I went and got DD2 book from over 20 years ago and at 12 months she was only 9.450kg and at 2 11.8kg. She was a very active child. Nothing was ever said about her weight as she ate well and was growing.
Remember every child grows differently so it's not fair to lump them all in the same basket.
I don't like the correlation between birthweight and weight as they grow. Some women grow beautiful babies in the womb, but once they are out, they grow as they are meant to grow. If it were me, I'd be looking at her immune system as the cause of the problem - being sick time after time seems to me like her immune system is really low - otherwise she'd not get sick all the time. By all means go to the paed to rule out an actual medical cause (because it could be an intollerance?), but in the meantime, I'd be looking at ways to support her immune system because the gut is the key to so many health problems. Perhaps start with a probiotic made for children and see if it helps at all.
I think what Trill says about growth rates is exactly right.
I don't suppose you have any other childcare options? FDC even (so exposed to fewer kids)?
Bookmarks