I don’t think you do measure it. You observe it, you converse about it, you probably produce some evidence of your learning. But why the obsession with measurement?
Well, let’s assume the best of people: we want to make sure that our tax dollars are being at least adequately well spent. We want to make sure that students who don’t have parents who can provide such a rich experience are at least receiving an adequate experience. And a quick way to check in on millions of students is to measure some measurable things.
The irony with this line of thinking is that measurement itself has produced a disparate impact on the most marginalized members of our population, where urban youth are subjected to 266% more test prep than their suburban peers (and probably 10,000% more than your son!). This has led to a de facto second-class education, one centered around lower-order cognitive skills and algorithm use. This is one of the points I make in my post “School Without Scoreboards.”
By the way, we also homeschool our 7 children. A lot of what I write is informed by the commonsense ideas gleaned from those experiences. But I still hold on to the idea that school can be a place where children flourish, not be subjected to this misplaced obsession with measurement.