Congrats!!! My advice:
Be REALISTIC! There are tons of little talented gymmies as young as your dd, but that doesn't mean that they will all (if any) go to the olympics.
Try and stay away from the DRAMA in the viewing area. You will start to hear lots of parents talking about their dd's skills, and what they do at home to "additionally train", meet scores, and what level they are being moved up to...yada yada yada...IGNORE THIS.
If you have questions ask the COACHES and DO NOT base your dd's progress on what other kids in the gym or on team are doing. Every gymnast is individual and progresses at different rates and speeds.
Be your child's advocate. YOU are the parent and you know what is best for them. Go with your gut instinct if something doesn't feel right.
IGNORANCE is BLISS. Sometimes "knowing too much about gymnastics" will only cause confusion and lead you to become obsessed with the sport and how your dd is doing all the time. This will drive you crazy and yes--you will become "one of those parents"
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HAVE FUN!!! She is only 3 1/2 and any activity that she does should be primarily FUN.
Gymnastics is a MARATHON. In order to get the full experience and go the distance in this sport the gymnast must pace herself and as a parent of a young gymnast this is your job. Gymnastics is a great sport and teaches kids so many life lessons including discipline, work ethic, dedication, teammwork and goal setting to name a few.
Just take it DAY BY DAY and step back once in a while to view "the big picture". This will help put things in perspective for you and keep you and your gymnast GROUNDED.