One question which is often raised with regard to innovative teaching materials is one of evaluation: how can one properly test students' understanding? Indeed, it is much harder to test students on their understanding of how to apply mathematical software to non-standard problems than it is to test them on conventional pencil-and-paper computational techniques. We have experimented at Maryland with two ways of testing students on the CLeaR Mathematica materials in multivariable calculus:

1. Adding one or more problems to "conventional" calculus tests that require students to interpret and analyze the output from a Mathematica notebook;
2. Giving "on-line" exams in a computer lab, where the student is given a Mathematica notebook that is only partially complete, and is asked to complete it with appropriate calculations and interpretative comments.

Sample Problems Using Paradigm 1

(for pencil and paper exams)

1. Answer this question with the help of Notebook 1 (also available in Mathematica Notebook format). Consider the curve parametrized by r(t) = cos t i + sin t j + Sqrt[4 - sin^2 t] k.
   1. Why is it clear that v(t).a(t) = 0 whenever v(t).v(t) = 1?
   2. Determine the curvature and torsion of the curve at the points r(0) and r(pi/2).
   3. Determine the Frenet frame (the unit tangent, normal, and binormal) at the points r(0) and r(pi/2).
   4. It seems clear from the picture in the notebook that the curve parametrized by r(t) is not planar. How is this observation confirmed by the computations you have done in parts (2) and/or (3)?
2. Answer this question with the help of Notebook 2 (also available in Mathematica Notebook format).
   1. What is computed on output lines 2 and 3 of the notebook?
   2. What is computed on output line 4? What is the significance of the I's that occur?
   3. List all critical points of the function f(x, y) = x^2 y + 2x y^2 - 3y + y^4 in the xy-plane, and classify each as a relative maximum, a relative minimum, a saddle point, or none of these.
   4. Locate all the critical points of f on the contour plot in the notebook.
3. Answer this question with the help of Notebook 3 (also available in Mathematica Notebook format).
   1. What simple identities about the curl and gradient are being proved in the notebook? (State the result in standard mathematical notation, not in Mathematica input form.)
   2. Use the identities to compute curl[(x^3 + y^3 + z^3)(x i + y j + z k)].

Sample Notebooks Using Paradigm 2

(for on-line exams)

• Here is a sample Exam 1 notebook on the first part of the course (also available in Mathematica Notebook format).
• Here is a sample Quiz notebook on the multiple integrals chapter (also available in Mathematica Notebook format).