CMP Review 2024-11-14

November 14, 2024

Ever since he was just starting algebra, my youngest son was fascinated with calculus. He overheard family conversations and wanted to know more. I realized that there was some I could teach him using even the little algebra he knew, and the weekly calculus lesson became for him the highlight of his school week.

Somehow he inherited a screen shot from his older brother which depicted an essential calculus theorem. Even with the little calculus we had done, this theorem was completely over his head. He knew he couldn’t understand it. But someday, he believed, he would. So, unbeknownst to me, he held on to that image.

Algebra turned into geometry and trigonometry and Algebra II and then calculus became a very serious study. It was time to fill in all the gaps that remained from our splashes of calculus over the years. It was time to get to all the hard stuff, to leave no stone unturned.

Working on the white board, I developed for him one of the great calculus ideas. Well, he developed it too, because he was discovering the idea for himself. He was seeing that a sum of infinitely narrow rectangles was none other than his old friend the integral. It was an exciting lesson and I knew he would love it. But I was in for a surprise.

“I need to show you something,” he said. He browsed through his photos until he found a screenshot he had saved from at least four years before. Then he showed me a screenshot of a theorem.

I stared in disbelief. My eyes went from the whiteboard to the photo and back again. There on the whiteboard was the idea that eluded him so long. All those years he remembered, and now he got it.

Sometimes we think that a lesson is not complete until it is fully understood. We worry that a narration missed a point, that a line in Shakespeare was too hard, that a detail about an artist was overlooked. We think that understanding is the greatest gift we as parent-teachers can give.

But sometimes it is a gift not to understand. Sometimes it is a gift to hold on to a question, in private, in secret, in hope. Believing that someday the question will be answered if only we persevere. And when that moment comes, it is a taste of eternity.

@artmiddlekauff