Students have a hard time with first derivatives and second derivatives in general. However, putting it to real use in graphing makes it even more complex.
Many students in Calc I struggle with the different ways to take a derviative and the algebra that needs to be done to clean up the problem in order to solve for graphing is more problematic
I gave the students much more practice in their groups this semester doing derivatives before even getting to the use for the graphing. They were able to work together and talk them through. We then slowly merged the ideas into how we go about using the derivative to graph. The algebra was the hardest so again they worked in groups to talk them out. It also helped having Chris McNeal in my class as an extra person to answer and ask questions. They had numerous practice problems graphing
I compared the exam scores this semester to previous semesters to determine if having more practiced helped. Overall out of 30 students only 3 students were below a 90%. When I asked the class as a whole how they felt about the material, they all agreed that they felt better prepared for the exam because of all of the group work and practice.
I will try this again next semester and see if the same results occur. I will continue with the introduction of each step slowly.
I am hoping that having this extra practice and group work in the long run strengthens not only their ability to do derivatives but also their algebra skills. Unfortuantely I do not have enough time in one semester to do this in every topic we cover, so I am hoping to come up with more creative ways to have them do this practice outside of class, that does not make them complain to me that I am giving them too much homework.