If you stopped my students on the street and asked them to find the area of a rectangle, they'd do it: no problem. If you ask them on an exam, all that previous knowledge disappears. One year a student got stuck on area of a rectangle because for reasons unknown the student insisted it must have a height. It must be a three-dimensional object, and the height was not given, so the problem was impossible to her. More recently I have avoided such discussions during the exams, but have still gotten A = lw^2 or A=2wl a number of times when asking the classic optimization question involving fencing a rectangular area next to a barn.

This is not so different than physics exams on which the earth has radius larger than the distance to the edge of the solar system, or temperatures occur in negative Kelvins, or the bicyclist goes 16,000 miles in one hour if the speed is 12 mph. I know that students get anxious during exams, in particular. These "brain farts" or -- more chillingly, the real lack of understanding revealed -- lead to poor grades. Students get Ds or Fs. They may fail a class because of their difficulty keeping a clear head or relating concepts in a classroom to real life.

As I'm recovering from grading, though, I'm the one who feels like the real failure. Let's be honest: as teachers, we do measure ourselves by our students' success. If I can't get them to understand or can't get them interested or can't coach them to success, I feel like a failure. Rationally I know that there are many factors involved. If they're in college and can't reliably come up with the area of a rectangle, something went wrong far before I came along. Students who tell me they spend about two or three hours on calculus outside of class are certainly setting themselves up to evade success. As I learned from some wise friends, you can't care about students' grades more than the students themselves do. Students are entirely surprised that the class might be difficult: one student asked me recently if this was

*supposed*to be harder than high school -- she thought she was doing something wrong. There are a lot of things outside of my control here.

It doesn't change my feelings, though. Failure. Whose?

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