As someone mentioned, a lot depends on the intent of the student. Perhaps they can’t prove (ab)^{-1} = b^{-1}a^{-1} in the general case, but they figured out a way for hte Abelian case. If, however, they don’t realize they haven’t proven the real deal, then I would wack them.

To me, what I really hate is when students just sort of tail off and keep going until they stop. There isn’t really a conclusion and they don’t even know if they’ve done the proof correctly (even if they did so). A clear focus on what they need to establish and then realizing when they have done so is huge. If you are going with a harsh rubrik, I might be tempted to deduct several points for a “correct” proof that doesn’t realize it’s correct.

I think if you’re looking for a good grading rubric for proofs, you can find some inspiration in competitive events such as the USAMO or the Putnam Competition.

@isabel: Yes, if a student said something like “Here are a few examples; I can’t figure out the rest of the proof but here’s what I think I’m seeing” or in some way acknowledges that examples do not prove theorems, then that wouldn’t get full credit, but it’s not a complete screw-up. But honestly I almost never see that — I see much more of students simply giving the examples and then saying something like “from the examples, it’s obvious that the statement always works”. As my old linear algebra prof wrote on more than a few proofs of mine, if it’s so obvious then you ought to be able to prove it easily.

Everyone: I threw in the biconditional statement error kind of at the last minute. It’s not as bad, perhaps, as the others, but I have just seen it perpetuated so much that I feel like I have to put a stop to it.

How about another fatal error: Introducing an unwarranted assumption that trivializes the problem? For example, if you’re proving that in group theory and you start by assuming the group is abelian. Is that a complete screwup of the problem, or a serious but lesser error?

If I’m correct in making this generalization, then in my field one of the worst logic errors you can make is a non sequitur–assuming two concepts are necessarily deductively linked when they are in fact not. Is there a math equivalent?

Your other two proposed “fatal mistakes” seem perhaps a bit harsh to me, though; in those cases a lot depends on the intent of the student. If the student seems to legitimately believe that giving a few examples is enough to prove a general statement, or proving one direction of a biconditional suffices, then I’d call that a fatal error. But it may be that they know what a correct proof would entail and just can’t do it, which seems less serious and might deseve more credit. Perhaps you should be more lenient in cases where the student explicitly indicates that they know their work is incomplete?

A lot of this boils down to the students knowing what a proof is, which may be the most important thing they learn in their mathematical education.

Having said that, I don’t think it’s fair to give a 1 out of 10 to someone who proves only one direction of a bidirectional theorem, if they’ve done it correctly. A score of 5/10 is a bad score that also emphasizes that they only got it half-right.