What can a professor do about an exam that was too long for the allotted time, after the students have completed it?

This is an excellent example of when a professor may choose to grade on a curve. Grading on a curve normalizes the exam against observed performance rather than an absolute standard, and is a good way to re-normalizing when an exam proves unexpectedly challenging.

Your professor, however, might not choose to do so, or might choose to set the curve in a way that doesn't help you as much as you might like. The key is what's happening in the class relative to the actual educational goals of the professor.

• If the exam was harder or longer than intended, and the professor believes that the lower grades reflect that, rather than the knowledge of the students, then they are likely to grade on a curve.

• If, on the other hand, the students are showing evidence that they do not understand things as deeply as they are required to, then the professor may not curve or may adjust with a curve less than you hope.

It's really impossible to know which case without knowing the details of the exam and peoples' performance. A "long but easy" test sometimes is testing for deep facility that allows people to be extremely fast. For example, think of the arithmetic exercises sometimes done in elementary school, which can only be solved quickly enough if the student has effectively memorized and internalized arithmetic, rather than working out answers from first principles (e.g., counting on fingers). The same principle can sometimes apply in undergraduate and graduate education as well.

Bottom line: maybe the professor will choose a generous curve, but there are reasons that may argue against it as well, depending on the circumstances.

I found myself in a position similar to your instructor's. I considered and rejected several possibilities:

• Adjusting the curve or grading scale to compensate can be unfair to students who cannot read and understand English quickly, especially if the exam is text-heavy or the students are not native English speakers. The result of this approach is that the exam grade reflects speed of English reading comprehension more than mastery of the course material.
• Dropping the lowest exam grade, or allowing a grade on another assessment to compensate for the midterm exam grade, only works if there are other assessments that cover the material that was on the midterm exam. Otherwise, a student could get an A in the course without having mastered some of the learning objectives.
• Adding another assessment that wasn't on the syllabus, or adjusting other assessments to cover the material that was on the midterm (in violation of the syllabus), isn't fair either, especially to students who have other commitments and need to make plans in advance and schedule their time very carefully.

Ultimately, I decided that the fairest approach is to give students another opportunity (but not in the form of a required assessment) to demonstrate mastery of the course material. Depending on the course format, the size of the class, etc., several possibilities are:

• Allow students to submit a correction to their exam, and earn partial credit for answers that they didn't complete correctly on the exam but did successfully in the correction. There is a possibility that students will receive unauthorized assistance on this correction, though.
• Same as previous suggestion, but have the students explain their corrections in an oral exam. This reduces their ability to benefit from unauthorized assistance, since by questioning the students it is easier to see who really understands what they are saying. However, it is unfair to students who get nervous or have trouble expressing themselves in an oral exam.
• Grade the original exam on a curve, but allow students to take a makeup exam covering the same material, if they feel like their "curved" grade doesn't reflect their knowledge of the material. For students who take the makeup exam, the makeup exam grade replaces the curved exam grade. This may be unfair to students who have limited time to review for and take another exam, possibly because of outside commitments, and have to manage their school and other commitments very carefully.

As you can see, none of these are perfectly fair, either. Depending on the particular circumstances (and possibly with input from the class), an instructor may decide which is the least unfair.

I have faced this a few times. (Once there was a general insistence that there was not enough time, though the number and difficulty of the questions was not noticeably different than in previous years; another time large numbers of students emailed me after the exam to say they had not known there was a question 5 on the back of the last page, despite the obligatory "THIS EXAM HAS FIVE PARTS" boilerplate on the first page.)

The first thing you do is to mark the exam in the usual way. Then you observe. Did many people leave the last question blank? Or just scribble a few quick points in a way that suggests a lack of time? If someone did the questions out of order, did they do worst on whatever question they left for last? Is there any kind of noticeable pattern to the exams that is not like the pattern you usually see for that course (eg every year, the X question might be the one everyone finds hardest; if this year the Y question, which was last on the paper, appears to have been more of a challenge then you have something observable.)

If you see no pattern and the average mark is about what you expected you don't need to do anything. If you see no pattern and the average mark is low, you can either work harder at getting the material through the heads of this particular cohort, or set an easier final to keep the average up (an approach I reject, but mention because some people do it.)

When I did see a pattern, I made the following offer to my class:

Do you think your performance on the midterm truly reflects your knowledge of the material? If you do not (for example if you feel you were constrained by a too-short time limit) then you may use your mark on the final exam as a replacement for your mark on the midterm. You must request this accommodation within one week from today.

That last constraint was very deliberate. You have written the midterm. You have received your marked midterm. We have taken up solutions to the midterm questions and discussed particular areas where you may not have known something or may have been in error. You have a good grasp right now of what part of your mark deficit (my students always seem to have some mark they believe they deserve, and want to know why they didn't get that mark, as though I start at 100 and subtract) comes from "not getting it" and what part comes from "running out of time". If you think the real issue was running out of time, then you know that right now.

There was always someone who wanted me to mark their final exam and then only use that mark if it was better than the midterm. No way. This isn't some sort of bet or optimizing technique. This is a one-time offer: if you're so sure that midterm was not a reasonable instrument for assessing you, I'll throw it out. I'll use my remaining instrument, the final exam, for assessing that part of your mark. (Assignments in my class were group work and in any event assignments and exams assess different skills and knowledge; I would only be willing to substitute exam for exam.)

In years where there were many complaints, some (but not all) of the complainers would take the deal. Rarely, it would be really good for them. Say 50% on the midterm and 90% on the final exam. Often, it would be a small improvement - 60% on the midterm and 70% on the final exam. Over half the time, it didn't help them at all. They got 50% on the midterm and 50% on the final exam too, even though there were no time complaints on the final and they may have left early (I write finishing times on all final exam papers as I receive them.) So I'm not entirely sure this approach solves the actual problem of some students getting lower marks than they deserve on the midterm due to lack of time. However it completely solves the problem of students complaining because they believe they got a lower mark on the midterm than they deserve. And where there has been a true mismeasurement, it does fix that.