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In the linear algebra course I taught, I have three exams. In the first exam, which is probably too hard, students did poorly. So many students said they have lost confidence. To encourage them, I announced that if their grades in the final exam is better the average grades of the three exams, I will use the final exam's grade on their report.

In the end, some students performed much better in the final than they previously did and got very good grades. This caused other students who did better on average, but not particularly well in the final, to complain that it's unfair. I also realized that some students can game the system by intentionally under-perform in the second exam.

It is a bit late for this course. But what should I have done after the first exam? Perhaps I should just announce that I will reduce the difficulty of the second and final exams?

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    $\begingroup$ This doesn't seem to be math specific, maybe better suited for academia SE. $\endgroup$
    – Jasper
    Commented Mar 13, 2022 at 9:38
  • $\begingroup$ How can students "game the system by intentionally under-perform in the second exam?" In what scenario do they gain from under-performing? $\endgroup$
    – Erel Segal-Halevi
    Commented Mar 16, 2022 at 17:14
  • $\begingroup$ @ErelSegal-Halevi The can do so in order to lower average grade so they can make sure the final is used as their exam grades instead of the average. $\endgroup$
    – user11702
    Commented Mar 17, 2022 at 2:32
  • $\begingroup$ Let the exam grades be $x, y, z$. From what I understand, your reported grade is: $r = \max( (x+y+z)/3, z)$. I do not see how a lower $y$ can lead to a higher $r$. $\endgroup$
    – Erel Segal-Halevi
    Commented Mar 17, 2022 at 10:26
  • $\begingroup$ @ErelSegal-Halevi If they have a low $x$, they can intentionally lower $y$ to make sure $z$ is used as the grade. $\endgroup$
    – user11702
    Commented Mar 17, 2022 at 10:58

5 Answers 5

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In the past, I've identified the problems on the final exam that review exam 1 material and offered to use them in place of the original exam 1 grade, provided this improves students' grades and they come in to office hours to review exam 1. This requires a lot of questions to get good data.

Nowadays, I use a standards-based grading system with plenty of retakes. It's a lot of work, but handles these sorts of problems naturally: students can make up missed points in office hours or on later assessments.

A trick to preventing these sorts of situations is to offer students a "practice exam" as an in-class activity just before an exam. This makes sure students know how much they need to prepare for the actual exam but also lets you sneakily gauge how difficult the exam will be.

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    $\begingroup$ (I don't think it's "sneaky", I think it's pretty obvious it has usefulness to both parties. It also gives the instructor a heads-up if many people do badly on some section that it was was badly taught/ misunderstood/ needs revisiting, and also it guides students on where they can usefully spend their revision). $\endgroup$
    – smci
    Commented Mar 13, 2022 at 20:34
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    $\begingroup$ @smci the point of being sneaky is to keep the students focused on how it helps them. Don't want them seeing it as busy work. $\endgroup$
    – candied_orange
    Commented Mar 15, 2022 at 22:13
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Corrective: The practical immediate answer is to do another test, covering the same material. Burn an hour. Not the end of the world. Let students keep whichever grade they do better on. Use your judgment if you water down the re-exam or if you just cover very similar material and the kids get kicked in the ass (calibrated) on level of effort needed. But do warn them, if you are retaining the level of difficulty (say nothing if weakening it). But do advise them on how to effectively study for the re-test.

Preventive: I love Tom's suggestion, but another similar one. Don't do 3 exams per semester. Do one per week. Had a teacher, for two years in HS, pre-calc and calc, who did this. Every Friday was a 50-min test.

When I first encountered it, thought it was a lot and how do we cover the material. But it ended up working great. Just like homework problems are the most effective studying, tests are basically sets of compelled homework problems. The regular routine made things very easy to plan in my life. And having more moderate stakes (not high stakes like a final/midterm, not low stakes like turn in your homework) felt like a reasonable amount of stress.

By the way, this will also allow you to get better at calibrating things yourself (since you have more tests). And it makes a random poorly designed test less critical in students' end of grade experience (just "one of those things", like a stochastic variable in a trend). But those are tangential benefits. The main driver is better training of the trainees by having more evaluated performances.

I'm sure I will hear squawks about lost lecture time (from people paid to lecture, go figure), but there is more and more proof that people learn from practice (from doing) better than from hearing/observing. And tests are a performance by the student.

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  • $\begingroup$ Frequent test seems to be a good idea. Do you think we can replace homework with it? $\endgroup$
    – user11702
    Commented Mar 14, 2022 at 3:54
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    $\begingroup$ I don't think doing another test is a good idea. The students already feel the marking is 'unfair' so changing the exam structure again isn't going to make them feel any better. Further, it may only be 1hr of exam time, but it will require them to commit a lot more prep time and stress for something which isn't their fault. $\endgroup$
    – David258
    Commented Mar 15, 2022 at 11:24
  • $\begingroup$ @ablmf: You don't have to replace homework, but you can make homework cover similar questions to the upcoming tests but optional; if they hand in any homework you will give feedback but it would not count towards the grade. The compulsory frequent tests allow you to evaluate the students, and the optional homework allows students to prepare for the tests. $\endgroup$
    – user21820
    Commented Mar 16, 2022 at 11:17
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Students call anything "unfair" when they don't get the grade they want.

Yes, students will game your system. It doesn't matter as long as they leave with the required information in their heads. Students need to realize that writing exams are tools for their own improvement and that grades measure performance, not effort.

If other students can blow off the mid-term but still ace the final that represents an unequal effort, surely, but the goal of your course is not to make your students expend effort, and certainly not for them to expend effort equally - it's that they learn the material. It's illogical for a highly capable student to invest effort in an exam that won't have weight in their final grade if they don't need that exercise to master the material - it's just a waste of time and they can make better use of that time by applying it elsewhere.

Maybe it is unfair that some students can learn quickly, skip the midterm, and still achieve on the final. But that's just reality - some people are more capable learners than others. What matters is that they leave with the skills your course aims to provide them. The only disconnect is with students who feel their efforts entitle them to grades regardless of their actual achievements.

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As a student, I appreciate that you care about students' performance and experience in your class.

What many professors I've had have done is average the lowest test score with the highest and replace it with the result. So if they have 50, 80, 100, it goes in as 75, 80, 100. I always appreciate policies like this because in the span of a months-long semester, especially during COVID, there will usually be a period where I underperform just because of issues outside of school.

In my experience, as long as you are a good teacher, communicate well with students, and maintain a well structured class, nobody will complain too much about how extra credit is done if it's given. Then again, I don't hear every complaint on the teaching end so what do I know.

My linear algebra professor this semester does a HW and quiz every week, which helps prepare students for tests, and also spreads the course grade weight across more assignments so having a bad day on one test won't kill your grade for the entire class.

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  • $\begingroup$ Happened to see this answer of mine, and wanted to make a comment: The linear algebra professor I referenced at the end ended up making our final worth 40% of our grade, and although it was supposed to be cumulative, they made 80% of it about the final third of content, which everyone failed because frankly she didn't teach it well. I, and most of the class, went into the final with a B, and despite studying for weeks ended up with a C- in the class (not good enough for course credit) after the final. Don't be like that professor. $\endgroup$
    – Goel Nimi
    Commented Jan 5 at 1:26
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All great answers, another thought.

Sometimes a student has external factors going on or is not good at taking tests (Nero Divergent students especially)

So instead of another test I give them a project that covers the test scenarios. Ex: instead of 20 questions about the Python programming language i will give them a scenario where they have to cover 20-30 items in a project program.

This allows me to still test their capabilities and evaluate their score accordingly.

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