This is my fifth year calculus, and by far my best. I have a gloriously small class of six motivated students. It's the first year that I have the privilege of teaching students whom I previously taught (I had most of them three years ago in Math 9).
Setting the Stage
From Day 1 I have created an ethic of "we don't use things we can't prove." I encourage them to use their intuition and estimation to make conjectures about new ideas, but I don't give them formulas to plug and chug on. I do my best to eliminate or limit lecture and instead give them guided explorations to work through.
In the last several weeks, they have proven: the power rule, the product rule, the quotient rule, and derivatives of sine and cosine.
Begging for Proofs (yes, really!)
What's so amazing this year is that students are begging to prove more things. When I mentioned that I might go through the proof on the board for the quotient rule one student said, "what, do you not trust us to figure it out ourselves?" is a gloriously sassy voice! Another student has been e-mailing me photos from home as she figures long proofs on her whiteboard in her bedroom. When she was home sick for a day she asked, "is there anything else I can prove, this is a fun game?"
Often some students will figure out how to prove something before others do. I've started to say "don't ruin the punchline," as if we're talking about a joke. If someone finds out what to add and subtract to prove something, they may give small hints if their classmates ask, but they must keep in mind the joy that they had with finding it on their own.