Why you and not the mom next to you?
If I take your suggestion, and word gets out, I've pretty much lost all claim to control over my curriculum or choreography, unless you have an unassailable credential that everyone will accept.
What if I was going to do that anyway?
Now I'm afraid if I do it, because you'll think you I did it for you and I'll never get you off my back. So your great idea, which I had already thought of, just got trashed.
What if your idea is not technically feasible?
Either I have to bring you up to speed on a lifetime of figure skating, I have to be patronizing, or I have to call you an idiot.
Would you let me walk into your office and tell you how to do your job?
What in the world gives you the idea that you know better than me?
You have one, maybe two skaters-- your kids. I've known thousands. Do the math.
What have I done that makes you think I'm not paying attention?
Did another problem you noticed not get fixed? Did a child get hurt, or ignored? Is the class or rehearsal out of control? Are other mothers complaining?
At our old rink, we did it this way
Right. You want me to do it the way they did it at the rink that for some reason you don't go to anymore.
Update: Anonymous' comment, (first comment below) has made me think a little more about this, so I'll amend a bit. Verbally assaulting a coach with a long string of suggestions about how the choreo or class can be improved, even with a smile on your face, the second she steps off the ice, is not going to elicit a positive response from the coach. You are basically marching into my "office" and proclaiming that you know better than me. You have given the coach no time to process or consider the class, and now you want to "fix" it. Write it down, send me an email, ask if you can talk to me.
In general in negotiations, simply saying "here's my idea, it's better than your idea, why aren't you doing it this way" is not going to get you the result you want. Ask a question--why is it like that? Can you tell me what effect you're going for? State your concern "I'm afraid it won't be polished by the end of the rehearsal period." Simply saying "do it my way" is sure to elicit a defensive response, and then the problem, if indeed there is one, doesn't get fixed, the coach has now flagged you as "one of those moms," and you don't get what you want. Everyone loses.
I probably have taken lots of parent suggestions, when the parents exhibited some professionalism, sensitivity and good timing in how they approached me. Parents expect coaches to behave in this way, after all.