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I am marking an exam for a course I tutor. One student has given two answers for a particular question in the exam – one answer is fully correct, while the other is completely wrong.

Each answer has full working and explanation of how the answers were derived. It appears that the student wasn't sure which method they should use, and so hedged their bets by attempting both and hoping one of them was right. (The student presented their answer as "it is either X or Y", and in later questions gave parallel answers depending on whether X or Y was correct - I don't believe it was reworking that they forgot to cross out.)

How should I mark this question?

  • Full marks since they produced a right answer?
  • Partial marks because they are "half-right"?
  • Or no marks because they haven't demonstrated full understanding of the problem?
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    Regardless which way you go, you may consider to advise the instructor to make clear to students for future exams that, when multiple answers are given, only the first (or last) answer will be taken as the one that will be graded. Mar 16 at 15:47
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    Was the question - in the opinion of several students, not in yours - anyway ambiguous ? If so, then you will have to give full marks for the correct answer.
    – Trunk
    Mar 16 at 16:26
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    Is it clear to the student that they gave 2 different final answers? Did they not make any comments about getting different answers?
    – Kimball
    Mar 16 at 16:42
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    The edit helps understand the situation quite a bit. Exams that have this kind of chain of dependencies are problematic because one mistake throws off everything later. The student has done the right thing in order to earn credit on the corollary questions.
    – Ben Voigt
    Mar 17 at 17:42

15 Answers 15

98

Sounds like the student has demonstrated partial mastery of the material: they were unable to select the correct method but were able to correctly apply the method once selected. So, absent other information, partial credit seems appropriate. The amount of partial credit to give will depend on the relative difficulty and importance of method selection vs. method implementation.

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    With this approach, I would also consider whether the incorrect method they used for their alternative answer was nevertheless executed correctly.
    – towr
    Mar 16 at 15:40
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    This is what I ended up doing for this case, as they did have all the correct working for the right method and knowledge of how to use it, but clearly not a full understanding for why that method is chosen (and not the other one). I understand that partial marks won't be the best solution in all situations however.
    – Jedf
    Mar 17 at 4:30
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    I think this is fine, but it raises questions about 'show your work' and 'partial credit'. If the purpose is to mitigate the consequences of making an error in executing the solution by giving credit for understanding how the problem can be solved, then it would not necessarily be applicable if the student does not know how to solve it - and not being able to pick the correct solution, even after having executed two candidates, is a case of not knowing how to do so (this assumes there was no plausible way for a knowledgable student to see ambiguity in the question.)
    – sdenham
    Mar 17 at 14:32
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    Wondering how many points they would have gotten if they had only indicated the method to be used without working the problem. Sounds to me like that is how many points should be lost.
    – Mike Wise
    Mar 17 at 15:44
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The wrong thing to do is give full credits. Imagine a much simpler question that can be answered with yes or no, with a student providing both those answers. In such a case, I would not award any points.

Answering a question correctly does include choosing a correct way to do it. The student clearly failed at that. It is not fair to others who chose one way to solve the problem they deemed to be the right one (and ended up chosing the wrong one). If the student had chosen the wrong answer out of the two, they would have been awarded no points. To not be able to chose the right answer shows an inability to solve the problem at hand, so it would be completely OK to not give any points.

Otherwise it is an incentive to just write down any possible answer that comes to mind in the future (for questions where the student is unsure about), hoping that because the right answer might be somewhere among them, they will get at least some points.

But no matter what you do, I think it is a good idea to present your suggested way of dealing with it to your professor as Roland suggested, asking if they agree.

