After five years of homeschooling we, my husband initially, decided to enroll the kids into our neighborhood public school.
My husband felt it would be best, so I could have a "break" from all the responsibilities I was taking on. His reasoning was that I had just lost my mom after a very intense 18 month bout with leukemia, and I was in college. He wanted me to take the time to ease up a bit after that experience and take more college courses if I felt I was up to it.
It took a bit of time, but after a couple of months of working through grief, homeschooling, and juggling college I realized he was right. I am only one person and I needed time to process things and take a breather.
However, just because they are in school does not mean that I shoulder off everything to the school. It is still my responsibility to introduce them to good solid literature, help them develop a strong vocabulary, do math drills, and include or expand upon any areas I feel are important that the school does not. I do love their school and their teachers are wonderful, but after Christmas break there was what seemed this big shift and mobilization to get ready for the standardized tests. Which means more focus on math and the mechanics of reading. I think that is fine, give them a solid math base and reading and writing base, and I will cover history, poetry, the great books, and natural sciences.
This said, it does not mean that I am burdening them with tons more work. "Afterschooling" for us, is an extension of our life like homeschooling was. They are given books to read each week from a list here and asked to write down any vocabulary words they come across that they don't know, and write a quick couple of lines summarizing their reading for the night. My boys like to read before bed, so they fit in 15-30 minutes of the book I give them, and then move on to whatever other book they are reading. Poetry, I recite to them, we discuss it together, and this can happen over dinner or while Im cooking. History- Im a history geek and topics in history come up in several conversations during the day. Sciences, same thing.
Really, afterschooling is just doing what is a parental duty and truly does not even need it's own name.