How do I get my daughter to stop basing her performance on how good this other child is doing? My daughter is constantly berating herself for not performing as well as the new girl in her class. Her comments include "This girl is showing off," and "I am stupid and not as good as her."
Whether or not your daughter's neighborhood friend is really doing better in her schoolwork than your daughter is probably irrelevant, your daughter perceives she is. I'm guessing that your daughter's former sense of security within her school social world and her former identity (before the new girl arrived) have been thrown for a loop. She needs to find a way to reestablish that sense of place, belonging, and being appreciated.
Please ask her teacher for a private meeting to discuss your daughter's emotions. Don't accuse the teacher of playing favorites or contributing to the problem. Ask her for her help in brainstorming ways you both can help your little girl feel better about both her school performance and her school social life. Meanwhile, it may be a good idea to invite a few of the friends your daughter has made through kindergarten and her public school activities to come to your house to play, thus showing her that she does still have this connection with her former friends.
Mirror your daughter's concerns, empathize with her feelings, and continue to offer her encouraging words about who she is and what she does. Explain that it's normal for a new kid to get a lot of attention and for kids to compete for who will be the new kid's friends. Try to get her to empathize with this new girl's/friend's position and how easy it is to be caught up with all the attention she is receiving. If you are friendly with the new girl's mom or dad, discuss the situation openly without blaming their daughter. Maybe there are subtle ways you parents can help these kids remain friends, if that would be the best situation for both. Helping your daughter deal with her anger, her fears, and her measurement of self versus the new girl will take considerable abiding and creativity. I am convinced that with help from the teacher, the other girl's parents and a few of the suggestions I offered, you will begin to make inroads into these problems.