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Updated: The first version of this question was implicitly assuming that we know in advance roughly what the distribution of grades in a class will be (for instance, roughly the same as last year). This assumption wasn't actually essential to the question, and was bothering some people, so I've updated it to remove that assumption.

Inspired by this question, how difficult should an exam be? For instance, assume you've decided about what skills should denote a B- student, and that you're free to norm exam scores to course scores as needed. By adjusting the questions you put on an exam, you can adjust what score a student with B- skills will typically get. Where should you aim for?

Some answers I can think of, which some plusses and minuses:

  • Aim for the score traditionally correlated with that grade. For instance, in the US there's a custom that a B- corresponds to a grade in the 80-82 range, so you could try to write an exam where a score of 80-82 will typically reflect the skills of a B- student. (Actually, if it's a class where non-exam items will tend to increase grades, aim a bit lower.) The advantage is that, if the exam, there's "no curve", and students tend to feel like they've done well. A disadvantage is that if you accidentally make the exam too easy (say, a question has a short cut and doesn't reveal as much as you intended), your options are limited.

  • Aim a bit lower. Say, aim for an exam where a score of around 70 reflects a B-. More importantly, this gives a bit more separation at the top of the class (you have room for a question or two to distinguish the top students from the very good). On the other hand, students seem to feel like their grades are arbitrary, and they may feel more competitive (and less willing to work and study together) if they feel like their friends doing well will cut into their curve.

  • Aim much lower---say, put a B- around 50 and curve heavily. This makes the cut-offs between final grades much wider (the gap between a B- and a B might be something like 7 points instead of 3), and therefore a single small mistake has a smaller impact on a score. Students get quite discouraged by these, in my experience.

My question is whether there's any actual research suggesting that some of these are clearly good/bad for student learning, and whether there are ethical or practical issues I'm overlooking.

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  • $\begingroup$ This isn't a full answer, but an easy way to offset the problem you pointed out with the first solution is to aim slightly higher for a future exam (or the final). This way, you make students happy (and get your goal) in the average case, but in the case where you mis-target, you have a less "mean" way of fixing it. $\endgroup$ – adamblan Apr 22 '14 at 16:46
  • $\begingroup$ Perhaps slightly related: matheducators.stackexchange.com/q/1112/262 $\endgroup$ – Benjamin Dickman Apr 22 '14 at 17:20
  • $\begingroup$ @adamblan: That's a good point, and an option if you're taking route 1 and miss, but it has its own problems---doing that implicitly reweights the exams so that the later ones are more important. $\endgroup$ – Henry Towsner Apr 22 '14 at 17:43
  • $\begingroup$ @MichaelE2: Just to be clear, my question is about the desired numeric average to aim for given an established expected letter grade. But as for the underlying question: for any given course, I'd like everyone to get an A, and that's what I'm aiming for, but I have yet to achieve this objective, and part of teaching a course is giving an exam and final grades which fairly reflect what students have actually learned. (As an aside, if my students were consistently getting A's, I'd have to start wondering if the course shouldn't cover a bit more.) $\endgroup$ – Henry Towsner Apr 22 '14 at 18:16
  • $\begingroup$ Wouldn't this be better on academia-SX? I'm not sure how much of this is related specifically to mathematics education. $\endgroup$ – Loop Space Apr 22 '14 at 18:41
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In my experience, many students will just coast along the second half if they get a good grade in the midterm. So I make the midterm a bit harder, and it counts less than the final in the overall grade. This makes sense otherwise too, as the final covers the whole course, and the midterm just the first half or so.

There should be no "desired grade" for an exam, I consider it my duty to certify that whoever passes my course has shown a sufficient mastery of the subject matter, as defined by the syllabus. If everybody passes with A or B, great! If few pass, there is something wrong (too ambitious syllabus, exams were too hard, bad teacher, ...).

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  • $\begingroup$ But how do you determine what constitutes passing with an A or B? Or, put the other way, when you write an exam, you presumably expect that even people who have mastered the material will not necessarily score 100%; instead you write it so that people who have mastered it will score...what? $\endgroup$ – Henry Towsner Apr 22 '14 at 19:30
  • $\begingroup$ @HenryTowsner, our grades are currently 0-100, with 55 as passing grade, not letters. They used to be A = 90-100, B = 75-89, C = 60-74, D = 55-59, F = 0-54; C was "pass," D was "pass with reservation." I give leeway for mistakes, my exams typically add up to 120 points. Yes, it has happened that people got final grade over 100... I tell them I'll keep the extra grade for the next time they take the course. $\endgroup$ – vonbrand Apr 22 '14 at 19:34
  • $\begingroup$ If I understand what you're describing, you're in the first camp---you arrange your exam so there's an a priori correspondence between score on the exam and score in the course. What would you do if you misestimated an exam---say, a problem turned out to much harder than you expected, and no one got more than an 80? $\endgroup$ – Henry Towsner Apr 22 '14 at 19:43
  • $\begingroup$ @HenryTowsner, it has happened to me, catastrophically. In that case I'd either regrade the offending question (i.e., give full grade for a certain part of the work; and if somebody did more, give extra credit), or just adjust grading for that question according to the maximal grade (i.e., if it was worth 20 points, and the maximal points were 10, multiply that one by 2). $\endgroup$ – vonbrand Apr 22 '14 at 19:52
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    $\begingroup$ @HenryTowsner, but done only in case I discover a fatal mistake on my part too late. I define the exams and the grading beforehand. Doing the exam myself helps big way in weeding out badly set up problems. $\endgroup$ – vonbrand Apr 23 '14 at 14:51

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