Just before Spring Break, we were finishing up a unit on sequences and series in my pre-calculus classes, and I started thinking about review problems. I thought about the Handshake problem, the diagonals of a convex n-gon, constructing a Sierpinski Gasket and related figures, patterns in Pascal's Triangle, and a few other "puzzle-style" problems. Some of these showed up in some form in the homework problems I had assigned during the unit, but I realized that I had not really used them in class. Looking back through my plans for the unit, I found I had pretty much followed the textbook, and in the process probably reduced a really important set of ideas to a bunch of formulas, and lost some of the excitement I usually had for the subject.
I had taught the unit wrong.
I didn't teach any of the content incorrectly. I didn't leave out important ideas that the students will need as they move into the next course. I tried to make sure everyone learned the content, I checked for understanding as we went along, and I gave opportunities for students to process new ideas.
But I had lost the "story" of the unit. A colleague once told me that she did a lot of up-front planning on a unit, and on a course, because she really wanted to understand the content as a story, with a beginning, middle, and end, including plot twists and cliff-hangers. My sequences and series unit had all the important pieces, but lacked any narrative thread to bring the ideas to life. The result was I had students really worried about memorizing formulas, asking "Is this on the test?" or panicking because they were having trouble with patterns outside the typical arithmetic or geometric ones.
I can't go back and reteach the unit a better way. We don't have time left for that in the school year, and once the students have seen the major plot points, rearranging them or weaving in story details isn't going to make the story more coherent. So, I now have a big note on the first page of my planning notes for the unit to get back to the story. And I'm working on the narration for our next unit.