What to do about "grade grubbers?"
I usually say that my rule for regrades is like the NFL's rule for replay challenges: there must be "indisputable visual evidence" that the original grade was incorrect. For instance, scores were added incorrectly, or a correct answer was marked wrong.
If the score was a "judgment call" (for example, I deducted two points for some error and the student thinks it only deserved one point deducted), I won't change the grade, unless I did something really egregious.
I've seen the policy "We regrade the entire assignment and your grade could go up or down" but it makes me a little uncomfortable. It suggests an element of randomness or caprice in the grading, which I don't want to promote, especially for younger or weaker students who may already feel like their grade is random. I don't think students should feel like they have to roll the dice to get a genuine error corrected.
I never had issues like that but I've heard the following solution by a colleague: The students inspect their works in the presence of the professor of TA. If they have any complaints they have to write them down and hand them in. There is never any discussion going on, but only written complaints are accepted. All written complaint are treated carefully and there may be a next date for inspection and even a next round but this can only consider the thing which have been addressed in the first round. I.e. you can't "grub for a point at exercise 1" in the first round and then try to "grub for another point at exercise 2" in the second round. All complaints have to be submitted in the first round.
What I've heard is, that this dramatically reduces grade grubbing and especially makes "grade grubbers" think hard about their mistakes (which can be considered as a good side effect). As a matter of fact, it's much harder to articulate why you think that you deserve another point for some exercise if there is no clear "misgrading" if you have to do this in written form.
Edit some years later: Now I have some personal experience with this system. I did this at least six times and it works awesome. Were I am we have an official "grade grubbing date" called "Klausureinsicht" i.e. there is one date at which all students can have a look at their exams. I usually have 10 students in one room who can have a look at their exams at the same time. I answer all questions related to the content of the exam, but as soon as the grading is in question I hand out a paper and a pen and ask the student to write down their complaint. I even encourage writing complaints down. It further helps to cut off the discussion if I add that "I can't answer questions on the grading as I would have to look up how our general grading for this kind of solution/error is" (I actually have one but do not bring it to the Klausureinsicht). I collect all replies, keep them with the exams, check them and reply to the students via email on the spot or the next day. Answering the complaints is usually pretty easy. Most of the time I just write "The grading/deduction of points corresponds to our grading system." or "What you have written down does not show what you are complaining about/what you may intended to write and we can only grade what you have written." Only one time (out of several dozens) I had a student trying to continue grubbing.
Also, the fact that you can not change the grading system at your place is a big plus in my eyes, at least when it comes to grade grubbing. (One the other hand, you have to be more careful and work harder when producing the exam so that you can be pretty sure that the difficulty of the exam is ok.)
I have three primary methods for dealing with this behavior:
On the first day of class, I mention that I won't tolerate grade-grubbing. I say that I am firm and will treat everyone equally. What I don't say is that students are much more likely to see leniency if they showed a good effort in class, and are nice about asking me to review a grade.
I admit that I make grading mistakes (and so do the TAs), and we'll fix the problem if they occur.
I have a policy that if you ask for a re-grade, I'll look at the whole assignment and if the end grade is actually lower, that's what you'll get (I don't think I've ever lowered a grade this way).
I would like to avoid having to deal with the student who tries every which way to get the few points they need to move up a level.
You'll always have to deal with a couple. If you set the ground rules on the first day, you'll see fewer on average.