Really. They do. Even, maybe especially, when you're not trying to "teach" them.
I just heard my seven-year-old tell my ten-year-old "we need to max out our chao's running stats." I'm not sure what that means -- it's a video game thing -- but I'm sure he knows what it means. They're both smart kids, and if they don't understand something, they either ask or they research it themselves. Right now, that video game happens to be their obsession, and they're burning up YouTube's stock of tutorial videos on how to beat it (while starting to talk about making some videos of their own about it -- they both got cameras for Christmas).
We've been homeschooling Liam (seven) for a year now. Daniel (ten) just decided to take the plunge (we've always encouraged the idea, but he had to make the decision on his own -- he really liked the "social" aspects of "public" school until the down sides became burdensome).
I started the homeschooling thing with the idea that I needed a very specific curriculum. I made Liam miserable for several months with daily worksheet assignments and such to build the "portfolio" and flesh out the logs required by our state's laws. I'm still doing battle with those record-keeping requirements, but over time we've shifted toward, and are now in the process of fully adopting, the "unschooling" approach.
I just procured a copy of Mary Griffith's The Unschooling Handbook, and it's already proving enormously useful in terms of helping me get over the panic aspect of not having a set-in-stone plan for each day.
I haven't given up "goals" and "lessons" entirely. I throw random math problems (mostly multiplication right now, but I'm looking forward to geometry -- Daniel just got a book full of skateboard ramp blueprints and is awaiting improved weather to build his first one) at them throughout the day. I purposely insert unusual words in our conversation so that we can discuss their meanings. When we decide to watch a movie, I try to find something with real historical content or a hook worth discussing in terms of some subject area. I'm also setting up some mechanisms for keeping track of what they're into online (the obsessions change -- sometimes daily, sometimes weekly) so that I can plug it into the paperwork, and expand on it offline. Within the next month or so, I expect that they'll both be blogging regularly as well.
Thing is, they're both already at or above "grade level" (as defined by the state standards) in all subjects. The curriculum approach wasn't helping them advance, and in some areas it was holding them back and boring them. The point isn't to get them ready for standardized tests, it's to get them ready for life.
They read like fiends -- when they find something they're interested in to read about. Their vocabularies are constantly expanding because when they don't understand a word it gets discussed (Tamara and I spent 20 minutes discussing the variants of "dedicated" and "dedication" with Liam the other night after he got independently interested in the difference between "dedicated server"and "this book is dedicated to ...").
Math's a little harder to work in at this point, but the "I'm trying to do this, how do I?" situations come, and when they do we show them the concepts instead of just giving them the answers. They can both add, subtract and multiply, we're starting to work on division, they've got a fair handle on fractions, and Daniel's doing some elementary algebra. Tamara and I are looking forward to having to refresh ourselves on geometry to keep up with them just for the next year or so ... and frankly I expect them to leap ahead of me and require outside help in the not too distant future (I've forgotten most of trig and never took calculus).
Science is easy. Not only do we work everyday situations into science lessons (bathtub displacement and volume! Why does a lightbulb glow -- which kind of lightbulb? We did incandescant vs. fluorescent yesterday), but they've got constant questions of their own that turn into "lessons" learned, and they inherited the space obsession from both sides. We watched the first half of Apollo 13 last night, pausing frequently for discussion; second half tonight; then it will be on to The Right Stuff and From the Earth to the Moon, hopefully with a segue into politics and The Moon is a Harsh Mistress resulting).
And, of course, social studies is a snap. They've already had more community involvement than most adults. They've attended state and national political conventions. They've marched in numerous protests. They've canvassed for ballot issues and for their parents as candidates. On the non-political side, we make a practice of helping them buttonhole hapless victims to discuss "what do you do" with them. That will probably turn into a series of "cultural journalism" video projects in the future.
The hardest part of this whole schooling thing is staying the hell out of their way so they don't run me down.