"why is the child not able to perform at school in the way you have witnessed at home?"
Because they are not taught to that level. They cannot show they can do the work if they are not given it to do.
Bert, she is now at secondary and things have got a lot better. The contrast is amazing.
Yes, I did think she was above level 6 by the end of year 6, mainly because she got A* at GCSE at the end of year6, having taken her from school for just over a year when she realised she was never going to learn anything new in maths at school.