I like, very much, Artur Schnabel's description of the work of his teacher, the great pedagogue, Theodor Leschetitzky. "It was a current which activated or released all the latent vitality in a student's nature. It was addressed to the imagination, to taste, and to personal responsibility ..What he arrived at was truthfulness of expression .."
This recognizes the teacher's ability to find the individuality of a student, and lead the search for convincing music-making. These qualities, plus the ability to encourage the student, and to explain specifically how to make things happen, are among the most important aspects of teaching an instrument.
Music is about language and communication. The earlier one starts listening to classical music the more one hears it as a native language, though one can also learn to appreciate it later. So I encourage my students to listen to classical stations (in New York it's WQXR-FM) and go to age appropriate concerts. And while the more one knows about music the more one can appreciate it, listening should be pleasure, not work. One shouldn't need an advanced education to appreciate it!
I am a pianist who has played lots of Beethoven, Schubert, Chopin and other "standard" piano repertoire. I love to discuss how this or that pianist handles various masterworks that all musicians know.
But I enjoy the freshness of reaction to music by many people who were not music majors! One lawyer with little earlier musical education loves 20th century opera. Another young man I know can enjoy, in one day, listening to a recording of Rubinstein playing Chopin Polonaises and another pianist playing a melodramatic ballad of Bartok and a quirky 40 second long piece of Schnabel. And, of course, children often react with a spontaneity to music before they develop tastes and prejudices.
Which is not to say I emphasize contemporary music above other parts of the repertoire. I don't. But I ask my students to keep their ears open to music from different eras. Artur Schnabel said "There are only two kinds of music: good and bad!"
I am a lifetime member of the Leschetizky Assocation as well as a member of the Music Teachers National Association and the Associated Music Teachers League. I encourage my students to participate in their programs and concerts.