Doodle told me this morning that he has been "really enjoying triangles", but he had some inquietude about the Law of Cosines ... he couldn't quite get why and how it works.

It is one thing to be able to use a formula, yet another to prove or derive it. But often far beyond both of those is the ability to internalize it -- to have a mental map of its operation.

We hacked around with it for a while, trying to get some traction on it. After about 30 minutes, we both felt satisfied that we had our heads around it.

It struck me that he is habituated to understanding everything he learns, and it is always a let-down to him if we have to just accept something, lacking the ability to see inside some "black box" of knowledge. I could hardly have formed such a habit in a school setting, because the proofs and formulas came on schedule -- too fast to internalize -- and no one prepared me for such a task.

I never even internalized the Pythagorean Theorem, yet supposedly I was one of the top math students in my state.

(Can you use the images below to do this? See here and here for more.)

"Other folks trying to understand the Law of Cosines would really benefit from a little visualization app, like you did for the eclipse shadow," I said.

He agreed and seemed to think it wouldn't be too hard. Then we pondered about how it would be for him to do a family of apps devoted to demonstrating complicated concepts.

With limited help from the local web goddess (Sudoku), he recently started into the long slog of setting up for coding Android apps. Many headaches and investigations later, he can make a smiley face app and send it to my phone.

Yet, I have no doubt that he will soon be blessing the app store with some little treasures. He has a calm persistence and a real love of creating these works. And he's eager to be working in a serious coding environment where he has good tools at hand. (unlike his current situation, pouring his work into the beginner-oriented Khan Academy platform)

What is the common thread of all of the above? It is that he is blooming in a low-structure "pull environment" where he determines his pace and has room for chasing his curiosities.

So, being Thanksgiving 2017, I declare that I am thankful to the modern pioneers of homeschooling (parents from the 60's and 70's who took big risks) who helped make our wild, wonderful unschooling world possible.

Thank You! ... and happy Thanksgiving to everyone.