This makes total sense. I remember learning the ideas of calculus for the first time and thinking "wow, that makes sense" and also realizing how incomplete my picture of the world was without that coherent thought framework.

And I love that article. She really captures the damage that my early math education did (which I've been working the last year to overcome).

"Unfortunately a lot of what little children are offered is simple but hard—primitive ideas that are hard for humans to implement,” because they readily tax the limits of working memory, attention, precision and other cognitive functions. Examples of activities that fall into the “simple but hard” quadrant: Building a trench with a spoon... or memorizing multiplication tables as individual facts rather than patterns."

My oldest, while in second grade, learned enough calculus to determine the _location_ of a train given it's acceleration and since it started. That's because she was interested, and asking questions, and luckily I had explained to her the slope of a graph just a few days before the train ride. She didn't learn the formula to memorize, but rather the concepts. Only then did we do the calculations the long way, on paper, pen in _her_ hand.

why?

The question my three-year-old son asks over and over each day. It's exhausting, and I love it. I do my best to provide the answer instead of simply stating "because" or "just do it" as one of my greatest fears is to suppress his natural desire to understand as much as possible about the world. Also, as a child I hated memorization, yet loved delving into a subject that intrigued me.

Calculus. Surely any child who has ever ran, or ever seen a hill, knows about velocity and slopes - concrete instances of the abstract concept.

But I would say that that's not math, any more than speaking means you know grammar, or digesting means you know biochemistry. Math is formalization.

For calculus, it's slopes at a point without dividing by zero (using the cheat of limits).