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It often happens that students answer (partly) a questions in an exam (lets assume this part is okay), but then add something completely off-topic or something very wrong in their answer (see examples below).

How would you deal with such an issue and why? Should one ignore the wrong things and only grade everything which is okay? Or should I bring to account that the additional part of the answer shows that he/she hasn't understood much?

Here are a few examples I have in mind:

  • Assume the question is split into two parts. The answer for the first part was okay, but then on the second part the student discloses big deficits in their knowledge (e.g., he/she uses the fact that $\mathbb{R}$ is compact; or he/she calculates $(A+B)^{-1}=A^{-1}+B^{-1}$, etc., i.e., wrong arguments where you see that not much was understood).
  • Assume someone answered a question right, but then add something is is which has nothing to do with the rest of the question (once when I was a TA, it happens that a question was just an exercise, the student copied the solution which was given in the exercise, but he did not finish when he arrived at the end and he also copied some part of the next exercise into his exam solution).
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    $\begingroup$ Assuming that a student managed to write down all mathematically sounding paragraphs (I wonder if it is possible to pull this off before the heat death of the universe), including the one which is a correct proof, would you give him all the points? $\endgroup$
    – dtldarek
    Mar 26, 2014 at 22:28
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    $\begingroup$ I remember with a wry smile being asked to prove something on an exam in a topology class in college that I didn't know how to prove. I therefore wrote down everything I could think of that might be relevant with the hope that something would stick and I would get some partial credit. I'll never forget the prof's written comment when I got the exam back: "This is true but completely irrelevant to the current problem." $\endgroup$
    – jtbrasel
    Mar 28, 2014 at 11:30

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Grading a random argument.

This is my interpretation of your question: The student in my above grading example achieved the correct answer, and therefore feels cheated at losing points for the extra (incorrect) simplification. Should we be sympathetic to the student's complaint and give full credit, excusing the extra work done?

My Best Argument Against Taking Off a Point

If you take off a point, the student learns his or her lesson, and you find a later quiz with the following answer:

enter image description here

Now you are in trouble. If you take off points for not completing the question, the student feels cheated again, and believes that you are just being arbitrary. They were playing it safe since they got in trouble simplifying before, and the question didn't say to simplify! This kind of thing can cause a student to "check out" of the class entirely.

My Best Argument for Giving Half Credit

If you do not take off a point, that student loves you and takes your class again next semester. They turn in a quiz with the following answer.

Lesson not learned.

You would love to curse that student's college algebra instructor for not emphasizing taking care with arithmetic, but that college algebra instructor is you. So you curse yourself.

In conclusion,

I've felt both of these bad outcomes. The second bad outcome feels much worse. I'd stick with always taking off points for incorrect things. The student's future instructors will thank you -- and the student's future instructor might be you!

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    $\begingroup$ With a little more care in stating the question these issues can be avoided. For example, in the arithmetic examples, have as a default policy that all answers are to be expressed as fractions (improper, if need be) in reduced form, unless an integer, in which case they are to be written in standard integer form. Of course, don't spring this on them out of the blue. From prior discussion in class they should be familiar with what you want and with the formal "legal statement" on your tests that enforce what you want. To me, simplify completely and calculate just doesn't do it. $\endgroup$
    – Dave L Renfro
    Apr 8, 2014 at 18:15
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I would absolutely take points off in those situations. If I ask "what's 2+2?" and one student responds "4" and the other responds "4, because $\pi$ is rational", the first answer is definitely better than the second, so it should be scored higher.

On a more idealistic level, my job as a grader is to determine the student's understanding of the material as precisely as I can. If the student has revealed that he has a big gap in his knowledge, or that he's just memorizing someone else's words without the slightest understanding of what's being said, then his score should be lowered. One could argue that the student who answered "4" might have also thought that this was implied by the rationality of $\pi$, yet simply didn't write it, but even the fact that he had the good sense not to write this nonsense means he's a little suspicious of it too, and that's worth something.

By the way, in the second case you listed, where a student bleeds memorized verbatim solutions together, I would be inclined to take off more than the first, since there's a hint of dishonesty there (or at least bad faith). I don't think demanding an honest effort towards a genuine understanding the material is unreasonable. Conversely, not enforcing this with grading tells the student it's okay to mindlessly memorize everything, even if the teacher doesn't like it.

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    $\begingroup$ And what would you do with a student who responds "4, because $\pi$ is irrational? $\endgroup$
    – mweiss
    Apr 8, 2014 at 17:13
  • $\begingroup$ Part of a "good answer" is to stick to the relevant facts. So a right answer adorned with lots of true, but irrelevant, facts gets less credit. If combined with blatant nonsense, I might just give zero points. $\endgroup$
    – vonbrand
    Jun 16, 2014 at 12:07
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When I first started teaching and I was grading homework or exam solutions, I would often write something like,

I don't understand your argument here.

at a place where the student's argument was incoherent and made no sense (and they would get a correspondingly low mark). I think I probably wrote that phrase out of a desire to be polite or to protect the student's feelings. I had felt that it would be too harsh to tell a student that their argument was totally wrong.

