Ah geez - I want to go get that teacher fired.
Flashback - MY 3rd grade teacher, Miss Price, who gave a math test, and 2 people (the smartest girls in the class) made in the 90's. All the rest of us failed the test. Now what does that tell a normal person? It tells us that the teacher didn't teach the math lesson in a way that ALL but 2 kids could understand. Did this teacher care? NO. She lined us all up except the two girls and told us all that we were going to get paddled for our bad test results. All the kids were trying to get to the back of the line, and she said "Well, since everyone's trying to get to the back of the line, I'll just start the paddling at the back of the line." Those poor kids at the back of the line were just terrified. A tall, big boy was in front of me, and I was clutching his shirt and had my face in his back, crying. I had never been paddled in school and I was devastated.
She was never actually going to paddle the class. She just wanted us to believe it so that we'd be "scared" into learning the material.
It wasn't long before I was throwing up at school, J.. I was not a shy kid like your daughter, but this kind of stuff was too much for me too. And yes, I started to hate math in the 3rd grade. She made me feel like I was stupid. For the first time, I thought I wasn't good at an academic subject and that feeling never went away after that.
Fast forward to when my son was in 2nd grade - one of the 3rd grade teachers was like what you are describing. I knew all about the teachers because I was at the school a lot and I wanted to know exactly what my kids were getting into. The 3rd grade teacher who was notorious for being mean to kids was Mrs. Smiley. And there was NOTHING smiling about that woman. One of the moms I worked with on the fundraising told me that her kid had had her the year before and I pressed her for details. This is what she did and I think that you should consider doing this if you can't get your child in another teacher's class. She called a meeting with the other parents in the classroom at a local coffee shop, and almost half the class parents showed up. They discussed the problem, and they decided on the strategy. Then as an entire GROUP, they showed up at the school at the woman's classroom and told her that they needed to speak to her. She came out in the hall and this group of parents told her point blank that if she didn't stop being mean to their kids that they would go all TOGETHER to the principal about her. And if that didn't work, they'd all go to the school board.
This worked, J.. She changed overnight. She also worked with those parents, treated them respectfully, was good to the kids. It must have about killed her to do it. The next year she was back to being awful to the kids, including throwing a paper at one kid and giving him a paper cut to his face.
A new principal came in and quickly, she had that woman's number. She started riding her about her performance, and it wasn't long before she pushed her out of a regular classroom and put her into ESL. I felt sorry for those ESL kids, but at least there weren't many of them. She watched her like a hawk, too. The next year, the woman was no longer at our school.
So, what I'm saying is that there are different ways of handling things. I cannot imagine even thinking of this scenario until this other mother told me that she did this. This gal had chutzpah, I'll give her that. Without the group of parents, it wouldn't have worked. They totally faced her down. As a substitute teacher, and seeing the school world from both sides, I can only imagine how mortified I would be if I were the teacher and had this happen to me. But then again, I don't act like a beotch to my students.
The child whose face was cut by her throwing his paper at him? He was my friend's child. She tried going to the ineffectual principal (before the new one came along) to get her child into another classroom, and couldn't get anywhere with it. He had a medical problem that made it so he had to pee often, and the teacher wouldn't let him go to the bathroom. He would sometimes pee his pants and she'd be awful to him about it. I told her to go to the doctor and get a letter telling the school that the way the teacher treated him was bad for his emotional health and he needed a different teacher who would not tear him up emotionally. It wasn't until she presented this letter to the principal and said she'd take it up the chain to the superintendent, that she got results. Interestingly enough, they moved another little girl at the same time, one who sounded JUST LIKE your daughter. I wonder if that mother got a letter from the doctor as well...
As far as you emailing the teacher, I promise you that she isn't going to "get" your "underhanded way" of telling her that it's not cool to get angry at 8 year olds because they don't understand math. She doesn't care to "get it". She's going to treat them the way she does because that's the way she runs her class - by fear. You will have to be straight up with her and tell her that you do not want her yelling at your daughter. Yes, it's going to put some tension between you two. But she isn't going to be any nicer to your daughter if you tiptoe around this issue.
Honestly, I'd be in the principal's office so fast, trying to get her moved to another class. I'd be at the doctor's office too, asking for a letter. Your daughter is suffering emotionally and it will not stop.
The last thing I will tell you is what a different mother (from the ones I've talked about so far) told me about this woman teaching her son. She told me "That woman CHANGES children. My son loved school, his mind was open to concepts and learning, and she CHANGED him into a kid whose attitude toward school was no longer enjoyment of learning. Instead, school was something he dreaded and he had a hard time learning because she made him nervous and so unhappy."
Needless to say, I made sure my child was NOT in this awful woman's class.
I know this has been a long post, but I really think that you need to move heaven and earth to deal with this. They will tell you that she cannot be moved. It's not true. She can. But you will have to push until it gives to get it done. It's worth it, no matter what you think other teachers will think of you. (Honestly, some of them may respect the heck out of you for having done it. They know, THEY KNOW what this woman is all about...) So don't worry about being a pushy parent. Stand up for your daughter so that she has a chance here...