When I started making math movies with the boys my goal was to show other kids what kids doing math can look like. There are examples everywhere of adults doing math, so kids can see those examples with no problem. There aren’t nearly as many examples of what it looks like when kids work through problems, though.
So, 5 years into it we are all pretty comfortable in front of the camera and my younger son – just by luck – is making exactly the videos that I was dreaming about in the beginning.
Below are the last two ones we’ve made. They show him working through algebra problems. Nothing fancy, nothing speedy, but really nice work through the problems. I love the way he thinks through problems and think that other kids might enjoy these examples showing what a kid doing math can look like.
Mike's Math Page 
It’s also really nice for us adults who haven’t seen this stuff for a long time to see a young ‘un working through these problems.
I told someone just a couple of days before you posted this that my role models were a 10- and a 12-year old. Fearless. Tireless. Curious. And yes, there are thousands of vids of students, teachers, and who knows who doing everything right the first time, knowing exactly what to do – like being back in high school, where I never really got a chance to try to figure anything out since someone would always know the right answer immediately (which, I’ve heard, is how people learn to be “bad at math”, the one math lesson I seem to have excelled at). I love your “What a kid learning math looks like” because… that’s exactly what a kid learning math should look like.
There’s an interesting mooc on edx right now from UTA, just started – Michael Starbird’s “Effective thinking through mathematics”; I took it a few years ago. He isn’t teaching math per se, but approaches to problem solving, skills for figuring stuff out. He sits with a couple of non-math students, and lets them thrash out a variety of very basic problems from lots of different areas – puzzles, the infinitude of primes, Simpson’s paradox, a 4-d cube, “a bug’s view” topology. In the IBL tradition, he gives very little math guidance (occasional facts), but instead reminds them of the “5 skills” – make it a simpler problem (which I see your kids doing all the time), try something you already know how to do, etc. Unique concept, very different from all the mastery-based and example-based courses around.
We’ve got lots of how-to-do-this-procedure videos – now we need more “here’s one way to struggle through figuring out how it works” models, more “what it looks like to learn math”.