Miss Frances (Frances Horwich) was the teacher and the television audience were her students. The series began, as noted, as a local program based out of Chicago before being picked up by NBC. However, the first run lasted about 4 years or so before returning for a syndicated run in 1959, which lasted relatively around the same length of time.
Let's take a look at a sample episode from 1955, courtesy of Internet Archive:
I guess now we know where Elvis Presley got the idea for peanut butter & banana sandwiches becoming his favorite delicacy, according to legend, at the end of his career.
Could Ding Dong School exist today? Not under its old title, certainly.