The teacher I went to last spring was a good teacher in many ways but I am actually not that happy with certain things she did or did not do. In a previous thread I asked about singing without accompaniment. This teacher did never let me sing without it. The problem I experienced was: I never learned to feel or hear the intervals. Only when I play eg a an ascending fourth (eg G-C) and then try to repeat without accompaniment do I really learn how to sing it. This is a great exercice. I never learn by repetition alone as some do. My question: why would a singing teacher never ask a person to do this?
When I started studying music, I thought my instructor was going to teach me everything I wanted to know about music. My lessons were once a week for thirty minutes and I practiced each day to prepare for my next lesson. Things progressed nicely and I thought I was on the right track, but then I started to realize there were holes in my learning. My solution was to study on my own in addition to the lessons. I found printed materials and did my own research, studied music theory, found additional exercises to practice on my own and eventually departed formal lessons when I realized I could study at my own speed, (which was much faster) and I could take things in the direction of my own choosing. Learning with an instructor got me started, and that was important for me, but it wasn't the complete answer. It was my own responsibility to find my direction, devise methods to pursue that direction and then go for it. I've accepted that personal obligation to myself and will probably still be pursuing musical knowledge, understanding, and growth for the rest of my born days. And I've come to realize the difficulty an instructor might have in teaching each individual student every thing they need to know and knowing all the ways to help them learn it. I've discovered we all have our limitations, and so do our instructors. You are free to pursue your musical education in any manner that you choose, you just need to realize that fact.
At a lower level of singing, it's a much higher priority for most vocal instructors to teach a pupil how to sing in tune than how to test them with unaccompanied performance. Yes, acapella singing can help one get better with relative pitch, but at the level where the pupil may have difficulty staying in tune even with a piano, often acapella instruction turns nightmarish quickly: no beginner is going to magically learn how to sing in tune if they never hear a pitch reference!
Also, it's totally possible to learn relative pitch and intonation with a pitch reference; that's actually traditionally how this skill is trained in beginner/amateur musicians. Once a pupil gets to the point where they are good enough with intonation that they can sing it unaccompanied reasonably enough, it makes sense to spend some time practicing without a pitch reference in order to strengthen their sense of pitch and remove dependence on a piano. However, this exercise is unlikely to be useful if they still rely heavily on the pitch reference in order to sing intervals (are unable to internalise the music well enough).
Would you tell your maths professor to let you do your homework without checking your answers? Sure, you'll not be dependent on the answer key, but you also lose the critical step of comapring yourself to an objective standard. It hardly seems to make sense for a vocal coach to immediately focus on being able to sing acapella by the same logic.