I was working with my AP Calculus students on understanding limits graphically. With all the closed and open circles, left- and right-sided limits, they were feeling really lost. I had given them two mini quizzes on the graphical interpretation of limits, and the second one went a little bit better than the first, but not by much. So what I decided to do was have them create their own graphical limits problem in groups. They were tasked with sketching the graph of a function (piecewise defined) and coming up with questions about limits and evaluating x-values that would challenge their classmates.
The conversation surrounding the properties of limits, when a limit exists and when it doesn't, and how to know when a function is defined for a certain x-value were amazing. The students were debating about all of this in their groups and discussing what the answers were to their own problems, justifying their answers with theorems and properties. And when we did a gallery walk (so that students would work out and answer the problems that their classmates had created), students were talking out the answers as they looked at their classmates posters, helping each other understand why the answer was this, and not that, showing each other on the posters how to evaluate one-sided limits, and so on.
Although not everyone got all the problems from their classmates correct, they really felt much more confident and seemed to have a stronger understanding of graphical limits after the activity.
Why was this activity so effective? I'll do the research and revisit that question soon.