Calculus I—Day 9

I gave an exam during the last class, so rather than the usual format of “What I Taught”/”What Worked”/”What Didn’t Work”, I am going to talk about the exam itself.

The exam consisted of 8 multiple choice questions, 8 true/false questions[1], and 6 free response questions. The free response questions were all meant to be fairly straight-forward computations, the true/false questions sought to highlight theory, and the multiple choice questions had a mix of both.

Overall, I think that the exam was fair. Before writing the exam, I looked over my notes and the homework to compare what I had planned to teach to what I actually taught, and created a study guide that emphasized the topics that we had actually covered. The exam then essentially wrote itself, in that most of the questions were either directly copied from the homework (with slight modifications such as changing numbers around or reframing the question slightly) or from my lecture notes. Moreover, each question was specifically tied to one or two items from the study guide. Thus, a priori, I had reason to believe that the exam was fairly constructed.

Having graded the exams, I continue to believe that the assessment was fair. Out of 26 students who took the exam, there were 3 As, 8 Bs, and 7 Cs, which seems to be a fairly reasonable distribution for a summer class. Students seem to perform better in summer classes as a general rule—in a regular semester, I would be concerned about the relatively larger proportion of Bs, and worry that I had made the exam too “easy,” but I see no reason for concern in this case, as the tail to the left looked pretty typical. Basically, the middling students are better motivated and have more time to focus on this class, hence the middle of the pack is a bit higher than one might expect during a semester long class.

On the other hand, the exam was too long. Generally, when I write an exam, I let it sit for a while, then come back to it a few days later (after forgetting exactly how I wanted the questions answered) and see how long it takes me to get through it (thus creating my answer key at the same time). As a rule of thumb, I figure that anything that takes me one minute to complete will be finished by my fastest students in four or five minutes, and require nine or ten minutes for my slowest students. Thus if it takes me 10 minutes to write a key, then I will have some students done after 40 or 50 minutes, and only one or two students still in the room after 90 minutes. Since we have 95 minutes of class, I was shooting for a 75 minute exam, and was happy when the key took 8 minutes to finish. Unfortunately, this metric seems not to apply in this particular class. After 50 minutes, no one had finished, and the last student finally turned in his exam after more than two hours. Obviously, that was too much. I’ll have to cut that down for the remaining exams that I need to give.

As to the actual details of the exam, there were some real positives, but also some really surprising problems.

First off, I am very happy with how the students performed on the true/false questions. These were tricky, and often turned on little subtleties, such as whether or not a particular function was even defined. I was also quite pleased to see students take ideas that we had discussed in class and extend them to moderately novel situations on a few of the free response questions. The majority of my students are clearly thinking mathematically, which is heartening.

On the other hand, the lack of basica algebra proficiency was appalling. On more than a few exams, I saw the simplification
\[
\sqrt{16x^4 – x^2 + 2} = 4x^2 – x + \sqrt{2}.
\]
I guess I am going to have to remediate, but golly, it is depressing.

I also had a few students complain about the inclusion of a natural logarithm in one problem, because none of the homework problems included logarithms. Of course, we had discussed logarithms in class (and worked a couple of examples), they should have been covered extensively in pre-calculus, and their calculators have an “LN” button that completely dealt with the particular problem in question. Perhaps it was a bit unfair to ask the question, given that none of the homework problems included logarithms, but that was a balance question (asking questions about more material vs making the homework too long) and a question of fighting the online homework system. In the future, I think that I will assign those extra problems.

Notes:

[1]
The true/false questions require not just an answer of true or false, but also an explanation of the answer, both allowing for partial credit and a better demonstration of a student’s mastery of the theory in question
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