Handed back exams yesterday. Man, was that demoralizing. :\
What I Taught
I started the evening by handing back exams and going through the problems. It took quite a bit longer than I expected.
When I actually got to teaching new material, I started a three-day sequence of lectures that revolves around curve sketching. My goal for the day was to discuss the first and second derivatives, and what they tell you about the shape of a graph.
First, I made formal the idea that a positive derivative corresponds to an increasing function, and a negative derivative corresponds to a decreasing function. This matches the intuition that derivatives correspond to slopes and that positive slopes correspond to increasing functions, but I think that it was important to state this relationship explicitly.
This gives the idea that places of zero slope are interesting (which confirms our look at critical numbers earlier). In particular, we can ask “What is happening near a critical number?” There are a few pictures that we can draw: assuming that a function is continuous and that the derivative is zero at some point, then the derivative on either side of that point must be positive, negative, or zero. Drawing an exhaustive set of graphs (changing the sign of the derivative on either side of a critical point), we got the idea that we can characterize local minima and maxima as places where the sign of the derivative changes. This is the First Derivative Test.
After stating the test, I spent the remaining time going through two examples. In each example we determined the critical numbers, classified the critical points, and determined the intervals on which the functions were increasing and decreasing.
What Worked
I got to work examples! Yay!
I didn’t get through nearly as much material as I wanted—discussion of the exam took a while—so I got through as much material as a could, found a good stopping point, and had enough time to work a couple of examples, which was nice.
What Didn’t Work
As I stated before, the exam was difficult, and many students thought it was unfair (for the record, I don’t agree with them as an overall assessment of the exam, but there are a couple of questions that might have been a little unfair). I had intended to simply go over the questions, but at every turn there were complaints about how things got graded. My refrain is always “If you have a specific question about your grade, talk to me in private,” but this didn’t seem to dissuade anyone. I really should have been more forceful about it.
I am also really disappointed in how I let the time get away from me. The clock in the back of the room (i.e. the one that I can see from the front) doesn’t work, and so I generally set up the computer at the front to display the time. I forgot to do this, and so didn’t really know how much time was passing. This, coupled with a desire to go over the exam to everyone’s satisfaction, led to a lot of time being spent without going over new material.