When I was in my AP Calc class in high school, the teacher began by explaining what a derivative was. He drew a picture with a curve, drew a line across two points on that curve, and explained that the derivative is the slope of that line as the two points get closer and closer together. We knew how to calculate a slope, so it was easy to build on that knowledge and take it further. It’s been 20 years and I can still do this, after having seen it that one time. I don’t know if this way of teaching derivatives is unique, but it felt like a far cry from memorization, even back then. It was effective because it was visual, because it explained something intuitively, and because it added a small piece of information to an existing foundation of knowledge.

Incidentally, the only things I remember from biology are pictures, or the demonstration that a teacher made when someone “stretched” her “intestines” across the room.