I've been thinking a lot about how I teach during Covid, as doing stuff online has of course stopped my in-person ensemble teaching, and all my one on one lessons are via zoom. For no particular reason, I've been thinking of the different teachers I've had or observed others having. They mostly break down into these 4 types, and obviously many teachers are hybrids and don't easily fall into one category. I'm probably missing a few types (like the "just intimidate your student" type):
1. The "This Is My Personal Model" Teacher. These are teachers whose approach is basically to show you the model they use to navigate their instrument and music, and they mostly assume you have chosen to study with them because you like how they play and want to know more about how they think about playing. This kind of teacher generally has no interest in showing you how to do whatever it is you want to bring in or what you are doing in your ensemble, they've got a model that worked for them and they will show it to you. Inevitably you will learn a lot from this person if you stick with them, but you may get frustrated if you are looking to them to guide you through repertoire that they aren't interested in.
2. The "Observe and Fill In The Gaps" Teacher. This teacher will listen to you play and will assess what they think is missing from your playing or repertoire or musicianship, and they'll work to fill in the gaps. This teacher is great if you want to be a well rounded musician, but it does require that the student has the patience and humility and faith to accept that the holes in their playing are worth fixing. Some students have a narrow field, in terms of what they aspire to play, so they may get restless working on stuff they don't think they need to.
3. The "You Need to Learn This Repertoire" Teacher. This kind of teacher thinks that the best way to learn music is through repertoire, because the repertoire they select adds to your vocabulary and also helps you confront different challenges along the way. The pieces may require different technique, or provide a way to apply theory (in the case of improvised music), or have rhythmic challenges, or simply help get you ready for doing gigs etc. This method creates well rounded musicians who understand the language of music, and it does put them on a path where they can play music that their teachers and professional peers take as a given. Of course, students have to see the benefit in the teacher's chosen repertoire, and sometimes accept playing a piece of music they aren't drawn to. And the student has to hope that the teacher's chosen repertoire gets updated to stay relevant and isn't stuck to out of laziness.
4. The "Tell Me What You Want To Do And I'll Show You So You Can Shred Quickly" Teacher. These teachers are the ones who want you to bring in your favourite song or favourite lick, and they will show you where to put your fingers, and they thrive on getting students doing flashy things quickly. They are all over You Tube as well. The repertoire is student driven (which makes them happy) but the teacher is basically teaching you how to build the facade of a house, which looks good, but with no foundation or walls. It's impressive from one angle, but missing a lot of what you might need in the long run (like knowing where the note Ab is if I ask you to do it on the 2nd string), and these students rarely read music well. This kind of teaching is pretty much unique to the guitar, or maybe drums, as I can't imagine a violinist who can play a flashy Paganini lick without knowing the notes they are playing or the scale that is the basis of the piece. Because it often does keep the student happy and engaged, this kind of teaching is attractive, especially for adult hobbyists. But for any younger students this kind of teacher can set up a whole range of problems and knowledge gaps should that student want to study music or simply work outside of a certain kind of band later on.
If I'm being honest, I think I'm probably a 2 + 3 Teacher most of the time, with a little bit of 1 and 4 (as in, I have no problem showing my students how to play their favourite songs) except I never have allowed my regular students to skip learning to read music because I didn't want to be cursed by every future post-secondary teacher and I hate tab.