On the purpose of examinations

I have just finished the second round of mid-semester exams in my calculus courses. As I may have mentioned before, I’m teaching two sections of “Calculus I” using a traditional grading scheme and one section of “Calculus II” using standards based grading. Both courses encountered their tests last week and had them returned with feedback this week.

While grading, I mentioned the following on Twitter:

Joe Heafner then replied: 

 

Since then, I’ve spent quite a while thinking about my purpose of giving them exams. I have the sense that students think the purpose of exams, in way exams are traditionally graded, is to compile as many points as possible. Here is a common approach:

  1. Write something down for every problem, whether or not you know how to do it, because you might get partial credit points for having at least something right.
  2. If you know how to do a problem, do it as quickly as possible; so long as you get the right answer at the end, there isn’t a need to write clearly or keep track of notation or show your chain of thought. If the answer is “7” then so long as you have “7” on your paper at the end, you’ll get full credit.
  3. Once the exam is returned, look it over. Ask questions of the form, “Why did I lose three points on this problem?”

I really dislike every step of this approach, but I don’t think I have ever communicated this well to my students and I have not been able to reward a change in behavior away from this strategy when I’ve used traditional grading. On the other hand, with standards based grading, I’ve tried to put effort into motivating my students to follow these steps instead:

  1. Keep track about what you know how to do and where you still need work. If you encounter a problem that you aren’t confident you can solve, don’t worry about it until you have more practice, seek more guidance, or have more time to study that topic.
  2. If you know how to do a problem, show all of your work. This is your chance to show off! Write neatly, explain your reasoning, and demonstrate mastery of the process. Don’t fret if the answer is “7” and you got “8”; we are interested in conceptual understanding and will happily overlook inconsequential arithmetic errors.
  3. Once the exam is returned, look it over. Ask questions of the form, “I am missing something conceptual here. Can you help clear it up for me?” or “I wasn’t sure how to attempt this problem. After I have more time to work on it on my own, can we go over it together?

I suspect that the two different grading schemes would result in very different types of solutions for process problems. One such problem on the “Calculus I” test stated, “Use the limit definition of the derivative to find the derivative of the specified function.” I was not very successful at getting my beginning calculus students to understand that I am more interested in the process they are using instead of just their final answer. Standards based grading has allowed me to have conversations during class about the reason we ask these types of problems and what constitutes a solution versus just an answer.

The purpose of giving exams in my courses is to allow the students the opportunity to communicate their level of mastery of the course material. I’m looking for their ability to demonstrate conceptual understanding and their fluency with the technical processes needed in various problem-solving situations.

I think that standards based grading has made it easier for me to explain this to my students. I don’t think my students see exams as an adversarial process where I am judging them or their abilities. It is my hope they see exams as an opportunity to show what they know, to discover what they still need to work on, and to give us both a clear picture of where we should go from here.

Postscript. One issue I need to work on in upcoming courses is motivating students toward mastery earlier. Some SBG students are a bit behind in showing mastery of the standards, and while this hasn’t been a problem for them yet, I expect they will begin struggling quite a bit very soon. Now that our course material is building on itself at a swift pace (integration by parts into improper integrals into the Integral Test for Convergence of Series) I worry that their [lack of] progress on our “integration by parts” standard will cause them difficulty keeping up with the course. My students have been great and I feel like this was my failure at putting together an accurate timeline of what they should know and when they should know it. I think maybe I focused too much on “you can always improve later” that the message “…but there’s no time like the present, so do it today!” was lost.

1 thought on “On the purpose of examinations

  1. Points=problems.

    Thanks for another insightful post. In particular, I have been really emphasizing the idea of “you need to know this by the end of the semester,” and now I need to consider how much I should temper that message with “the more you know now, the easier the rest of the material will be.” Thanks.

Leave a Reply

Your email address will not be published.