Say I went to a dojo (or gym or whatever the place where you train is called) and watched a class. How would I objectively judge whether or not the teacher was proficient at his martial art?
What do I mean by proficient is left open on purpose. By leaving it vague, I risks the questions being closed and attract down votes. By giving a strict definition, it would answer the question. However, I am really not interested in semantic arguments about the definition of proficient or competent.
You will be able to tell a good teacher by evaluating the students. Students should be:
If the students display the above, then the instructor is presenting the material in a way that is interesting and informative, is able to make corrections in the form used by the students, and able to address a wide range of abilities.
Effective teaching is all about communication, and disseminating information to a wide range of physical abilities and learning styles. If the vast majority of the students are "getting it", then the teacher is effective.
First off, there's a difference between teaching and doing. Depending on the art you are in, there may not be a big difference (e.g. Aikido). In different arts, particularly sport or combative arts, there's a large gulf between being a good teacher and being a good practitioner (e.g. TKD). It's not impossible to be both, or find someone who can do both well, but you have to know what you're looking for.
If you want to learn how to fight according to that school, look for a good practitioner. Although they may not teach well, they know what to do, so it'll be up to you to make up the difference.
If you are in a nonviolent art, or one where practicing and teaching are not so far apart, look for a good teacher.
How to tell if a person is a good teacher:
Ask their students (or watch them teach)
A good teacher is the same across all disciplines: they pay attention to their students and they keep their students engaged in learning. Better teachers can also analyze learning problems more effectively. But the only way to know if a potential teacher is a good one or not is to watch or talk to their students. Their students will tell you whether or not class is engaging, whether or not they learn new things often or focus more on basics (either is good, depending on what you want out of it).
Above all: know what you want, and take your time when looking for it. Don't feel pressure to sign up or commit quickly.
As an adult I've only studied Muay Thai and Brazilian Jiu Jitsu. Before starting a new gym, I do some research to see what the community has to say. For example, I would Google the gym and maybe ask for a review on a popular martial arts forum. In my opinion lineage is important as well, so before visiting I make sure they're a black belt under a reputable person.
My goal when I visit for the first time is to make sure that the instructor is interested in his/her students. Everything else (competition record, equipment, etc) is secondary. What I really like to see is the instructor paying special attention to a student who is struggling or if the whole class is struggling, stopping and breaking down the technique further.
I would propose to talk mainly with the sensei and find out what type of person he is. Does he inspire you? Is he a good instructor? Does his dojo belongs to a verified federation? In my opinion if he is a good instructor he wouldnt try to impress you. He would tell you to give it a try. Also you could talk with other people from the dojo if they are satisfied with their training and their teacher. For me talk with balck belt practitioners (20+ years old) cause they know him better. You cannot judge his proficiency cause you dont know much about karate, you can start and if doesnt match your expectations find another.
If the style is a sport fighting style, if they and their students win, its a good sign.
If the dojo or dojang or practice hall or whatever it is called in the relevant style is in a mall, or has nothing but children practicing in what is essentially an athletic day care, that is a bad sign.
If the dojo is part of a chain, that is a bad sign.
If the dojo has contracts, and a big focus on money, and slick advertising, that is a bad sign.
Otherwise, unless you are a senior martial artist already, you cannot judge the technical proficiency of the teacher--you haven't yet developed the eye and so you cannot judge them technically.
You can find people you trust who are senior level martial artists, if you know any. Here I am talking about 20-year people. Their opinion can help guide you.
You must ask your self why you want to start practicing a martial art. There are many legitimate reasons:
Technical proficiency of the instructor is only important for sport (where you can see the winning or losing of the students, and the trophies on the wall) and for self-defense.
There is a classic martial arts joke that goes like this:
A guy in New York gets mugged, and the mugger beats him up and takes all his cash. He swears this will never happen to him again, so he goes to his local dojo, and tells the instructor he wants to learn to defend himself so that he won't get beat up again. The instructor says, "Well, you live in New York. You might get beat up and mugged once every ten years. If you join my class, you will learn to fight and defend yourself, but it will take years. And we will beat you up every night."
Real self defense (I recommend the books, especially the early books of Marc "Animal" MAcYoung of nononsenseselfdefense.com if this is your goal, and you can take his NSFW writing style) is not really about physical fighting skill. Its about awareness, avoidance, and so on. So if the goal is self defense, you have to ask yourself seriously if martial arts are the way to get there.
All of this is to say, judging the technical proficiency of a martial arts instructor is either impossible, moot, or won't get you very far.
This is why all serious answers about choosing an instructor or dojo or dojang or whatever it is called talk about how the instructor teaches their students, the culture of the dojo, and so on. That is what you will live with year in and year out.
If martial arts interest you, pick an instructor for these reasons, and begin practicing. If they are not good, you will very quickly find your learning at an end (a year or two) and can try another. If they are good, you will be on the road, and will start to develop the eye to let you see, and can eventually evaluate other instructors.