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    Your yes/no example is irrelevant since that’s very different than the situation OP is dealing with. See my answer for an alternative analysis.
    – Dan Romik
    Mar 16 at 15:01
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    @DanRomik It is relevant. Each question has some maximum n of bits of information that it can reveal about the student's mastery. Once a student gives two answers, they are removing one bit of information (one bit is used up deciding which answer to take). Therefore, if they got everything else right, their response reveals n-1 bits of information. In the extreme, for a true/false question, n=1, so the students is revealing no information, and should get no points. While a true/false question is different from OP's situation, it is not irrelevant, as it illustrates the wider point. Mar 17 at 0:11
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    @Acccumulation to put it another way, the yes/no example is an appeal to extremes fallacy.
    – Dan Romik
    Mar 17 at 2:50
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    (to both Accumulation and DanRomik) Appeal to extreme applies if this answer is trying to say that the OP case should have the same treatment as the extreme case. Giving an extreme is valid if this answer is trying to show that [in one end we have zero mark, and on another end slightly reduced marks (for example). Therefore, something that's in between should be given marks in between as well.] However, this answer doesn't seem to do that, even though it starts with a hedge on "the wrong thing to do is give full points". So I would agree on Dan Romik that the extreme is used fallaciously here.
    – justhalf
    Mar 17 at 9:49
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    "Otherwise it is an incentive to just write down any possible answer that comes to mind in the future" -- on that note, I'd argue that an upper bound on the grade in this case should be 1/number of answers, assuming of course one of the answers is correct. The student may keep adding answers, but then their grade will be correspondingly diluted.
    – swineone
    Mar 18 at 14:09
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I was in this situation before, on the student side. I had time left after finishing the exam and I re-worked a problem that I had trouble with. In my excitement over solving it the second time around, I forgot to indicate which solution was my official answer. My professor called me over after the next class session and said something along the lines of "it looks like there's two different sets of work here. Was that intentional? Or am I reading this completely wrong?" After a brief facepalm moment, I explained what happened and indicated which version was intended to be my official answer. He didn't dock me any points for the confusion.

The professor was certainly under no obligation to do any of that, but as the student I really appreciated his effort to ensure my score was a reflection of my understanding of the material and not of my ability to draw a big "X" over old work. I suspect that he wouldn't have been as generous if I wasn't able to instantly indicate which answer to use (someone trying to game the system wouldn't have an answer for that question and would likely pause to think before responding).

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    Someone trying to game the system could often easily just ask someone which answer was right after the exam. Someone not trying to game the system could very well pause to think before responding - I, for example, often pause to think before responding to even the most basic questions. It's a flawed way to figure out intent. Your score is a reflection of your understanding, to the extent that you're able to demonstrate said understanding. Failing to properly indicate what your answer is is comparable to making a symbol that could be interpreted as either T or F to a true/false question.
    – NotThatGuy
    Mar 17 at 7:50
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    @NotThatGuy Possibly, but you can read many people's body language and determine if they're being straight with you. I, for one, have no poker face, and the professor in question was rather good at reading and understanding people.
    – bta
    Mar 17 at 14:51
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    But, assuming the grade carries some significance, the system has to be fair to all students. It isn't fair if the outcome depends on the grader's ability to correctly read a person's body language. Anyone who has played poker knows that you can't correctly read 100% of people 100% of the time.
    – JBentley
    Mar 19 at 0:09
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    It goes both ways. You could similarly say it's equally unfair to penalize a student who got the question right but forgot to cross out an answer with the same severity as another student who has no idea how to do the problem at all. Teaching/grading always requires you to make judgement calls at some level, you just have to try and be as fair as you reasonably can.
    – bta
    Mar 19 at 0:18
  • @bta Actually, I wouldn't take that argument at all. A truly fair system has a clear, unambiguous, and pre-published set of rules for determining the grade. In a formal exam setup (e.g. one which counts towards a final grade at a university or a formal qualifcation), failing to follow the exam instructions (e.g.. answering the correct questions in the correct way) does result in no marks being awarded. It might seem harsh to someone who fails to read the instructions properly or makes a mistake, but everyone knows they have to conform to the same standards as everyone else.
    – JBentley
    Mar 19 at 16:06
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Edit: OP edited the question to clarify the situation, and with the edit it looks like the scenario I’m proposing below is not what happened. So my answer is no longer applicable. I’m leaving the answer up rather than deleting it, since it may be applicable in other cases that are of interest to future readers of the thread.


