The video post from the other day about handling ungraded homework assignments went so well that I thought I’d let you all have another crack and designing my courses for me! This time, I have a question about really bad mistakes that can be made in a proof.

One correction to the video — the rubric I am developing for proof grading gives scores of 0, 2, 4, 6, 8, or 10. A “0” is a proof that simply isn’t handed in at all. And any proof that shows serious effort and a modicum of correctness will get at least a 4. I am reserving the grade of “2” for proofs that commit any of the “fatal errors” I describe (and solicit) in the video.

The only other errors I can think of offhand are leaving out the base case in a proof by induction or misusing a negative in a proof by contradiction. These mistakes demonstrate a lack of understanding of how the proof method works.

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.

Assuming what you’re trying to prove is my biggest pet peeve. As a TA I’ve mostly dealt with freshmen and sophomores in calculus classes, so I tend to be a bit more lenient than you’re planning to be, but the upperclass math majors should know better. (In calculus one sees this a lot; students often want to prove some algebraic identity, so they start from it and proceed until they deduce a triviality such as 0 = 0. Of course this is a good way to discover a proof, and in the case where all the steps are reversible it can easily be transformed into a proof, but it’s NOT a proof.)

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.

I could be wrong, since I’m not the math type geek, but this post seems to be seeking aggregious logic errors. Your first error seems to be a mathematical variant of a tautology–simply restating the theorem as a proof of it. The second is simply a a fallacy of composition.

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?

@virusdoc: If I understand you right, I think that would be a simple case of not using any logic at all rather than an egregious misuse of it. In some ways I would rather have students making spurious deductions or not showing their work than have students totally screw up the way logic works and think that’s OK.

@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?

Well, I’d think the first two (only giving examples and circular reasoning) can completely disqualify a proof from consideration. Essentially, the first two you mentioned did not prove anything at all. However, failing to account for the latter part of a biconditional statement isn’t so serious that it would warrant the minimum score (and yet serious enough that there should not be the equivalent of half the total or more points awarded for such a proof).

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.

I’m of mixed minds on some of these. In terms of introducing an unwarranted assumption into a proof, isn’t that something we do all the time in our effort to prove things? Then we see what we can take away…

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.

The only other errors I can think of offhand are leaving out the base case in a proof by induction or misusing a negative in a proof by contradiction. These mistakes demonstrate a lack of understanding of how the proof method works.

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.

Assuming what you’re trying to prove is my biggest pet peeve. As a TA I’ve mostly dealt with freshmen and sophomores in calculus classes, so I tend to be a bit more lenient than you’re planning to be, but the upperclass math majors should know better. (In calculus one sees this a lot; students often want to prove some algebraic identity, so they start from it and proceed until they deduce a triviality such as 0 = 0. Of course this is a good way to discover a proof, and in the case where all the steps are reversible it can easily be transformed into a proof, but it’s NOT a proof.)

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 indicatesthat 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.I could be wrong, since I’m not the math type geek, but this post seems to be seeking aggregious logic errors. Your first error seems to be a mathematical variant of a tautology–simply restating the theorem as a proof of it. The second is simply a a fallacy of composition.

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?

@virusdoc: If I understand you right, I think that would be a simple case of not using any logic at all rather than an egregious misuse of it. In some ways I would rather have students making spurious deductions or not showing their work than have students totally screw up the way logic works and think that’s OK.

@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?

I figured that the saying “here are a few examples” was more common than it is, because it’s something I always did. But I may be unusual.

Well, I’d think the first two (only giving examples and circular reasoning) can completely disqualify a proof from consideration. Essentially, the first two you mentioned did not prove anything at all. However, failing to account for the latter part of a biconditional statement isn’t so serious that it would warrant the minimum score (and yet serious enough that there should not be the equivalent of half the total or more points awarded for such a proof).

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.

I’m of mixed minds on some of these. In terms of introducing an unwarranted assumption into a proof, isn’t that something we do all the time in our effort to prove things? Then we see what we can take away…

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.