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In a quiz, there was a question asking students to show something doesn’t exist. A lot of them gave proofs by contradiction.

Initially, I designed the marking scheme so that an incomplete proof by contradiction would lead to 0 marks, because I thought that exhausting all cases is very important in a proof by contradiction. Otherwise, there may not be any contradiction at all (another case is valid). Some students may impose an (perhaps, obviously contradictory) assumption and claim that their proofs are complete.

However, I found that a lot of students missed some cases, or imposed extra assumptions which were neither given nor proved. Some of them were certainly with loss of generality. I gave them 0 marks for that question. I’m not sure whether marking in this way is too harsh.

Should students be given partial marks if they gave an incomplete proof by contradiction?

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    $\begingroup$ How many marks would a student get for a perfect answer? 1? 10? 100? $\endgroup$
    – JRN
    Commented Jul 31, 2018 at 9:52
  • $\begingroup$ That’s just one question, so something like 5 out of 60. $\endgroup$
    – tonychow0929
    Commented Jul 31, 2018 at 11:29
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    $\begingroup$ Primarily opinion-based .... unless someone can find some actual research done on this. $\endgroup$
    – Gerald Edgar
    Commented Jul 31, 2018 at 12:22
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    $\begingroup$ If this is primarily a proof-writing course, I would say that 0 points for an incorrect proof is fine. However, you should allow them to resubmit with corrections, possibly for full credit and possibly several times. Think of it like being a referee for a journal. $\endgroup$
    – Adam
    Commented Aug 2, 2018 at 14:11

2 Answers 2

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There's no abstract reason that an imperfect proof by contradiction should categorically fail to get credit.

A proof should generally get partial credit based on how much knowledge of the relevant material it demonstrates. An incomplete proof by contradiction could correctly get the main idea but omit some of the technical material needed to make the argument work; I would give that close to full credit. An incomplete proof by contradiction might lead to an argument that makes real use of material from the course, but is ultimately a dead end; I would give that some credit for demonstrating knowledge of the material. An incomplete proof by contradiction could also end up accidentally trivializing the problem by imposing a contradictory assumption, missing the point entirely; I wouldn't give credit for that.

I'd recommending avoiding the question "how far is this from a correct proof", which I don't think is a useful way to grade. I prefer "how well does this demonstrate mastery of the material".

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    $\begingroup$ +1 for last paragraph. Getting the "correct answer" is at best evidence of understanding, and it's understanding (or skill in doing something, etc.) that you're probably supposed to be grading. $\endgroup$
    – Dave L Renfro
    Commented Jul 31, 2018 at 19:47
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I agree completely with Henry, but let me try to mention some specific, practical advice that you or others may find helpful.

I strongly encourage you to adopt a more wholistic manner of marking/grading, based on the student's mastery of the material.

I use the following method of scoring student work in mathematics, where all problem solutions are graded on a five-point scale. It has worked well for me for about twenty years, and I would recommend it to anyone.

5 = solution is basically perfect, completely acceptable

4 = solution is marred by a few minor errors

3 = the solution has displayed basically the right idea or approach to the problem

2 = the solution has displayed some understanding of relevant ideas

1 = the solution has displayed some understanding of some ideas

0 = the solution has no merit

I find it easy to use this rubric when evaluating student work. I just keep the phrases in mind, and assign the corresponding score. Thus, the work is evaluated as a whole, on the basis of whether the student has demonstrated mastery over the topic.

My quizzes usually have just one problem, graded out of five. The final exam will have ten or fiften problems, each graded out of five. I often give extra credit for participation on math.stackexchange or whatever, and these are also worth up to five points.

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    $\begingroup$ I use a similar kind of rubric, but I arrange the scores differently (so that they better align with a standard 90-80-70% breakdown of letter grades that is common in US institutions): 5 points for a near perfect answer; 4.5 for an answer that is basically correct, but which might suffer from minor errors (of the kind that I often make a the board); 4 points for significant progress (basically, they have the right idea, but couldn't get to the end), 3 points getting started (this might be thought of as correct application of a relevant idea); (con't below) $\endgroup$
    – Xander Henderson ♦
    Commented Aug 1, 2018 at 21:46
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    $\begingroup$ (con't from above) 2 points for displaying some understanding of at least one relevant idea; 0 points for a blank answer, irrelevant work, or doodles. Basically, I really like your rubric, though I tend to adjust the point-values a little in order to give a grading scale that ultimately looks a little more like what my undergrads are familiar with from high school. $\endgroup$
    – Xander Henderson ♦
    Commented Aug 1, 2018 at 21:47
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    $\begingroup$ I've been using this since last year and it's been helpful. One question: I'm unclear on the difference of the 2-point "relevant ideas" vs. 1-point "some ideas" cases. Is the latter meant to include "some irrelevant ideas"? Something else? Any clarification on your thought there would be appreciated. $\endgroup$
    – Daniel R. Collins
    Commented Nov 6, 2019 at 5:43
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    $\begingroup$ I have used the score 2 when the student understood something relevant, and 1 when the student wrote something correct, even if it was not relevant for the answer. This is better, in my opinion, than a 0, which is for a student that writes nonsense or nothing. $\endgroup$
    – JDH
    Commented Nov 6, 2019 at 21:05
  • $\begingroup$ Just saw your reply, thank you! $\endgroup$
    – Daniel R. Collins
    Commented Dec 30, 2019 at 18:46

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