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CAM 1/26/11:

I did this activity as a substitute teacher in Tevian's Honors vector calculus class. It worked really well, particularly because it forces students to think geometrically about their algebraic calculations. I was surprised by the following things:

• Several students (and therefore two of five groups) got hung up because they did not believe that the two different expressions for the vector field were the same. It was helpful to them to actually show that $\hat\phi=-y\hat\imath+x\hat\jmath$ explicitly with a diagram, working out the components of $\hat\phi$.

• MANY students were troubled with the examples where they calculated the integrand to be zero. If you do not think of integration as chopping and adding, then the integral of zero is a strange concept. For some of these students, they are still unaware that zero is a number and therefore a possible answer to a calculation. See our article about Students don't believe that zero is a number.

• One group calculated $\vec B \cdot d\vec r$ generically, without having a particular path in mind. Because of the dot product, their answer was proportional to $d\phi$. They then had a lot of trouble thinking about this when they did choose a radial path. Here again, they were trying to integrate zero and had not thought geometrically about why that would happen.