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When giving a lesson, do you

A) assign an exercise that is clearly beyond the student’s capabilities, but within reach with a lot of work, and spend multiple lessons working on technique over the same thing

OR

B) assign an exercise that is a little out of the student’s capability and spend one or two lessons on it, knowing that the focus will still be on technique

OR

C) assign an exercise that is within the student’s capabilities and work the musical aspects and development (as opposed to focussing all on technique)

This is my opinion, and I'm by no means a "professional". This also depends on the target age group you're referring to... but in my experience getting lessons and giving middle/high school lessons option B seems to get the most traction. Something to challenge them a bit, but also set the bar obtainable. They usually don't hold interest in playing the same thing over and over, especially after that's probably exactly what they do in class. If I can challenge them a bit and keep the pieces rotating then it seems to keep their attention a bit more. Again, my opinion...

The answer to all is "Yes"; depends on the student, level and capabilities.
That's why they have you teaching them.

You need to teach the student you have in front of you. When you are working with a student, you need to constantly gather data that determines your strategy for how you are going to teach them. If they are an unmotivated student, you might have luck with assigning them an easier etude, with a side project of bringing in a selection from a piece/ song they enjoy, just to keep them interested (or something like this, just keep them motivated). If they are an exceptionally motivated student, then you could potentially have a combination of a+c, where you assign them something way too difficult however they bring it
Back completed and are ready to move on.

There is never going to be a 1 size fits all, and if you teach like that, you are teaching wrong. (Not to say you are teaching wrong, as I’m sure you aren’t looking for an end all be all, more of a guideline).

There are teachers who give students music they do not have the technique to play, without teaching them the technique they need. That leads to unnecessary student failure and reflects badly on the teacher. I've never understood the stance of assuming that the student will somehow figure out how to do something they are currently incapable of, just by assigning them an "exercise" that requires that technique.