Monthly Archives: February 2014

On the purpose of examinations

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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.

The Adventure Continues

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I am now four weeks into my adventure in standards based calculus for this semester’s “Calculus II” class. Over the last week, I’ve given the semester’s first round of exams, both in Calculus I (using a traditional grading method) and in Calculus II (using standards-based grading). All of my students have received back their exams with my feedback. In this post, I’m hoping to reflect on my experience with both sets of exams, and give an update on how things are going.

About the Exam Grading Experience
Something I’ve struggled with using a Traditional Grading [TG] system is how grading exams makes me feel. Sure, no one enjoys grading exams, but I’ve found it can be a really miserable experience. For instance, when I see a solution that has a bunch of algebraic errors, instead of noting, “This student needs more practice with algebra” I have thought, “I didn’t explain the algebra very well” or I wonder, “Should we have gone over more algebra review? Should I have assigned more homework problems on this topic?” etc.

A second thing that has bothered me is that while I can easily grade an “A” paper, and I can easily grade an “F” paper, it is somewhat time-consuming to assign grades to the in between cases. For example, on a problem graded out of 14 points, I have to make lots of decisions of the form “Is this solution worth 3/14, 4/14, or 5/14?” — and this feels really subjective. I also believe it sends the student the message “try to get as many points as you can” rather than “try to master this topic perfectly”

The last big thing that is bothersome about the TG system has to do with what happens after I hand back an exam. In the instance of a student who has done poorly, I have seen them stuff their graded exam into their bag, and it is never to be seen again. When students ask me questions about their exam, the questions are always of the form: “Can I have another point on this question?” or “Can I still make a B+ in the course?” This is unfortunate since I think better questions would be, “What idea am I missing that caused this error?” or “I missed a step in this line of reasoning, can you help me find where I went wrong?” or “How come 1/0 is undefined but 0/1 isn’t?”

Happily, my first round of SBG exams resolved both these issues. First, grading the exams was a lot easier on me, since I knew each student would have as many opportunities to re-demonstrate what they missed. Second, instead of figuring out if they “learned” 20% of the idea or 22% of the idea or 24% of the idea, I could simply suggest they practice more and try again later, so the exam grading process went much faster. Lastly, since I handed back the tests, the questions students have asked have been all about mathematical ideas, and not about trying to find the optimal point-getting strategy.

Beyond Grading
I’ve also gotten a lot of positive feedback from my SBG students. Several of them have mentioned that they appreciate having less pressure on exam and quiz days, since they know (a) their scores will be replaced later on and (b) they can bring up their scores at any time by doing a re-assessment. Also, I’m getting many more students during my office hours and I have a much better sense on where each student is with our material. This is great because I can offer better advice on how they can improve. I know that this student needs more practice on integration by parts, and another student is having troubles remembering all our trigonometric identities.

Quick Summary:

  • My SBG assessment is going faster than my TG assessment, (even though the number of problems I’m assessing per student has gone up substantially). The grading is much faster. Students want to learn how to do the problems after I hand them back, rather than just toss them out.
  • My SBG students seem happy with the way the course is going; many of them come to my office hours regularly and want to do more problems. They are asking better questions and no one has argued for more points or a better grade on anything.
  • I hope I am sending the important message that to be successful in mathematics, you have to get used to self-correcting. In other words, you don’t have to get a problem right the very first time; instead, the better skill is to have the patience and confidence to re-attack what you don’t know — even if learning it takes multiple attempts.
  • And now I really wish I had set up my Calculus I course with an SBG system, too.