Is it possible that your daughter is BORED? Is she just high-energy? Is she a little younger than the other kids in her class? Sitting still for an ENTIRE DAY is a lot to ask of some 6-year olds.
I get angry when I see that ADD or ADHD is being brought up when kids are just being kids. ADD and ADHD are very specific disorders and it's much more than not being able to sit still.
My daughter was supposed to be tested for ADHD in first grade because she couldn't stay on track, she was disruptive and didn't sit still. We yanked her out of public school (not that that should be YOUR solution, but it was ours) and she is doing JUST FINE. Her birthday is in July, so she was a lot younger than the other kids in her class and she just wasn't mentally ready for first grade. She would get bored, or she didn't "want" to do the work, so she would goof off. If you gave her the right incentive or discipline, then she could do it if she wanted to. She just didn't want to...yet.
An ADD or ADHD child usually can't behave, because they physically CAN'T. They also tend to have other issues, sometimes sensory, OCD, etc. When I see an ADD/ADHD child compared to a child that can't or won't behave/pay attention/follow through there is a BIG difference that I can see.
I also distinctly remember getting in lots of trouble for the same things in first grade, but I was bored because I was gifted. When you understand something the first time a teacher says it, it's hard to sit through it while the teacher explains slowly for the rest of the class. I was also high-energy. Fortunately for me, my pediatrician told my mom to get me more challenging schoolwork and sign me up for gymnastics :)
Hopefully your school psych knows what she's doing. I don't put a lot of stock in school psychs--but that's just me.
Now that I'm part of a homeschool community, I'm glad we can celebrate our children's spirit. We don't make them sit down and shut up to learn. They learn at their own pace, in a way that make sense to them. Some kids need frequent breaks. Some kids learn through doing. Some kids need high-energy lessons.
There are also other things you can do like eliminate artificial dyes and high fructose corn syrup. My daughter concentrates better if she doesn't have those things. She is a "normal" 11-year old who can concentrate, follow through on projects, etc. She needed a little extra help learning how to organize her thoughts--hey, not everyone is born knowing how to do that!
Good luck to you!