These topics are difficult for me to post about and I feel like I am trudging through mud to get concepts out somewhat clearly, yet I can't seem to completely refrain from commenting. So, here we go

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Dependence on teacher, how should it be ?
Well, even though faulty_damper's basic "outline" is pretty much fundamentally correct, there are many, many facets involved within the process of learning and teaching, and there is a lot that I think goes uncommunicated on this subject (and a lot that is not actually understood).
Teaching privately is similar to parenting, in my opinion. We could say :
Purpose of Parent : To get rid of every child.
Purpose of child : To get rid of every parent.
If a parent cannot get rid of his child, the parent is not doing his job.
If a child cannot get rid of his parent, the child is not doing his job.
Fine. But what does that all really mean ? I will start by saying that I don't think that the actual "purpose" of parenting is simply to get "rid" of a child, but that getting "rid" of a child is a
result of a parental purpose. And, what do we mean by getting "rid" of a child anyway ? Well, in the case of parenting, to "get rid" of a child means that eventually the individual has learned how to take care of oneself.
There are basic skills in being able to take care of oneself; like being able to keep a job and have an income, how to go shopping and buy clothes and other needs, how to keep a house, how to relate with other people, how to drive or find a way of transportation, how to cook and do laundary and clean, how to think things through and make decisions (the list goes on and on). Basically, a parent's purpose is to raise/teach a human being how to function in life/society as a human being. I don't think I need to say too much to point out that there are HUGE variances in the quality of parenting within the world, and as a result, there are HUGE variances in how people function in life and in society as human beings.
As it goes with music, the purpose of a teacher is not just to "get rid" of every student. Specifically, the purpose of a teacher is to "raise" the human being to become functional as a musician/instrumentalist in life/society. Similarly to a human being needing to learn specific skills in order to be a functional human being, a musician/instrumentalist must also learn specific skills in order to be a functional musician/instrumentalist. And, just as there are huge variances in the quality of parenting within the world, there are also huge variances in the quality of teaching within the world, and therefore huge variances in the quality of functionality in musicians.
With parenting, the backbone to raising a functional child (or not) lay within the values that parents hold as individuals, and it is very similar with teaching something like piano/music. And, even more important than that which is valued is whether or not the value has actually been understood by the child/student and passed along in a meaningful way.
As human beings, we learn skills from our parents that we learn to rely on for the rest of our lives. Maybe the role of our parents changes and our dependence on them for providing certain things for us dissolves, but our dependence on the skills that they taught us becomes more and more pertinent throughout the rest of our lives. In many cases, we may never grow out of our dependence on certain skills and values, and that is actually a blessing or a curse depending on what we have or have not learned. Again, it is very similar with teaching and learning about music.
Dependence on teachers for particular matters should last only as long as it is needed in order for the student to learn the skills that will be relied on for the rest of a musician's existence as a musician. A student will never grow out of depending on the musically functional
skills s/he has learned from a teacher, and these skills will become paramount to the student once they are effectively learned. This means that the role of a teacher will change over time.
With parenting, their initial roles include feeding and bathing and changing the child because the child does not yet know how to do these things for herself. As the child grows and matures, the parent's role along these lines changes with the child's growth. The child learns how to do these things on their own and so a parent's role focuses on "bigger" challenges that will come with the child's growth. It is similar with teaching. That which the student is incapable of providing for oneself, the teacher must take on the role of providing, until the student has grown to such a point where the role of the teacher can change in focus.
I do expect that my students will "move out" at some point (at which point my "role" will change), but I certainly aim for them to be fully functional as musicians when they do. There are other factors involved but perhaps I will come back at another point.