I really enjoyed reading this after finding you via typing key words that interest me into the Substack search bar.

I teach Calculus to high school seniors and juniors who, by virtue of their intense workloads and extracurricular commitments—our campus is fantastic, but many driven students mistake the buffet line of diverse participation options as a 39-course meal—often are pretty exhausted. Given the finite time I have with them and attention I can hold while presenting to them, my goals must include the necessary procedural fluency alongside the conceptual development that lets that have meaning to attach those procedures to.

Where I’ve landed in the last few years has been to diverge from that unrealistic goal of 100% understanding. My goal is to provide context, as you said. I want them to have a sense of how this new idea bridges old ones forward and of how a particular relationship confirms to intuition. I use many more metaphors—we analyzed functions via their derivatives’ graphs on Wednesday which I tied to figuring out if a person likes you from body language—and narrate the long proofs that centered so many college lectures, and then I walk them through problems while constantly pointing back to fundamental ideas, meaning, and the rich vocabulary being applied underneath the algorithmic procedure they follow. The instruction is richer—especially when Desmos lets me whip up an applet to make an idea or problem visual and interactive—and my students learn how to do the mechanics of the mathematics while developing fluency in the world. This strikes me as exactly what you’ve advocated.

Maybe I’ve gone astray from your piece; if so, I apologize. I appreciate your point, and I’d like to think that thoughtful design can accomplish sufficient learning on both fronts, but I am with you: if a student can solve that yogurt problem, they’ve bought time and opportunity to elevate their understanding later with prolonged exposure to it.

I really enjoyed reading this after finding you via typing key words that interest me into the Substack search bar.

I teach Calculus to high school seniors and juniors who, by virtue of their intense workloads and extracurricular commitments—our campus is fantastic, but many driven students mistake the buffet line of diverse participation options as a 39-course meal—often are pretty exhausted. Given the finite time I have with them and attention I can hold while presenting to them, my goals must include the necessary procedural fluency alongside the conceptual development that lets that have meaning to attach those procedures to.

Where I’ve landed in the last few years has been to diverge from that unrealistic goal of 100% understanding. My goal is to provide context, as you said. I want them to have a sense of how this new idea bridges old ones forward and of how a particular relationship confirms to intuition. I use many more metaphors—we analyzed functions via their derivatives’ graphs on Wednesday which I tied to figuring out if a person likes you from body language—and narrate the long proofs that centered so many college lectures, and then I walk them through problems while constantly pointing back to fundamental ideas, meaning, and the rich vocabulary being applied underneath the algorithmic procedure they follow. The instruction is richer—especially when Desmos lets me whip up an applet to make an idea or problem visual and interactive—and my students learn how to do the mechanics of the mathematics while developing fluency in the world. This strikes me as exactly what you’ve advocated.

Maybe I’ve gone astray from your piece; if so, I apologize. I appreciate your point, and I’d like to think that thoughtful design can accomplish sufficient learning on both fronts, but I am with you: if a student can solve that yogurt problem, they’ve bought time and opportunity to elevate their understanding later with prolonged exposure to it.

Happy Friday!