First Order

My son took calculus in high school. He was a math wiz in the earlier grades, but that was due to his innate aptitude for numbers, not a result of study. His aptitude was pretty impressive. He even proved some entry-level theorems in number theory. But he did not like to learn things, partly because he found learning hard. Back when I was his age, people who were smart (like him) but learned slowly (like him) were told, "You're not applying yourself!" (like me, and always with that exclamation point). But by his high school years we knew better. Not everyone learns at the same rate, which means even some smart people learn slowly. The smart ones, unfortunately for them, spend their early years thinking they don't need to learn. Which lasts until their innate aptitude no longer covers what they are asked to learn.

So when he took on calculus, he hit his limit (heh, see what I did there?). He did not innately comprehend what a derivative was. I offered to help him, but it turned out that was kind of the absolute last thing in the world he wanted. Still, his math teacher and I met with him and she straight-up told him he wasn't going to make it unless he took some help and that I was the best source of that he had. So he agreed. ("Agreed," the way the eldest son of a powerful emperor "agrees" to marry the unpleasant daughter of the king of a neighboring land, which typically isn't known for an overwhelming display of enthusiasm.)

We went right home and got to it. I figured if I could help him over that basic hump and get him to see how math could define a derivative for him, he might actually just take off on his own. Or, if not on his own, he'd see that help from me didn't have to be a crushing omnipresence in his life. He most certainly did have the smarts to comprehend it. For your amusement, here is what I tried to show him. It's the means by which you take the derivative of the function for a parabola:

He did not really look at it. He just wanted what we were doing (which was me trying to help him) to be over. When I realized that's where we were, and where we were going to stay, I told him he could ask me again any time he wanted some help, but he was free to operate on his own. He never asked me for help again. I don't think he knows what a derivative is to this day, although I'll be quick to admit that probably doesn't have any direct adverse effect on him. Most of the benefit of learning math or science for the average person is probably in their awareness that they can learn such things. Also, that such things are definite and reliable. You know, that science is real and math can answer questions for you. He probably got that much anyway.

Still, I wish he'd let me help him. He could have done it all on his own, after just a bit of that. That's what a good parent ought to do, I think. You try to help them just enough to let them see they can do things on their own.

Well, I did try.