It appears that the student wasn't sure which method they should use, and so hedged their bets by attempting both and hoping one of them was right.

This is a possible explanation for what happened, but not the only one. The student might have written the wrong answer first, and proceeded to solve other questions in the exam. Later in the exam they might have suddenly understood what the correct solution is and written it down. In their haste (and because of nervousness, time pressure, sloppiness, or some combination of those things) they forgot to cross out the previous, incorrect solution.

If you think such an interpretation is not unlikely, giving full or almost full points is something to consider. Think of it as a case when you’ll be applying the principle of charity.

Note also that I’m only suggesting this course of action because the correct answer includes a full explanation of its reasoning. Awarding full points to someone who wrote multiple answers without at least one of them having a full and correct reasoning would be wrong, among other reasons since it would give an incentive to students to try to game your exams by trying to make multiple stabs at guessing the answer without having any understanding of the material you’re asking about.

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    Mostly agree with this, but strongly think it should be “almost full” and not “full.” Mar 16 at 17:15
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    The student could also have given the right answer first and then lost confidence in their answer and given the wrong one. I don;t think the OP makes it clear which was first or second. Mar 16 at 23:09
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    The principle of charity does not apply here. The principle of charity is about assuming the other person is right until proven otherwise. In the case of an exam, determining whether the student knows the material is the whole point, and the burden is on the students to establish that they do. Mar 17 at 0:16
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    @Acccumulation the principle of charity is about giving people the benefit of the doubt in situations involving ambiguity. There is no obligation to apply it in any given situation; rather, you can choose to apply it or not as you prefer. If you are a professor and prefer to be a harsh grader and not give students the points until they’ve removed all trace of doubt that they may not understand the material, that is your right. I suspect however your students will not appreciate this pedagogical approach. In practice it’s pretty rare that test answers don’t contain any ambiguity of this type.
    – Dan Romik
    Mar 17 at 3:43
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    @StephenG-HelpUkraine sure, as I said in my answer there are several possible interpretations for what happened, including some which reflect less well on the student. OP can choose to grade according to the most charitable interpretation, or according to the least charitable one, or anywhere in between those extremes.
    – Dan Romik
    Mar 17 at 3:50
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Not an answer, but too long for a comment.

In the best of circumstances the student will provide an explicit meta answer too, something like

I can solve this problem two different ways. Here they are. Since the conclusions disagree, they can't both be right. I am pretty sure the first one is, because ... but I can't quite see where I might have gone wrong in the second.

I would award extra points for an answer like that.

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    Strong disagree with awarding extra points, and weak disagree on even awarding full points. The comment you have as example shows that the student was not able to determine which method is the appropriate one, hence lacking important knowledge. This should be reflected in a deduction of points. The best case scenario is when the student writes a correct answer, then proceeds to comment "I'm fully confident the above is correct, but I believe I found an alternative solution, see below" and proceeds with the wrong answer. In this scenario, the wrong solution should be dismissed as a comment.
    – Neinstein
    Mar 17 at 7:31
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    @Neinstein We have to agree to disagree. Thinking about how you think is a crucial part of learning to solve new problems. Evidence of that is worth more (to me as a teacher) than correct answers to the questions I ask on an exam. Mar 17 at 13:07
  • @Neinstein If it's a course covering the theory of alligator wrestling, then exam points should be awarded based on demonstrated mastery of the ins and outs of fighting a gator. Other coursework (graded assignments, quizzes, term projects...) can be used to reward creative thinking about what it means to wrestle an alligator.
    – Z4-tier
    Mar 18 at 2:55
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As a student, sometimes I notice at the end of the exam that I still have time and try to redo a task that I believe I have done wrong. Or do it in a different way to be certain. Of course it is usually expected that students clearly mark which answer is supposed to be graded, but I do not think it is fair to, as other answers suggest, assume by default the stance of "this student wanted to cheat the system by providing multiple answers".

