I teach as well, and see some excellent responses here, to a meddling question.
I would summarize by saying that the stroke is not so much as your arm stroking, as it is your brain stroking. As you approach the table, and decide how you will hit a ball, you need to solidfy, or "assure" (as RandyG pointed out) your brain, and allow it to focus on exactly what it is you are doing. This is now accomplished by "stroking" the shot.
The number of strokes an individual will become comfortable with is unique, and can change based on the mental rhythms of the player.
The key is to get in tune with your mental focus. If you are thinking about your arm at this point, all is lost...you are not stroking. The word triggers one to think of physical movement rather tham mental, but it is mental connection to the shot you need, the stroke should allow you to "feel" how hard and where you will be hitting the ball.
Once you get very accustomed to your own feel, the stroke will also allow your brain to trigger an alarm if something "feels" wrong.
Some students respond to trigger thoughts as they stroke..." I am going to hit it there, this hard..." "I am going to hit it there, this hard"..."I am going to hit it there, this hard"....I AM GOING TO HIT IT THERE, THIS HARD"....wham
The "stroke" of the arm is also crucial, but I am assuming you are past looking for a simple stance and stroke, and a totally basic concept compared to "stroking" the shot.
It is like "follow thru". Many people think they have to follow the cuetip thru the cue ball, when actually you only need it to be accelerating at the point of impact to have "follow thru".
Hope I added something to help you understand your dilemma.