Do you think that a person can truly appreciate something without fully understanding it?
Short answer: yes.
Longer answer: I don't believe in "true" appreciation of aesthetic or cultural artifacts as being distinct either practically or metaphysically from whatever other kinds of appreciation or quasi-appreciation there are.
Long answer: I certainly do consider some basic literacy in music theory or music history to be part of a good education, just because it is a major cultural product that nearly everyone consumes a lot of. In the course of receiving that education, some people will find that it enhances their enjoyment of music. (Some don't, though! And they always make sure to tell me about it! Including on my course evaluations!) People who are in favor of that kind of knowledge-enhanced enjoyment tend to elevate it with the prestige term "appreciation." The thing is, though, that people's aesthetic experiences and reasons for liking music and personal uses of music vary a lot. Since liking music in whatever way and for whatever reason really doesn't involve holding factual beliefs, I'm at a loss to see what good I'd be doing by going up to some guy who likes music and telling him that he's right to like it but that he likes it for the wrong reasons.
About as far as I'd go is to say that, since pleasure is good, most people should consider learning more about music to see if it enhances their pleasure in it. If it does, great. If it doesn't, they should stop. It's not a one-size-fits-all situation.
In school we learn wonderful things like how to find integrals, solve equations, and how to calculate valence electrons of elements based on their atomic numbers. Because, obviously, they will be very important in our futures -- especially if we become artists, musicians, writers, actors, and business people.
We learn so much in school. Yet, when most people look at paintings they don’t truly understand them. When most people listen to music, they don’t really know what they’re hearing. Most people would fail simple music theory tests, even though many have listened to music most days of the week since they were babies!
Similarly, if you have working eyes, you should ask “Why do shadows look like they do? What color is snow, really? Can I predict the colors of different colored materials at different times of the day? If not, why? I have been seeing them for years, haven’t I?”
I think the problem here is that people can’t understand what is really important. Calculus, mechanical physics, chemistry, microiology, etc. are interesting to learn, perhaps. But, they are relatively advanced topics. People don’t use them in daily life unless they are professionals. Why not learn things that we think about every day instead of those that will frankly be useless to most?
Why don’t we learn how to understand our senses?
Learning about sight, sounds, thoughts, etc. should fit in somewhere in the first year of high school. Everyone needs to learn the physics of art and color (e.g. this and this), music theory, rationality, and logic.
For example, why should people start learning (or pretending to learn) philosophy, the art of thinking, in college? Should we be able to make life-changing decisions without even knowing how to spot errors in our thinking?
As a science researcher, I know first hand how hard it is to find a good balance between being well versed in worldly topics and being focused on a field in order to excel in it. But, both of these areas of study should not be called the true basics, in my opinion.
As president of my school's philosophy club, I took a different approach to teaching the basics of philosophy and thinking than traditional classes do. Instead of asking students to discuss the lives and ideas of famous Greek philosophers, I asked them to analyze their own lives and make their own philosophies. As expected, they were terrible at it at first. But, by the end of the year people began to actually think about the world around them.
So, my point is that we should -- in life and in school -- emphasize actual everyday thinking more.
The biggest challenge is that it takes so long!