If the goal of the question was to see whether a student is able to find the correct answer, then the student managed this and deserves full points - perhaps minus a few as a gesture that they should in the future clearly mark which answer is correct.

If the goal of the question was to see whether the student is able to choose the correct approach among of few, and from their solutions it is not obvious (to them) which one must have been the correct one, it is not as clear and I would understand deduction of points. However, keep in mind that you generally don't want to punish students for writing down more - you want them to demonstrate what they know, after all. And in my opinion having one correct answer and one wrong answer is still better than having only a wrong answer.

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There is another option: Always grade the first answer (excluding those crossed out) and ignore any of the subsequent ones. Ideally it should be announced as a consistent policy beforehand. It can sometimes be hard on the student, but it really drives home the point of giving just one answer. It also avoids quite a bit of discussion.

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    Imo fully discarding submitted work is not ok if it was not clearly communicated beforehand. I'm reading your "Ideally" as disagreeing with that. If you do agree with me, you might want to reword that to make it clearer :)
    – lucidbrot
    Mar 16 at 13:58
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    It might make more sense to always grade the last answer and ignore any of the preceding ones. It's plausible that the student wrote an answer, then realized it was incorrect, wrote a second answer, and forgot or otherwise failed to erase the first answer. On the other hand, it's unlikely that a student would write an answer and then intentionally write a worse answer that they don't want graded. Mar 16 at 20:10
  • @TannerSwett They might write a worse answer later if they did not understand their first answer was better, which I'd be reluctant to reward. This is why the order of the answers is significant IMO. Mar 17 at 1:55
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    @lucidbrot Giving multiple answers when only one was expected/requested is also "not ok", and doing that should remove any reasonable expectation from the student to receive more than the minimum score between the two answers (although how exactly to mark it would be left up to the grader).
    – NotThatGuy
    Mar 17 at 7:47
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    @TannerSwett Always grading the first instead of the last answer is more of a pragmatic decision, otherwise you would always have to check every exam for additional answers to the same question on some later page, or you would end up grading the first answer and then crossing that out again, which would not help with discussions. I tend to tell students before the exam that I will always pick the worse one, which certainly helps scaring them into handing in only one and properly crossing out all other attempts.
    – mlk
    Mar 17 at 9:57
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Such things need to be clarified beforehand. Now you need to read the grading rules in detail, apply what is applicable, and probably do so in favor of the student.

For the next time, you should add a rule about this. A common rule is:

When more than one answer is given, the answers are not graded. Please make clear, e.g., by striking out the other answer, which answer should be graded."

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In this situation the only question for me would be to see if the student is being sneaky by putting in both answers or if they genuinely are unsure which one is correct. This can be judged by talking to the student usually. In the former case marks must be deducted (50% maybe) because the grading is supposed to reflect what the student has learned and they clearly haven't.

In the later case I'd give full marks because trying things shouldn't be penalized.

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I would not give many points to the student. This incentivizes guessing and just putting multiple answers.
Extreme case: Calculate 2+3
Student answers: 5, 6
Now, if the student gets many points for that, they might start guessing entirely and provide ansers like 1,2,3,4,5,6,7,8,9,10,11,12 and hope that whoever grades the exam will pick the correct answer.

On the other hand you could also argue that the student answering 5, 6 has executed two different approaches of what to do with two numbers: addition and multiplication. This shows that the student masters these two techniques, but it also shows that the student did not understand the fundamental part: what is the question about?

So in your case you could give some points for successfully executing two different approaches, but I would subtract a substantial amount of points for not even understanding the question correctly.

