I finally completed my K-W-L's with my calculus students. We spent one class day where I threw up a random graph and asked them to come up with some ideas about derivatives and how the graph looks. It started off simple, "The derivative tells you the slope of the function," to having a girl say, "The function goes up when the derivative is increasing." I put up her statement as said and then we had to explore some slopes on our graph and one student ended up coming up with the more correct statement "If the slope is decreasing, then the graph will have a maximum."
Wow! In only thirty minutes, we had moved through critical points, the first derivative test, and were talking about concavity and the students were creating the rules for the second derivative test all based on an initial incorrect statement (she knew what she meant, she just used the wrong words and it was a wonderful teachable moment).
This was in my first period of calculus. It was one of those magical classes that you always dream of, when the students see why things are working as they are and you are so proud of them because of how they are creating the most amazing ideas on their own. Then I had two more periods of calculus.
The second two times, the K-W-L was still much better than what I could have done lecturing by far, but we stayed with discussions of the first derivative. I just had the students take a quiz over the relationship between graphs and derivatives and hopefully the success of using a K-W-L for introducing this material will be evident.
Oh and that random graph from the first day? It was taken from the 1989 free response question highlighted in the movie Stand and Deliver. At the end of the unit when we were preparing for the quiz, I pulled out the question again and we walked through all of the parts. It was a great way to conclude the unit and reflect on what we had learned!