The thing with anxiety, and this is real anxiety IMO, is that it's a bit of a tightrope. Avoidant behavior makes it worse and "toughing it out" (throwing 'em in and saying "sink or swim") makes it worse. The hard thing is you have to figure out how to not completely avoid the anxiety provoking situation, but how not to push too hard. Sometimes schools don't make this easy, but it seems like you're in a good situation with the family daycare. I don't think what you describe is outside the realm of normal, but it is at the far end of the range.
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I'd sit down with the daycare person and formulate a plan. For my dd1 who had severe sep anxiety much like you describe she needed a teeny bit of structure to go into after she had warmed up, but not very much structure, really. Your dd might be different. Some kids thrive on structure. I'll tell you what I think worked best for us.
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My dd1 had horrible sep anxiety as a preschooler and we tried the leaving while she was hollering and crying and the staying, etc. It was really hard to know what to do and I don't think we really did it right, but she eventually got to where it wasn't quite as bad and seemed happy enough when we came to pick her up.
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I felt more comfortable with how we handled it in K. I read the book "Kids, Parents and Power Struggles" by Mary Sheedy Kurcinka and while it's not specifically about separation anxiety the overall theme of the book is recognizing what kind of temperament your child has and you have and what are trigger points for each of you. It does have a anecdote in there about a mom dealing with severe sep anxiety in her child and I adapted that to deal with dd1's sep anxiety. Overall, I think it's a great book and one of the few that I could find my dd1 in. I'd definitely recommend it if you've not read it.
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The way her class was set up was pretty loose and unstructured in the morning for the first 45 minutes and then the kids circled up for a more structured "morning meeting". For the first 45 minutes there were different centers and the kids were expected to choose one, but the centers were things like playdough or another material in the sensory table, dress up and pretend, building with blocks, legos, etc, the "listening loft" where kids could listen to stories on a tape player. It was a little chaotic and dd1 just felt a little bit overwhelmed. She really doesn't like a lot of structure and doesn't like being told what to do, but at the same time with this set up she just felt at loose ends and anxious because she didn't know what to do. She didn't know the other kids in the class before school started, either. Dd2 was about 2 at the time so she and I were both with dd1 (a little tricky).
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What I think ended up working best was just hanging out in the classroom trying to fade in the background and not particularly interact with dd1, but just to be there as a security blanket while she found her way. For several weeks I also stayed for morning meeting, but then when the time seemed right I was able to tell dd1 that I was going to take dd2 outside, but I wasn't leaving and I would come back in to say goodbye. This gave her a chance to experience the morning meeting circle by herself. She felt more comfortable there since she knew what to expect from the teachers and the other students. I would say goodbye after the morning meeting. I think eventually she got to the point where I could say goodbye before the meeting/circle time. It was a long, long, process, and certainly not without tears, but I really think that's what she needed.
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There are so many teachers and daycare providers who will tell you to just go ahead and leave while your child is screaming and crying. Tore my heart out every time I tried it and we did try it. The reason teachers promote that is because it does work for many kids, probably for most, but every now and then you get a kid like my dd1 and it just doesn't work well at all because it's a sink or swim scenario for them. It's too big of a push. When I picked dd1 up when I did this she would still have red eyes and look so sad. She just is very sensitive this way. Anyway, I think the long slow goodbye helped to make it better and having a definitive time frame, too. When she knew I was not going to leave in that first 45 minutes it helped her to relax enough to start to have fun.
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One other thing that helped us was setting up 1 on 1 playdates with compatible kids from school. This was incredibly helpful. My dd1 especially liked to go over to the other child's house, but your dd might like the chance to show off her room. It gave a common shared experience that carried over to school. At times it was uncomfortable for me to approach the other parents and ask for this, but it was just so incredibly helpful it was really worth putting myself out there.
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Good luck with it. I know it's hard, but dd1 is in 4th grade now and barely glances in my direction when I say goodbye now!!