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    giving even one point for "you answered correctly to a question you did not understand" is like giving a Nobel Prize to the lucky monkey that by punching randomly a keyboard will produce a masterpiece. Nobel Prize for Literature, clearly, the comment does not apply to the Nobel Prize for Peace.
    – EarlGrey
    Mar 17 at 17:24
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Between the bounds

If the student was clearly unsure, then:

  • Lower bound is whatever they'd get for selecting the incorrect method but otherwise showing correct workings, etc: ideally they should get more than that, unless there's a very strict existing rule for "only first answer counts".
  • Upper bound is whatever they'd have got for using only the correct method; they should get less than that.
  • Either way, subsequent questions' grades shouldn't be affected by using the incorrect values from this question: any penalty should be applied only once.

Rubric suggestion

S - any score for "showing workings" or other stylistic points. A base value that is given even with an incorrect selection of methodology, so long as they show good workings for some methodology.

C - any score for giving the correct answer. S + C should be the total points for this question.

N - number of guesses.

P - Problem space untouched by their guesses. Assume a value of 1 for now (very large problem space).

The rubric would then be S + P(C/N).

In the degenerate case where there are no style points to be had, P is 1, and they made only two guesses, you'd give 50%. They're not "entirely wrong", even if the wrong answer was the first; but they're not "entirely right" either.

P is, more correctly, the portion of incorrect problem space left untouched by their guesses. For large problem-spaces like "what method should you use for this question?" you can treat it as 1. But in a multiple choice question of A/B/C/D, with A correct, the incorrect problem-space is only three items long, B/C/D. So if they guess "A or B", this would use a third of the incorrect problem-space, giving P=2/3, for a total P(C/N) score of only 33%. And if they guess both possible answers in a true/false question, then P=0, so they get 0%.

Alternative possibility

It is possible, however, that by writing "it is either X or Y", the student was instead asserting "the question does not give sufficient information to select between these two approaches, both of which could be considered valid depending how you read the question."

If that might be a valid opinion about the question (if the question could be read so that the incorrect case might apply) then you'd most likely need to award full marks, plus extra credit for being the only one to spot that.

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I will first look the matter from the Student's side and hence I will give full credit for the correct answer. Regarding the wrong answer, I will be still happy for the wrong answer because the student is developing his/her analytic power to give a different argument, in other word, the student is trying to be creative. He might be wrong somewhere which led him/her to an astray but I will praise it. There can be multiple reasons as well behind the two answers. It might be that the student wanted to impress the tutor with two different argument of the same question. But I can not presume. So I will go for the student.

So I will give full marks to enhance and motivate the student.

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The wiser approach will help students to develop good habits of cleaning up their mess. In the long term the other approach leads to lower productivity for everyone including yourself. Don't give full credit.

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I think the second answer should be seen as a correction of the first, so the second answer replaces or overrides the first, and only the second answer is graded.

A simplified example, in an oral conversation:

Question: "What is 40 plus 30?"
Answer: "That is 80. Oh, wait, that is 70!" or just:
Answer: "80 [2 seconds pass] 70."

This assumes that there is no feedback from the asking person, like body language, that could indicate that the first answer was wrong. This is the case with a written exam.

0

If the exam was "you are in front of a fire, what do you do to extinguish it?" What mark would you give to someone answering:

  • I put some fuel on the left side of the fire and I put some water on the right side of the fire.

?

You have a student that provided two answers and could not decide which one was wrong and which one was right ... so for what it matters, the student did not understand anything about the question, because in their mind both answers could have been correct.

<//edit after your additional edit that provides the X-Y explanation//>

After your comments, I am still thinking you could have awarded full marking for giving only the correct solution ("I throw water to extinguish the fire"), some marking for the correctly motivated wrong solution ("I throw fuel on the fire so it burns quickly and it exhausts the oxygen available in the sealed room where it happened") and still no points for both.

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    I don't think academia is so harsh
    – MAS
    Mar 18 at 5:08
  • @MAS exactly so, and unfortunately so.
    – EarlGrey
    Mar 18 at 7:22

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