If you have a library close by, you might check out the book by Mary Griffith, "The Unschooling Handbook". There are a bunch of great books out there, but this one might satiate your curiosity about how specific "subjects" like math and reading are learned at home without a compulsory curriculum.
I was also wondering what exactly unschooling looked like in some families when I first started hearing about it and blogs and posts give a wonderful insight into it, but that book opened my eyes and was easier to find what I was looking for without having to jump around a million places. I was absolutely fascinated by the stories that were given and it bolstered my confidence in finding ways to facilitate my children's learning in lots of ways at home.
We have been homeschooling/unschooling from the start. I participated in two different mom-run, at-home preschool co-ops with my first son, who was very social and loved to be with his friends. He picked up lots of friendships and felt a great confidence in himself during that time, which I consider one of the best benefits from our time in those groups--but what I was mildly surprised to see that his learning was very internal and though his environment influenced his learning, it came mostly in his time and his own way.
For instance, he was exposed to reading and writing at home and through the co-op so often, yet didn't progress from writing his name to writing other letters until maybe 5 or 6, years after co-op. I wasn't worried, but I was fascinated that he soaked up the knowledge and when it was "time", he started writing more numbers and letters. He learns math by asking questions and thinking about it, very internal. He'll be playing with his video games, and then building with legos, and then come and ask me a few questions, like "2 and 3 are 5, right Mom?" Yep. "Then what is 4 and 1? Is that 5, too?" He was excited to discover, on his own, that different combinations of numbers could end in the same number, I guess. But the cool part to him was that it was his discovery. He wouldn't let me "write" the addition problems for a while, he preferred to think them out, but later after showing him 2+3=5 on paper, he became intrigued and began writing math out himself, and plays with a magnetic number kit we have with math symbols, making up nonsense and occasionally finding something amazing, to him at least!
The point in my very long-winded essay
is that learning happens over time, it's a process, and when the child is given time and trust (as much as we can give to them) and interesting environments to play around with, the learning will happen. It took time and watching my son learn (without
daily heroic, stoic efforts and hundreds of dollars of orderly boxed curriculum on my part
) that was proof to me that unschooling could work.
I might add I have seen how his contemporaries (other 6 year olds hehe) write and do math and think and speak, and I see his progression to be perfectly normal, nothing out of the ordinary, and though his little brother who is 4 doesn't do what his older brother did at the same age, I see them each growing in ways that are unique to them and see nothing that tells me they are "missing" something by not being made to learn X or Z. From how I see it, if it has worked this long, it will work as they grow older, too. Maybe it will look different, and I am sure it will, at different ages, but the premise remains true.
I emphasize that learning is a process, and even many homeschoolers, after learning at home for a few years, tend to ease up on the curriculum and start having more laxness when they see how easily and quickly their children are learning. We sometimes think it has
to be difficult (maybe it was to us, in certain areas like math or reading), and sometimes because we see it that way (we are seeing with "school eyes"), we make it more than it has to be. Reading about how learning happens can be quite the eye opener in that regard to understanding unschooling, and sometimes this is referred to as part of the process of "deschooling", which means getting away from thinking of learning in terms of school subjects and compulsions and lazy students and seeing it as a holistic experience, where the learner can be trusted to know what is meaningful and important to them, and follow that.
I know of many families (unschoolers) that do use "curriculum" to assist their children in learning. For instance, a set of history on CD or a foreign language like Rosetta Stone or, goodness, math books! or Math U See or a handwriting worksheet. How about music lessons? It is based on what the child is leaning towards or perhaps mom or dad want to share something with that child. I think the point is the world is our "classroom", and libraries, museums, books, parks, people, internet, DVDs, it's all fair game when a person wants to learn something.