But I no longer use that phrase at all. I stopped using it when a student came into my office and tried to "explain" his totally incorrect argument to me. Of course, he hadn't gotten my subtle drift, namely, that his argument MADE NO SENSE!

So now, when a student's argument makes no sense, I just write:

Your argument makes no sense.

And perhaps explain why it makes no sense. Although this kind of feedback is blunter, nevertheless the student learns more from this honest assessment of the situation. They need the accurate feedback that their argument was totally off-base.

So don't mince words, and tell your students that their argument is totally wrong, when indeed it is.

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    $\begingroup$ Thanks for your advise. Can you maybe explain a bit how you would grade then the homework or exam if there is something which makes sense, but is followed by something nothing making no sense? $\endgroup$
    – Markus Klein
    Mar 26, 2014 at 17:08
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    $\begingroup$ My point is that it is important to identify nonsense as such, so I would mark it clearly as nonsense. And the grade will not be a full grade, even if there also a complete correct solution there, although in that case I am a little more tolerant. $\endgroup$
    – JDH
    Mar 26, 2014 at 18:06
  • $\begingroup$ I think it is important, especially for beginning instructors, to understand the difference between "I don't understand your argument" and "your argument makes no sense". It is genuinely possible, even in a low-level class, for a student to give a correct argument organised in such a bewildering way that it seems at first to be nonsense, although I can actually untangle it after much effort. This frequently happens to me in Business Calculus, even after years and years of teaching it. Students think in weird ways. $\endgroup$
    – LSpice
    Mar 23, 2020 at 17:12
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I think an important consideration in deciding how to grade something is to ask yourself whether the answer even makes sense. I mean this in the more formal use of the word sense making in Mathematics Education; a few relevant remarks can be found in my earlier MESE response here.

As far as implications for your question: The decision as to how one handles a very wrong answer depends on numerous factors. At least one of them is the setting (i.e., the class and the student).

The way I would respond to a primary school student who makes an error shortly after learning about the multiplication of fractions is very different from how I would mark the work of an undergraduate engineering major who made an analogous error in a course on mathematical modeling.

To illustrate this point better, here is an excerpt from COMAP's Mathematical Modeling Handbook, taken out of Henry Pollak's introduction "What Is Mathematical Modeling?"

enter image description here

My own preference is for Side 2: when one can reasonably claim that a solution "shows no judgment at all," it suggests that something has gone quite awry with regard to the student's learning. (Though, if such mistakes are widespread among members of a class, then perhaps the true error lies in the teacher's instruction...)

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  • $\begingroup$ I am still bitter about a chemistry problem where I was asked how many atoms participated in a chemical reaction, or something of that sort, applied the correct formula, and lost points because I recorded that there were 3.4 atoms involved. Well, if you want integer answers, then include a rounding step in your modelling formula! $\endgroup$
    – LSpice
    Mar 23, 2020 at 17:14
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    $\begingroup$ @LSpice it is no doubt interesting which education(al) experiences stay with us from when we were students. I think about this not infrequently as a teacher, and wonder what students with whom I've interacted will remember n years later. $\endgroup$
    – Benjamin Dickman
    Mar 23, 2020 at 17:17
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    $\begingroup$ I did have an experience once the other way, though not in mathematics—I had an English teacher who absolutely persecuted me, for whom I could do nothing right, and who would always seek some excuse to take off points on an assignment. When I looked back at some old work 15 years later, I saw that what I had interpreted at the time as persecution was a teacher writing helpful and constructive comments, but straight-A me couldn't see past "I didn't get a perfect score." I felt so bad about how I'd misjudged her that I wrote her a letter about it …. $\endgroup$
    – LSpice
    Mar 23, 2020 at 17:20
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I usually give credits for everything that is right, ignoring what ist wrong, iff things are not contradictory (I wouldn't give credits for "answer is right or wrong"). If things are contradictory (e.g. two solutions given), I mark the first one only. This is told to my students.

I am not happy with that as I sometimes would rather like to downgrade people writing useless stuff. However in Germany (where you also work) some courts decided that in multiple-choice, additional information may not lead to less points (see here, line 72). In some sense, students have a right for guessing, as they practiced in school. You can't prevent it anyway.

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    $\begingroup$ On the other hand, there is an annoying tendency to interprete German court decisions out of context (though this does not necessarily need to be the case here). Do you have a link to said decisions? $\endgroup$
    – Wrzlprmft
    Mar 26, 2014 at 17:28
  • $\begingroup$ That's absoluteley right. I have updated my answer. $\endgroup$
    – Anschewski
    Mar 26, 2014 at 17:46
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    $\begingroup$ "Answer is right or wrong" isn't contradictory. Maybe you want to assign no credit for contradictions or for tautologies? $\endgroup$
    – LSpice
    Mar 23, 2020 at 17:16

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