Here is one thing that I have been happy with this year. I decided to give the students goal-less problems for quizzes this year. Here how the process would go in a simplified Calculus I class with the following three learning outcomes.

- I can take derivatives of polynomials.
- I can determine antiderivatives of polynomials.
- I can solve definite integrals using -substitution.

A quiz would then look something like this: “A car is traveling with velocity kilometers per second.”

Note that there is no question here. With a goal-less problem, the students need to supply both the question an the solution. So a student might do the following:

“Question: What is the acceleration of the car at time ? Answer: Acceleration is the derivative of velocity, so we can compute the derivative to get , which evaluates to kilometers per seconds-squared when .”

The student would then submit this to a Canvas assignment that corresponds to the first learning outcome (“derivatives of polynomials”).

But the student could write a second question/answer combination as follows:

“Question: How far does the car travel from to ? Answer: We can find the displacement by determining kilometers.”

The student could then submit this to a Canvas assignment that corresponds to the second learning outcome (“antiderivatives of polynomials”). Since this situation doesn’t lend itself to -substitution, the student could submit question/answer combinations to up to two of the three learning outcomes.

Here is why I am happy I am doing this: one of my unspoken goals (unspoken to myself, even, even though it makes total sense) is that students should not only know how to use calculus, but they should also know when to use calculus. The fact that my students really struggle (in fact, I had to add an entry into my FAQ page about this) suggests that I shouldn’t assume that they automatically learn this without being asked to do something like this. The fact that students *don’t* struggle with this after the first week tells me that this is useful.

Note that students don’t struggle coming up with questions for material that we learn in the third week of class. This suggests to me that the reason why they are able to develop questions after the first week is the fact that they are being told that knowing *when* to use calculus is important—it isn’t just that they understand the material from the first week better. They really seem to be telling themselves that part of learning material is learning when to apply it.

I am going to add this to my tool belt. I could see this being done in a bunch of courses. I could imagine giving the students a group in abstract algebra. The students could write questions like “What is the order?” or “Is this abelian?” My colleague has done similar things in linear algebra: he will give them a linear transformation. The students might then determine the kernel, the image, the eigenvalues/vectors, whether it is one-to-one/onto, etc.

I think that I ultimately like this because it breaks the input-output response cycle. The students have to think more—just a bit more—for themselves, but in a way that is not too burdensome.