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I am a TA currently grading a set of 80 midterms for an engineering math course for fourth-year engineer undergrads. The test had four questions, each question had around four parts.

Of course, typically a student can make a wide variety of different types of errors on the exam. But what I have noticed in this exam is that some of these errors are not necessarily conceptual, yet we are basically taking all the points away for these "minor errors" (per instructor's grading scheme).

For instance,

  • For a question with several parts. A student copied one of his own (correct) answers from the previous part incorrectly.

  • A student wrote down all the steps correctly, until the last step where he had to evaluate the expression at a number and the mistake was forgetting a negative sign.

  • After a long derivation, a student substitute formula incorrectly (forgot a square root in the formula, ahem, Gaussian). Everything else completely correct.

There was a litter of other minor errors. Now, these students are in their final year of engineering school, so I think I can give them the benefit of the doubt that they wouldn't make silly mistakes such as writing "0 - 1 = 1" or copy things down wrong in a non-time-constrained setting, which is the situation they will most likely face when they graduate and work in industry.

Is there a way to provide feedback for a student if he or she makes silly mistakes on the exam? I do not think that giving them a zero for writing a minus sign incorrectly is a good way of either providing feedback or preventing the same mistake from happening again in their (long) lives.

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This is mainly at the teacher's choice, in practice. Some deduct, say, zero or 5% of the grade for a silly arithmetic mistake (which I recommend); some deduct much more --- to the limit of those who look at the final result only and deduct 100% if it doesn't match the official solution (which I don't recommend).

It may be difficult to give an objective evaluation because in some cases a mistake at the beginning can turn an exercise into a completely different one, much easier or harder. If there is more than one person grading, you should discuss it with the rest of the group.

As a mathematician, an interesting point to consider is that not all silly mistakes are equal. In some cases, a student should realize they have made a mistake with some sanity checks at the end. Examples: if you have shown that a certain event has probability -3.72 to happen, it is clear that there is something wrong. If your symmetric matrix has non-real eigenvalues, you should notice it (if you were taught that it is impossible). More subtly, the eigenvalues you have computed may fail the trace test (sum of eigenvalues = sum of diagonal entries of the matrix); a smart student would make that check at the end as well.

In my view, submitting an answer with mistakes that fail obvious sanity checks deserves a more substantial deduction: maybe student A and student B both flipped a sign, but while student A got a perfectly plausible solution, student B really should have realized that. It's part of their job to check that the solution they find is reasonable. Especially in this age when computers do most of the work in practice, noticing when a computed solution is patently wrong is arguably more important than computing it in the first place.

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    This is an interesting point, and it leads me to the idea that a well designed exam could include some specific critical thinking problems of the type "What's wrong with this picture?" I'm actually starting to see that type of problem regularly in the New York State math curriculum ("EngageNY") for middle and high school, where the exam shows "Here's what Hani wrote in response to such-and-so problem; is his reasoning correct or not; explain your point of view." I like the idea of separating this out for specific problems that focus on this, because (a) it helps students see... Commented Oct 28, 2017 at 12:29
  • ... that critical thinking is important and should be practiced explicitly, and (b) it helps us deal in a positive way with test anxiety. Test anxiety can lead a smart, capable, skilled student to turn in a certain amount of nonsense, which the next day that same student would see through in a heartbeat. So when I look over an exam, I try to see the forest as well as the trees. If a reasonable proportion of the work turned in on the exam shows solid skills and reasoning, I'm basically happy, even if there is an element of kwatsch -- to quote my German spouse. Commented Oct 28, 2017 at 12:34
  • It reminds me of an exam where we had to calculate the height of some filling in a column. The correct calculation lead to a filling higher than the containing column, meaning that the setup is not possible. Apparently, this was not done by mistake by the teachers - but since all of our practice questions were plausible, everybody was looking for problems in our their calculations...
    – aqua
    Commented Jan 24, 2022 at 16:10
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First, notice that students will typically be able to tell which of their mistakes were silly and which more significant. So for silly mistakes you don't need to write feedback other than to circle where they went wrong, and indicate the number of points taken off (or granted).

For a more serious mistake you can write more detailed feedback. Sometimes feedback here isn't necessary, either, other than to write, "see solution."

Second, you can figure out what types of errors and what frequency of errors you want to bring the grade down from A to B, from A to C, etc., and decide how many points to remove based on that.

More importantly: what's a well written exam like? Example: I'm asking them to solve this eigenvalue-eigenvector problem. If the student has a vague idea what is being sought, I want him to get one point. If he sets things up well but doesn't know what to do next, he gets two points. If he follows through well but for whatever reason didn't quite get to the perfectly correct punch line, he'll get 3 points. Perfect, complete answer: 4 points. (This is just an example of the point distribution. You might end up with a different scoring structure.) In short, the scoring should be integral to the exam design from the beginning.

Most importantly: as a TA, you should be getting special guidance from the professor for grading a midterm exam.

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I've used a different color ink to distinguish feedback which does not contribute to the grade this time but which might in the future. One of the general problems with an educational system which uses grades punitively as "feedback" is that completely non-punitive feedback is almost always ignored.

