the teacher how he/she learns. We cannot just use one approach when teaching, and the student, by their response, will show us how best to teach them.
Over the years, I have become a better and better teacher, able to work with a wide variety of students and levels. I thank my students for teaching me how to do that!