.....It's hard sometimes to know whether to hold off with my stuff until the end, and risk losing them, or not doing anything with them, or ask about them at the beginning and then throwing the lesson off on a tangent (which it often does).
Always take stock of your own discoveries and questions. Keep a journal where you write your ideas, challenges and goals. I ask all my students to do this (not that they all oblige!) because it is a good habit to have when you are studying by yourself. Given you see your teacher only a small part of the week in comparison to the time you spend by yoruself studying your music, it seems silly not to take note of ideas that pop in your head while studying. If you simply keep these ideas in your head you may indeed forget about them, so write them out. I have always believed that "What gets written gets done."
Sharing your ideas and questions with your teacher might tangent your lesson, this is a real problem that the teacher must deal with not the student. However a good teacher knows how to solve the problem, give the student confidence to work on it with a method briefly explored in the lesson. Some teachers however get very distracted, almost as if they cannot answer your problem so spend a big part of the lesson trying to make sure that you understand it thus neglecting what the teacher has set out to teach you.
Sometimes I get problems asked which require more procedure than simply directly trying to solve the problem itself. Like how to make the expression better, how to make technical movements more effortless relaxed etc etc. Sometimes the solution to these problems should not be tackled immediately because of the inefficiency created trying to solve it. I almost think about riding a bicycle with gears. They are trying to ride in the highest gear without starting with the lower ones first.
Then as a teacher I tend to direct the student a little further back to where their ability works best at. I do have many very musical students who are technically incapable of always producing what they wish they could. It is best to move on not lament on small things, the more you move on and experience the better you simply become.
I find humans learn better with many musical ideas somewhat controlled in many pieces rather than "perfecting" a small number of ideas in few pieces. So if you find your questioning to your teacher is pulling the lesson to a hault maybe it highlights a deficiency in your ability to satisfy your wishes (given that the teacher is a capable teacher).
Your problems asked to your teacher should not confuse the lesson with an able teacher. They will either 1) give you the advice for the method for you to work on yourself. 2) Highlight the fact that you are trying to do too much in one step. I usually demonstrate that the student is trying to do too much by giving them the solution to their question and watching them struggle and drown in its difficulty for them. Sometimes they realise that they can do it but the progress is very slow and I encourage them to leave it for now and tackle other things which will make this task easier in the future.
....I always read about what the teacher wants to do with a student, but I never hear about what happens when a student comes in burning with new stuff that came during the practice. Lately I've thought of just asking for an occasional extra lesson to deal with those things so that my regular lessons can stay on track with what my teacher had planned.
I have three students of all my students who totally control the content of the lesson. This is only because they are very interested in studying the piano so generate a lot of questions themselves every week. Two of them are professional educators themselves so really do know the ins and outs of the path of learning. I find most students simply do not generate enough relevant questions themselves to control the direction of a lesson which would best suit their learning rate. But getting questions from students is always very much welcome. I am excited when they come up with questions much more so than when they just sit there like a quiet sponge.
If students do not ask me questions I usually ask them questions instead, trying to get them to actually ask themselves the question, yeah well how do I solve this problem actually, what do I think about the method required to solve this problem? I think a big part of teaching is how to teach to ask the right questions. If I don't have students asking me questions during a lesson I really feel that I have failed on some part because I am not making the student feel encouraged to investigate and test things with trial and error and get frustrated with difficulties and vent those frustrations to me in their questionings.
Usually I throw the student into the "deep end" asking them to solve a new or trouble passage on their own and letting me be an observer as they demonstrate their problem solving skills. If I see them completely flounder I try to get them to think about the right questions as to why they are not progressing as fast as they wished. All I usually have to say is "Do you think that part was right? What didn't you like about it? What do you think you did wrong? Then get them to ask me what they think might be their problems and I either confirm this or ask them to think again. So through constant questioning the student solves the problem themselves. Of course they required me to coach them through making the right focus on their questioning but this benefits the student a lot more than simply parrot teaching them with answers. It also allows the teacher to gain more insight into how the student thinks and how they deal with problems. You really get to know the thinking pattern of each student individually.