Yesterday I gave my students a quiz to test conceptual understanding of first and second derivatives. The maximum possible score on the quiz was 16. You can see that a few students had a strong conceptual understanding of these ideas, but many didn't. What type of conceptual difficulties did they have?
Let's look at the back of the quiz first. (Note: The front of the quiz was low level vocabulary questions and calculating the second derivative of f(x) = 1/x to determine concavity.)
The other questions I asked were:
- Based on the first derivative, when does the original function have tangent lines with negative slopes? Explain how you know from the graph.
- Based on the second derivative, does the original function have an inflection point? Explain how you know from the graph.
- Based on the second derivative, when is the original function concave up? Explain how you know from the graph.
For some reason students thought that the graph on the left was the original function and spoke about derivatives of the parabola. When some analyzed the second derivative, they said that there was no change in concavity, because the graph had a constant slope and since the slope was positive, the function was always concave up. Others talked about the function having a minimum at (0, -3). Although the first derivative has a minimum at that point, the original function does not have a minimum at that point.
Based on what students wrote, you can see that they have been in a calculus classroom. However, they had a hard time making the connection between the calculus ideas (concavity, inflection points, extrema) and the graphs of the derivatives.
Tomorrow, I will be starting class by assigning students to small groups of 2-3 students with one of the top scoring students in each group. Then, I'll have them do some similar questions using Plickers. Having students discuss the solutions with their peers should help some of them to get a better conceptual understanding of what is being shown on the graphs for the derivatives. I also have a derivative matching activity and I'll probably use that on Friday or Monday.
I called this blog post "Assessing Conceptual Understanding Fail", but actually it helped me to uncover some of the incomplete understanding students have. Learning about what my students know/don't know and understand/don't understand and modifying my teaching based on that is a win.