I think this is most likely normal. Both of my grandchildren have acted the same way when I left them at in home day care. My granddaughter went to a large preschool in a church basement and even tho she clung when I dropped her off she ran off to do one of the many activities available. She would then not want to leave when I picked her up. Does your provider have numerous activities that your daughter would enjoy? Does your provider help your daughter find something to do? Is there more than one provider? How many children are there. How much personal attention does she get? How organized is the school and day care?
My grandchildren have always ran up to me, glad to see me. When they were younger they would want to go home but now at 5 and 8 the 8 yo pleads to stay and the 5 yo is content to stay if I want to talk with the provider.
My grandson has been the unhappiest of the two. At 2, he would cry in the car on the way to in home day care, be crying as I left, and begin to cry when I returned. After a couple of weeks of this I stayed with him for a couple of hours. I talked with the provider, watched how she handled the kids and my grandsdon's response to her and the kids. Everything seemed OK to me. I decided that he was just having a difficult time adjusting. He still doesn't manage change very well at 5.
What helped him was for me to stay with him until he decided to get off my lap and begin playing. He would cry when I left but he already had an activity to catch his attention. At first I stayed for an hour or more. Note: His mother took him most of the time. I did once or twice a week. Gradually, I stayed a shorter time. At first I would give him a hug and tell him good bye. After awhile I told him I'd be leaving soon when he got down and gave him a hug then. In about 6 months he was OK with my leaving him. If he's busy when I come to pick him up he wants to stay until he finishes up. But first he runs at me to give me a hug.
He was the same way with a previous in home care provider but she quit the business after just a couple of months. He always cried in the car and at her house and wanted to leave as soon as I got there. I also spent time at her house and everything seemed OK with me as far as the day care and provider were concerned. She told me that after the first week or se he stopped moping and was happy the rest of the day.
I also think it's normal for a baby and a child to prefer to be with their mother or grandmother. My granddaughter, at 8, enjoys school and after school care but she regularly begs to go to my house instead. We don't waffle and her pleas are quickly over.
She is very verbal and has always told me when she had a bad day. (even while in nursery and pre-school) We would talk about her day on the way to her Mom's house. She would be OK by the time we got there.
Her mother and I focused with her on what was good about daycare or school. We sympathized when she said she hated it. We don't ask her if she wants to go to school because not going is not an option. When someone asks someone if they want something it gives them the message that there is a choice.
I recommend that you spend some time with her at the school so that you can evaluate the situation. I would ask the provider daily how
the day went, not just for your daughter but for herself and the other children. I would talk with other parents about how they and their child feel experience the school.
If after investigation you believe tht this is not the best placement for your daughter then move her to a different school. However, before you move her, investigate that school thoroughly. Visit with or without your daughter. Talk with other parents. Get to know the people who will be taking care of her. A few short conversations shoule give you a sense of who they are and how they'll act. Ask questions. Some conversations can be done over the phone. I talked with both of my grandson's providers over the phone before I visited their home.
When you decide to place her in that school stay with her for a couple of hours the first day. I've heard recommended to take a toddler to school and stay with her and then leave with her; then take her, stay with her, then return for her in a short period of time. Build up to leaving her for the whole time. Take a week or so to do this. While doing this reassure her that you will always come back. Play with her and some of the kids so that she'll know them before you leave.
I'm basing my recommendations on the hope that since she only goes 3 days/week you'll have time to do this on one of the other days.
Two and a half is a time for separation anxiety. You might find some clues while reading about separation anxiety.
I've found it very difficult to let go of my daughter and now my grandchildren when it came time for day care or school. We want them to be happy and they usually aren't with that sort of a change. It's up to us to show them the change is Ok and that we will return and listen to their stories.