Are you a "Teacher", or a "Coach"?

Those of us that have taught martial arts for a long time, know that teaching or coaching people isn't as easy as we thought it would be when we were still a student. We also are aware that Teachers can be coaches, and vice versa. But is there a difference between being a "teacher" and being a "coach?". I'd like to sound off with my views, and I welcome my reader's views as well:

Teachers:
Teachers are mentors, of sorts. They commit a sincere personal interest in you progress, act as a role model, gives you advice and shares their past and present personal experiences throughout your progress. Students know you have an expectation of them, and they trust that you'll guide them.Good teachers know that they have a big responsibility to their student and know that it is a two-way street.....As the teacher teaches, they too learn as well. As the student learns, they too teach the teacher.

Let's compare teachers to...let's say... theatrical directors. Teachers set the stage for the students, teaches them the script and performance, handles rehearsals, and motivates the students into realizing that EVERY day is "opening day".

Coaches:
The term "coach" is sometimes used to even the scales between student and the more experienced coach. In martial arts, many students tend to see the "space" between themselves and the "Master" as a very vast area, whereas the space between a coach and student is a bit less. That's not to say that coaches are "less than" a teacher, because teachers can be very effective coaches as well (and they should be!). The coach tends to focus on your performance, and students may find themselves shouldering the responsibility of performance more, as the coach fine tunes, gives advice, and constructive criticism.

While the Teacher teaches the gamut of "who what when where how why", the coach makes the student responsible for applying those facets into performance.

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In my opinion, these responsibilities are interchangeable, so a good teacher should be both the Mentor and Coach. What do my readers think? Would love to hear your thoughts!