First off, I would like to commend your desire to teach at a high school level. It is a very challenging and tiring profession but it is so rewarding and IMHO is one of the most useful settings that one can teach in (aside from maybe elementary school but that is a whole different ball game :) It sounds like you already understand one of the most basic fundamentals of high school teaching when you said "the main focus for me will be on teaching rather than the subject". Unlike at university, the actual content knowledge of the teacher is much less important than their ability to be organized, engaging, equitable, and controlling of behaviors and expectations and that is why it is crucial that if you want to be a high school teacher that you do your residency in a high school. Pedagogical approaches taken in high school vs those taken at university are vastly different and there is often little to no overlap.
That being said, I think that the most important thing you can do while trying to decide which school to do your residency at is research, research, research. As you can probably imagine, every high school has it's own culture, norms, and standards and it would be in your best interest to do research into your different options either by going to the schools' websites or calling and talking with administrators. Better than all of these is to actually visit the schools in person and observe a few classes. If you can, try to observe classes that 1) are in your subject area (math), 2) taught by potential mentors and 3) in different subject areas, such as history or language. I cannot stress how important this is as it will give you first hand experience with how the school, teachers, and students behave and interact on a regular basis, not just within the small bubble of your mentor's classroom, but from a wide array of perspectives. While you are in the buildings, talk to the teachers about what they like and what they don't like about the school. Talk to the students about what they like and don't like. You might be surprised at how candid teachers and students can be about their own school and you might learn something, either positive or negative, that will sway your decision. It is helpful to read up about different schools online, but nothing is as important as actually going in person to see what it is like and determine for yourself whether it will be a good fit.
As far as finding a mentor, I would say the most important aspect of a good mentor is one who gives you guidance and support when needed but will also give you the freedom to experiment and go about things on your own. It is important that they are moderately hands-off and will allow you to completely bomb lessons because that is the only way you will learn. What is important is that after you bomb a lesson, they are right there to tell you why they think it went wrong, offer advice for the future, and encourage you to keep on with it. What you want to stay away from is someone who wants to control every aspect of your teaching because you will never learn that way. Have these conversations with potential mentors beforehand and hopefully they will be honest with you.
As far as things that you should be focusing on during your teacher preparation, here is my list of things that I ended up focusing on the most during my time student teaching:
- Planning - efficient and effective planning is one of the most important aspects of high school teaching. With a solid plan, most lessons will go pretty well, all things considered. Without a plan, you will very quickly learn that "winging it" is never viable. At first, planning is long and tedious and the only way that it will get any quicker is with practice. Talk to your mentor, talk to other teachers, see how they go about planning a lesson, see if they have resources for you so that you don't have to plan everything from scratch.
- Assessment/Grading - I am not sure how the situation is in Finland, but assessment and grading is a huge part of high school teaching in the US. Whether or not there is actually any value in number/letter grades, does not take away from the fact that at some point you do need to assess your students' knowledge so that you can know where to go in future lessons. Assessment comes in many different forms (formative, summative, group, individual, projects, classwork) and it is important that you learn how to take all of this "data" (data, in teaching, does not have to be numerical, it simply refers to anything that can be learned about students from their work/discussion) and use it to influence future lessons. Blindly planning and assessing students without any purpose or goal is useless. Always use your students' progress and abilities to drive how you teach them.
- Classroom culture - Another important aspect of high school teaching is the atmosphere and culture of your classroom. Try out different things such as changing seating charts around, moving the desks around, putting up posters, having a wall for student work, etc. All of this helps to cultivate the physical environment of your classroom and while it might not seem like a big deal it can actually have a lot of influence on how well students learn in your room. A quick story: While student teaching, I had the opportunity to observe at a bunch of different schools in my district. At one school there was a teacher without his own classroom and as such had to move throughout the day to different rooms. I observed him in the morning in this tiny little classroom with no windows, barely enough room to walk around, and nothing on the wall except for bland grey paint. The students were raucous, unfocused, and jittery, constantly trying to get up and walk around even though there was no room. The lesson was not terrible but the teacher had to continually struggle to get the students on task. Later in the day, I observed the same teacher teaching the same lesson to a different class in a different room. This was more like your typical classroom: windows, posters on the wall, ample room to move around, etc. and, compared to the morning class, it was like night and day. The students were engaged, they were not disruptive, they were asking questions and having discussions. Now this may have just been because of the students, but it really struck me that day how different the classrooms were and how much of an effect this had on the teacher, students, and me. The more effort you put into making your classroom comfortable, personal, and safe, the more your students will appreciate it and the easier it will be to engage them (in general, of course)
- Reflection - it is super important at the end of each lesson/day to think back on what you taught, how you taught it, and how effective or not effective you think it went. It doesn't take long, just a few minutes to jot down your thoughts in a notebook or in an electronic document. It sounds obvious, but I have met countless teachers who I know do not think back for one second about their practice and it shows. They continue to make the same mistakes over and over again and get frustrated with the students because they do not understand that if they just changed one little thing about their lesson things would go a lot better. It is extremely helpful to get in to the practice of annotating lesson plans, i.e. going back and taking down little notes about what worked and what didn't. That way, when you go to teach that same lesson in the future (which you undoubtedly will) you know what to modify. Reflection also helps to ground your perspective and is certainly something that can be done with other teachers. Some of my most useful reflections have actually come from other teachers who have observed me or tried to teach one of my lesson plans.
- Getting to know students - honestly, one of the most important things you can do as a teacher is know your students. Not only should you know their abilities in your subject, but also get to know them personally. What sports do they play, what movies do they like, what music do they listen to, what makes them laugh, what makes them upset, etc. Knowing all of this will directly make you a better teacher because it allows you to relate to your students and make them know that you actually care about them, and not just their performance. I can't tell you how many times a student's entire disposition and performance in math has changed almost overnight because I started casually asking them about their day or their plans for the weekend or what type of music they were listening to. It has probably been some time since you were in high school so it will definitely be a bit awkward interacting with your students in this way but I promise it will get much more comfortable as time goes on. I would highly suggest that once you learn about your students more to incorporate their interests into word problems, tasks, exercises, etc.
- Have fun! - too often I see teachers get stuck in this rut of needing to make the most rigorous and challenging lessons and lose sight of the fact that these students are more or less required to be here and no matter what you do, they will not always want to listen or follow along with a lesson. I strongly encourage you to incorporate games, puzzles, challenges, videos, songs, drama, and anything else you can think of to make your lessons fun. Obviously, you can't do these things day in and day out but making them a regular part of your lessons will pull in kids who you never thought you could pull in.
- Classroom/behavior management - Personally, I am under the belief (and much of my experiences show) that if you do all of the above things correctly, then behavior management is just something that will ever really be an issue; the students will be engaged because of your good planning, data-driven instruction, and classroom atmosphere and wont even have time to think of ways to be disruptive. However, I know this is an ideal and classroom/behavior management will assuredly come up in your teacher education program. As you will soon find out there are about a million different ways that teachers monitor student behavior and some of it works and some of it doesn't. This will really be up to you to figure out exactly what works with you and your students. The one thing I will tell you is that I have overwhelmingly found that compassion is much more effective than discipline. Other than that, it will just be trial and error for you to figure out a system that works for you.
Ok, I think that is mostly it. ":) I know it is a lot but hey, teaching high school is a huge responsibility and it is better to be aware of all of this before you jump right in and try it out. I wish you the best of luck with your teaching and hope this information is helpful!