Wow, tough question. Alot depends on the child and the coaching style. Some kids are more visual learners and others can just be talked through routines although in gymnastics there has to be some demonstration of what the coach means. Usually to learn compulsory routines, they group the girls by "righty" or "lefty" since they will start on a different foot----very hard for a girl who naturally starts on her left foot to be in a group of girls that all start on their right.
From what I've seen at both gyms we've been at is they break the routines down. Teach the beam routine in 1/3s and about the same with floor. Once they get the 1st 1/3 down then teach the next section. Bars for L5 and up is taught alot by learning the low bar portion and the high bar and then putting them together. I would imagine for L4 there is some logical break in the routine---teach 1/2 and then the next 1/2. Once the complete routine is learned, then its repeat it, repeat it, repeat it. As the girls get better at doing the routine without any prompting then the coach adds the polish---straight legs here, point toes there. Lots and lots of repetition and correction.
Floor is usually taught without music and once they have that down, then they add the music.
That seems to be the way our gym works, too. They started by learning the individual skills, including the leaps, jumps and dance portions. Then they put several pieces together and then put those together until they had the full routine. And they didn't add the music until they had done the full routine without it.
Our gym seems to do the same as everyone mentioned. My DD was fine learning the floor routine because they learned by having the directions called out to them. She is not the type of kid that can learn by watching others. She has to be told and do it over and over and over herself.
Beam is proving to be very frustrating to her. Bars was no problem, but she loves bars. I think beam gives her problems because she gets bored learning the parts and can't see it all fitting together. I wish wish wish there was beam music. LOL. Beam is also her weakest (and she has had a lot of falls, etc) event, so that could be as well. Plus, until recently, the coach she had for beam (who was the HC, who has since left) had her close to tears on beam many times and I think any hope she had for loving beam was being squashed. With new coaches on beam, she is becoming much more confident on beam, even got her BWO a few times on beam alone after a fall/hitting her head trying one with her coach, so I think beam has a brighter future in her mind now.
I think it also depends on the gyms focus. Many gyms just practice routines, over and over and over during meet season. Routines and parts of routines. DDs gym still uptrains and sometimes it seems they don't work on routines as much as they could, especially during the week before a meet, but I imagine they have reasons and my DD is happy there, so I am happy.
Also depends on the kid big time. DDs teammate, is very good at just watching other girls do a routine and then she can figure it out. She learned the floor and beam routines really quickly. She teaches herself skills that way as well. My DD isn't like that at all. Different learning styles.
My daughter is 7 and is level 5 this year. Their first competition is October 24th and they still have yet to put their floor routine to music. Last year as a level 4 we worked on "perfection" up until our final meet at States. Best advice I could give is what our coach told us last year....."They will get it, don't worry." I know that is easier said then done. I am still a bit freaked out because we only have 8 practices before our first meet. I don't let Emma know that because she is so not worried. I will ask Emma if she has any hints. I know they learned it in sections and "perfected" the sections before they added on more. Then it was just doing them over and over and over. Emma does them at home all the time. She uses a line in the wood floor as her beam. I think the younger they are, cognitively they have a harder time remembering. Emma too has a friend that just picks things up by watching. Her coach always tells me that she is a kid who needs hands on to learn things. They will get it. It just needs to be repetition. Your DD will do great! I will let you know when I talk to Emma if there is anything else she can add. I know one thing she will tell you is to "stay tight". She hears that in he sleep. Good luck!!!!