M.P.
I agree with Shane. First the mother needs to relax and stop trying to change her daughter. Instead of focusing on changing her daughter's mind about the mistake, briefly sympathize with her feelings in a calm accepting manner. Example: I see you're upset. I'm sorry you're feeling so bad. and then move on. When her mother focuses on getting her to change her daughter's mind the mistake becomes a big deal when what you want is for her to realize it isn't a big deal.
It could be that the daughter is "chasing perfection" because she's found this is a way to get attention. If the mother isn't already spending time playing with her, talking about other things, showing her love by giving her attention then I suggest she try focusing on having a good time with her daughter when she's not upset.
Wanting to be perfect can also be a sign of low self-esteem; a feeling of not being good enough. Perhaps the mother could praise her more. It's been said that we all need to hear words of praise many times/day.
I also suggest that, if she hasn't already, the mother talk with her daughter's teacher and with the school counselor.