Perhaps due to my classical training, I feel best about lessons that include technique, such as scales, as well as learning pieces. Here is a conversation that often occurs as I am working on a study or scale with a student:
Student: “What is the name of this piece?”
Me: “D Major Study.”
Student: “So, is it a song?”
Me: “No, it is an exercise.”
Student: “If it isn’t a song, why do I have to learn it?”
I would usually tell the student that you have to practice your studies/scales so playing pieces will be easier, but one day, in a moment of brilliance one day I told the student I was teaching at the time that learning the violin is like going to a hockey practice: You don’t just play the game, but you also have to work on drills. The reason for doing these drills is so that you can develop all the skills you need for playing in a game situation. By already having the skills, you can react as needed when a certain situation arises. That seemed to make sense to him.
In music, you should practice your scales and arpeggios so that when you come upon one in a piece, you don’t have to think about every single note individually. The pattern will already be familiar to you. You should also work on exercises that will develop your technique. This is again so that when you run across a specific technique in a piece of music, you will already have the skill needed.
Having said all this, there are some teachers who don’t believe in using studies, and prefer to teach technique by picking pieces that include whatever it is they want to teach. There is definite validity to both techniques. My teachers were great believers in scales and studies, and it is these habits and techniques that I aim to pass along to my students!