The right way to do it is "if you don't agree with your grade, give me your paper and I will look at it more carefully. Of course I'm not just going to look at the question you're contending, but all of them. Your grade may go up, but it also may go down. Do you wish to wager?"
A very good high school math teacher of mine did this.
I think he did not get very many takers; the students were never confident that they would not lose more undeserved marks in the other question to more than offset any gain.
Look, I'm a student. I'm supposed to make mistakes. It's natural for me to be wrong. I should be wrong many times now so that I can learn and not make those mistakes in the future.
But you're a teacher. You are supposed to correct mistakes. You are supposed not to make mistakes, or at the very least to be honest about those mistakes. You're not supposed to try to intimidate me so that I stop questioning you; you're supposed to encourage critical thinking and healthy scepticism.
In short: if a teacher hears that he made a mistake when grading it's his duty to check it again, carefully; and if it happens to be true he should be honest about it and admit it instead of threatening that he'll lower the grade even further. What your teacher said could be read as: "Ok, so I may have fucked up, but that doesn't matter, because I'm so much smarter than you and I can fuck you up in any way I'd like; do you want me to show you?" Your teacher probably didn't mean it that way, but I met a fair share of teachers who did; it would be best to stay away from such ambiguous phrasings altogether.
The problem with being too open to regarded is that the teacher is inundated with frivolous requests. Grading is very time consuming, and regrading a test because a student simply isn't happy with the grade they earned is a horrible waste of time.
When I was teaching my policy was to reconsider anything a student felt was a mistake in grading, as long as it was accompanied by a written memo explaining precisely what the mistake was. I found this to be a good way to ensure I was getting legitimate regrade requests, since students would only put in the effort to write the memo if they really felt something was wrong.
Basically, the idea is to discourage essentially spamming regrade requests as a zero cost effort on the part of the student that occasionally pays off.
That the teacher has made one mistake implies the teacher may have made more than one mistake, and there's some probability that those errors are in your favor. And if its really correctness you're worried about, rather than just going for a better grade, then you should be okay with a full on review. But you should be confident that there are errors, rather than just winging it for the occasional payoff.
There is a deeper lesson and that is that correctness is what you should be worried about. If you think that the paper was not graded correctly, then let it be graded again. You don't get to cherry pick instances of the truth that benefit you.
What's so great about this strategy? When a teacher does this to me I assume he/she is more interested in doing his research instead of leading a class.
I have actually mostly gotten this strategy from very dedicated teachers. It's meant to address a strategy clever and grade-seeking students work out very quickly: without something like this, constantly asking for regrades has zero cost, and occasionally benefit to you.
This moves regrade requests more toward "No, I checked over my exam again and I'm sure this is a mistake..." rather than "Maybe I'll pick up another point or two..."
This was what we taught to say when I was a tutor for a university. The point was the student could not come and just point out that something was wrong with the marking and they deserve more points. If they thought the grade was not fair it could be remarked, but the whole thing would be remarked and a totally new grade assigned. You might come out better or worse for it.
The right way to do it is "if you don't agree with your grade, give me your paper and I will look at it more carefully. Of course I'm not just going to look at the question you're contending, but all of them. Your grade may go up, but it also may go down. Do you wish to wager?"
A very good high school math teacher of mine did this.
I think he did not get very many takers; the students were never confident that they would not lose more undeserved marks in the other question to more than offset any gain.