I'm a mom of a 2.5 year old and these 3 sources have been invaluable to me:
Janet Lansbury (janetlansbury.com or Janet Lansbury - Elevating Childcare on Facebook)
Dr. Laura Markham (www.ahaparenting.com and on Facebook as ahaparenting.com) and
Patty Wipfler - www.handinhandparenting.com (on Facebook as HandinHand).
All of their websites have tons of free articles that specifically address these questions or you can go onto their Facebook pages (or Yahoo group in the case of Hand In Hand) and ask for guidance. They do not advocate any sort of punishment or artificial consequences but setting gentle but firm limits with very practical step by steps of what to do in lots of different situations.
For me personally Janet Lansbury resonates most deeply with me, but all 3 of these sources recommend very similar things to do in this situation - it's about the parent (or caregiver) stepping in and staying calm but in control and not letting the child hit or push - you validate the emotion behind it ("I know you're mad but I won't let you hit.") I've used this with my daughter when she was very angry at me setting limits back around 20 months old and I only had to do this about 3 times - it really seemed to resolve it for her that "Mommy won't let me hit" so she would usually throw a fit instead to get her anger or frustration out. I would let her do that as long as she wanted as long as she didn't hurt herself, someone else or break something. The fit would only last 2 minutes MAX and she had it out of her system and moved on to doing something else as if nothing had happened. I might just be very lucky that she resolved herself so quickly
I know you said your daughter does understand know what she is doing (gentle touches vs. hitting) BUT the part of her brain that helps her rein in her impulse to hit isn't developed yet so keeping herself from acting on a very strong emotion, even if she knows it is wrong, is very difficult until that matures.
Out of curiousity since you are also a teacher in her classroom, your daughter can't be the only kid to have this issue so is there a specific policy that the school or you follow when it happens? If not, maybe if these techniques work for you you can pass them on to the other parents/teachers.
This is an excerpt from Janet Lansbury's article "Common toddler discipline mistakes":
There was a toddler in one of my parent/toddler guidance classes whose behavior could be considered “bad”. He was compelled to push limits, probably because his adoring, gentle mother struggled to set them confidently. She admitted that his behavior unnerved her. That, in turn, unnerved him, and “acting out” was the way he demonstrated it.
Some days I would have to calmly follow this boy, shadowing him so that he wouldn’t push or tackle one of the other 18 – 24 month olds. When I sensed an aggressive impulse coming, I would place my hand in the way and say matter-of-factly, “I won’t let you push” or gently move him away from the friend he was tackling and say, “That’s too rough.”
There was no point in reminding him to touch gently (in fact, that would have been an insult to his intelligence). He knew exactly what ‘gentle’ meant and was clearly making a different choice. But what I would often end up asking was, “Are you having a hard time today?” “Da”, he’d answer a bit wistfully, a hint of a smile on his face, recognition in his eyes. This simple acknowledgement coupled with my calm, consistent limit setting would usually ease the behavior.
Toddlers love to be understood. They also need to know that their discipline “teachers” are calm, unruffled and understanding, not thrown or upset by their behavior. And that is the way that I have come to understand misbehavior. It is not intentionally bad, mean or a way to upset parents. It is a request for help.
Help me, I’m tired. Help me, I have low blood sugar. Help me stop hitting my friends. Help me stop annoying or angering you… better yet, stop me before I do those things. Help me by remaining calm so I sense how capable you are at taking care of me. Help me by empathizing, so that I know you understand and still love me. Help me so that I can let go of these urges and distractions and be playful, joyful and free again."
Hope this helps!