I don't really think there is a name for our particular style of homeschooling. We have textbooks, and use the math book every day. I tell my kids when to do math, and how much to do. I give my son books to read in X number of days, and demand at least a 3-paragraph book report due the day after that. We discuss the spelling, grammar, and sentence structure in his reports and he corrects them if necessary. I've even been known to make him practice a few spelling words he frequently gets wrong by writing sentences and quizzing.
We also own a social studies book which has never been opened. We did the first chapter of the science book before deciding it was silly and retiring it to the shelf for the rest of this school year. My son has a reading and language book which is so dry and lame I can't imagine making my son use it.
We watch biographies and historical documentaries on television. We visit the science museum, the zoo, and the library. We get a map at the mall or the amusement park and play orienteering-type games. Most importantly, we talk. My son is a neverending well of often unanswerable questions. I explain complicated things to him that most kids don't learn about at his age, like how electricity works, the causes and benefits of evolution, and the effects of our everyday activities on those around us. Why beavers have flat tails. How long elephants are pregnant. How birds know how to migrate. Anything I don't know the answer to, we look up on the internet or check out a book at the library.
I love the idea of unschooling. I just don't fully support the idea of leaving my children's education entirely up to them. Sure, if they realize they need to learn math to count money, they'll learn it. But will they ever learn how to do anything but count money? I want my children to have the opportunity to do anything they want as an adult. So if, for example, they never learn advanced math, they're limited in their career choices to things that don't require that knowledge. I agree that kids learn better when it's a subject they're interested in, but I also think that an 8 year old isn't capable of understanding the long-term benefits of multiplying fractions.
Next year, I plan to work in geography, and teach some history at the same time. I think studies of overseas voyages can really only be appreciated when you understand how far they travelled. I'm just wondering what the hell I'm supposed to do in a couple of years when the district requires "proof" of his studies in language and social studies.