A Series I ALMOST Remember

I've been trying for AGES, or so it seems, to remember the name of a series I read in a magazine (Humpty Dumpty's Magazine for Little Children? Children's Digest?)
in, I think, the late 50s or early 60s. The series was about a little girl (it would help if
I could remember her name!) who lived a long time ago. I think she was spending the
summer with her grandparents; I do remember that they played a prominent role in the stories.
The funny thing is that I really remember just one of the stories. There was an Ice Cream Social, for which all the ladies made ice cream. The girl wanted to be sure to get her Grandma's ice cream, so she went up for ice cream again and again and again and....!
Of course, when the ladies serving got to her Grandma's ice cream, Grandma brought her a dish.
I remember that I wasn't surprised! However, the kid was in NO condition to eat another SPOONFUL!
Oh, how I'd love to read that series again, if I could just remember which magazine it was printed in, and when!