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I believe I may be more severe in grading than some other instructors. I usually find so-called "silly" mistakes to be indicative of serious underlying issues, and I usually take off around 1/3 of the points for a whole problem. Here are two justifications:

One, this is part of the lesson: you simply can't afford "silly" mistakes. Especially in an engineering discipline, if you switch a sign or move a decimal point and fail to check for that, you'll wind up building a self-destructive and possibly life-threatening mechanism. See many stories of major projects (buildings, spacecraft, etc.) blowing up due to such errors.

Two, there is a question of efficient use of the instructor/grader's time. If there is a long problem and a "silly" mistake occurs near the first line, then it seems egregious to expect the grader to follow through a unique train of logic from faulty initial assumptions, in order to award partial credit. As an instructor, I tend to have a solution sheet and score up to the point where the logic goes off the rails, then stop. (If the other steps are absolutely self-evident, then more grade may be awarded, but this is not assured.) This way I can grade the tests for any one course section within an hour. If this labor is handed off to TA's who don't have authority to set their own protocols, then grading may wind up taking many hours, and this inefficiency may be hidden from the people with responsibility to manage the process.

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    See many stories of major projects (buildings, spacecraft, etc.) blowing up due to such errors. The life-threatening errors weren't the fault of arithmetic mistakes, which get made all the time. It's a subsequent chain of system failure that ultimately causes a catastrophe. Grade however you like, but please put this ridiculous straw man to rest. Commented Oct 28, 2017 at 4:15
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    But an exam's environment is very different from what engineers would be in. You're not required to rush as much. If you do, I put the blame on the company for these errors, rather than the individual who do these silly mistakes. Commented Oct 28, 2017 at 11:14
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    -1 1) I don't think this really answers the question 2) As others mentioned, in real-life there are procedures in place that should catch "silly" mistakes (reviews, tests, etc). If something goes wrong, those procedures should be blamed. 3) You not wanting to spend time grading an exam seems like a very poor reason to reduce points. I agree with Elizabeth, grade how you want (and a third off doesn't seem that unreasonable), but these seem like bad justifications.
    – tim
    Commented Oct 28, 2017 at 11:49
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    @ElizabethHenning: You make a good point, but it seems a bit exaggerated to me. I don't think review failures can be said to be the cause of the catastrophe to the exclusion of the initial error. If the initial error is not to be considered as a cause/fault, how could any subsequent error consisting of a failure to detect it then be considered a cause/fault.
    – Ben
    Commented Jan 24, 2022 at 2:21
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    I'm surprised at all the downvotes this answer is getting. It seems pretty irresponsible to me for academics to take the attitude that accuracy doesn't matter because review procedures should catch the errors.
    – Ben
    Commented Jan 24, 2022 at 2:23
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The appropriate feedback on an error depends on how obvious it is to the student and how severe it is to the answer. It is usual to penalise "silly mistakes" on examinations, both for diagnostic reasons and to create an incentive for students to improve their work. In cases where there is annotated feedback, the degree of gentleness/harshness is generally calibrated by considering what stage of the degree the student is in, and the corresponding expectations.

For "silly errors" (spelling/grammar errors, calculation mistakes, transposition errors, copying errors, etc.) it is usually sufficient just to circle the error (I use red pen) to alert the student to the problem. The error should be penalised appropriately, with consideration given to the totality of the question. In most cases like this the student will be able to self-diagnose the error without any textual statement explaining it; they will see what they did wrong and why they lost marks, so there is no real need for an explanation. For larger errors, or errors on more subtle aspects of the material, one might circle the error and then provide an accompanying textual explanation. In some cases you might also annotate the number of marks that were lost for the error.

As to your proposal to give students the "benefit of the doubt", and avoid penalisation, that sounds silly to me. Firstly, there is no "doubt" for which to give benefit --- they made an error and it should be penalised as such. Secondly, marking examinations necessarily entails a judgment of work under time-constrained conditions; the mark is supposed to reflect the quality of the student's work on the exam, not your premonition of how they might do in practice if they were working without a time constraint. And finally, because they are in their final year it is all the more important that they are becoming more accurate in their working and purging "silly errors" from their work. Hell, some lecturers would probably go further than just circling errors at this final stage of the degree, and start writing some more blunt messages that alert them to the fact that it is unacceptable to make "silly" calculation errors in the profession --- "Wrong; your bridge just fell down and killed three people."

As one final point, if you mark work regularly, it is useful to make up some custom rubber stamps for common feedback you want to give on your exams. When I mark assingments/exams I often penalise an initial error but then treat that error as a premise for the remainder of the question, so that even if the final answer is wrong, it doesn't get penalised a second time. I have a little stamp made up that says "Error carries through: No further mark deducted" for this purpose. I also sometimes point out an error that is too minor (or below the level of expectation for the course) to warrant a marking penalty. I have another stamp that says "Error: No mark deducted" for this purpose. This saves me getting carpel tunnel syndrome from writing the same things out over and over again on exams.

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