Helium. I’m a section hiker and a school administrator, after being a teacher for 20 years. I only get to hike a few weeks every summer. I will say that you have selected a much in demand teaching position. Almost every school in the country wants and needs good math teachers. It looks like you have already decided on teaching in high school, which is a good start. You need to think about private or public schools, urban, suburban or rural schools. First, I would make sure teaching is what you want before going to grad school. Try substituting for awhile, visiting schools and talking to local administrators, they should let you sit in on some classes. Teaching is not for everyone. It can be a great job or an awful job, depending on how much you like it. Remember all those terrible teachers you had, will they went into teaching and never did enjoy it. The good teachers are the ones that love teaching and it shows.
If you do go into teaching you can later try and talk an administrator into allowing you to teach an elective class on camping. I did this for awhile, I loved it. I taught the basic camping skills involved in the outdoors, we or at least I, had a great time. We would also take kids out hiking, camping, kayaking and canoeing.
I would imagine you will find alot of retired teachers/administrators, that may be hiking this spring. Remember, that everyone has a different hiking style, you may not hike for too long together. Good Luck with your teaching decision.
Bilko