How much credit to give a short exam question with one error?

In comments here there seemed to be significant difference of opinion on how much credit to give to a short exam question with a single critical error in it.

Consider the example of student work below, from a community-college College Algebra exam. The only real mistake is the pair of sign errors in the first line of work:

On a percentage basis, how much credit should be awarded to this student on this problem? And, related: How many discrete gradations of quality are warranted for a problem of this extent (e.g., as they might appear in a grading rubric)?

(P.S. It may be helpful to note that this kind of mistake is probably the most common source of errors for students in all the courses that I teach.)

• Just a comment: For future exams or homework -- in case you don't have such problems already -- I think it could be good/helpful to present work like this and ask the student/s where the error is... Jun 27 '17 at 23:18
• @BenjaminDickman: Subject to time limitations that's a great idea. On many occasions I wish I had space for inspection/correction exercises like that. (In the past I've added things like that to my lectures and then had to cut them for time.) Hopefully the moments when I intentionally or not make an error on the board and have students call it out give a sense of that. Jun 28 '17 at 15:01
• Do you allow your students to use calculators during exams? Jun 28 '17 at 18:02
• @Jasper: In algebra classes such as the example, no. Jun 28 '17 at 21:51

I'll try to make this answer a little more general than just telling how many points I would give for this particular error (if interested: I'd give 5/10 at most, most likely less).

For that, let's discuss three different kinds of computation errors (i.e. not including logical errors, wrong proofs, etc.). The given error in your image falls in the third category and thus should lose lots of points...

1. Careless mistake, e.g. due to missing concentration.

This is the most common mistake, the one you might have thought about when asking this question. The student simply put a wrong sign/a wrong digit/..., but did all the right computations. Here, the general rule is: Give almost full points, unless the problem got a lot easier through the mistake. As an example, consider the task to differentiate $$\frac{5x^4 + 7x^2-3x}{x+3}.$$ If a student forgets to copy the three from the question sheet and instead differentiates $$\frac{5x^4 + 7x^2 - 3x}{x} = 5x^3 + 7x^2 - 3,$$ this is a lot easier and can only give few points, even if it was an honest mistake of just forgetting to copy this one digit.

1. Results that can be verified.

There are many examples of exercises where the results can be verified quite simply, e.g. computing zeros of a polynomial, computing the kernel of a matrix, etc. If there is a small error here that goes unnoticed, it should give more points off. On the other hand, if a student writes something along the line of "I put this value into the polynomial and it turns out to not be a root, however, I can't find where my computation went wrong", then doing this test should be rewarded by only taking off points according to case 1. This is also important in other classes (physics, etc.), if you for example make an error in computing the speed of an object, it should at least give partial points if you notice that this car, driving at about twice the speed of light, is highly unlikely. :)

1. Mistakes due to not understanding the topic.

This is the gravest situation, and the situation you have in your image. Thus, it should give the most points off. Let's take your image as an example of what I mean. The question to write this as a single polynomial asks the student to perform two steps:

1. Use the distributive law to eliminate parentheses.
2. Use the distributive law to collect the powers of $x$ together.

Note that collecting the powers of $x$ can be done without understanding the distributive law.

Now this student did not understand this law and failed at the first step. It might look like only two wrong signs, but looking more closely, the student simply dropped the parentheses without doing anything else. Thus, this error is just as bad as saying $$5(x^2+3x+7) = 5x^2+3x+7.$$ As the distributive law seems to be the current topic in class, there should have been quite a few examples discussed before the test and this student simply didn't understand this topic. Thus, you should only give few points here.

Note that it is sometimes difficult to distinguish between the first case and the third one, especially if only looking at a single question, e.g. you might consider it case one instead of three if the student did all other problems with distributive law correctly... Furthermore, it also depends on the current topic taught in class: if your given mistake happens in a graduate course in mathematics, it might be considered case one; whereas in a first year course, discussing the distributive law, it is a very, very bad mistake.

edit: Regarding the comment(s), I'll try to make the argumentation for how many points to give short(er than it was before). When deciding on points for an exercise, you have to see both the whole exam and make sure that points are fair, and you have to look at each single exercise and make sure that it can be properly graded. If you give not enough points, then you might be forced to go down to half points or even quarters when grading, because "There is an error, but still, most is correct...". If you give too many points, you might end up giving points to a completely wrong exercise, because you can only take off so much.

Looking at the exercise you posted, I'd give 4 points. There are two major tasks here, as explained above: Removing the parentheses and collecting the $x$. For each step, two points is a good number. This will allow to take one point off for minor mistakes (e.g. $5+4 = 8)$ and two points off for major mistakes (e.g. $2x^2 + x^3 = 2x^5$).

• +1 and I'm leaning towards picking this as the correct answer, but I think it could be improved. One of the things I'm surprised by is the majority of answers here defaulting to points-out-of-10 when that wasn't part of the question. Maybe you could address the secondary question on granularity in general: Are there really 10 discrete points on which to grade? More? Less? Jun 29 '17 at 18:44
• That is not so easy to answer. In your special case, I would give 4 points: There are two major tasks here, collecting powers of x and getting rid of the parentheses. With four points, you can take one off for small mistakes and two for big ones. I edited my answer to show the questions I ask myself when putting points on problems. Note that the ten points in other answers is most likely only due to the fact that ten is such a nice number, I don't think they suggest that this exercise is worth ten points.
– Dirk
Jun 30 '17 at 10:16
• I think that the comment above is ("I would give 4 points...") is a much more direct way of addressing the secondary question, which is specific to the example problem given. The long edit for a thought-process on making that decision seems far less clear. Any chance you could replace that with the last comment? Jul 4 '17 at 1:11
• @DanielR.Collins Sorry, seems I got a little carried away on writing the previous version. I tried to make it short and clear, hope it's more understandable now. :)
– Dirk
Jul 4 '17 at 6:35
• Great, thank you! Jul 4 '17 at 14:08

I like to grade questions like this on a five-point rubric. The main aspect of this problem is distributing the negative sign into the second group. Presumably students had practice problems similar to this. I would score this as 3/5 or 60%. It is D work. It is not satisfactory.

• +1 for addressing the rubric granularity part of the question, thanks! Jun 28 '17 at 7:51
• +1, I'd like to see a sample Rubric for this. Jun 28 '17 at 14:41

Without knowing anything else about the exam (which class it was, when in the term, etc.), my vote is 7/10 points.

If this is a basic/intermediate algebra class, then I see at least four things being assessed here:

1. Identifying and combining like terms

2. Signed number arithmetic

3. Distributive property (or "subtraction being the addition of the opposite")

4. Writing a polynomial in standard form

As given, the student's answer demonstrates three of these four items. Whether these are "equal" in importance is debatable.

As instructors, focusing on one error is tempting, because we might look at a problem worth 10 points and say "I can't give this problem 70% -- it definitely doesn't show a passing knowledge of the material." However, if I take a good look at any one problem I assign and ask "exactly what knowledge or skills are necessary here", I may find that students are having to show a lot more understanding on skills I didn't plan to assess. Therefore, I would focus on what things students must do correctly, and then award points based on a correct demonstration of those discrete skills necessary to get the right answer ("points earned" vs. "points lost").

In the particular problem posed there are essentially two tasks. The first is to distribute the sign over the second term (this implicitly requires understanding that negation is involutive). The second is to correctly pair the coefficients of like powers of $x$. The student has failed completely at the first task, and has completed the second task successfully. How much credit to give boils down to how much importance one places on each of these tasks. The first seems slightly more difficult (however, I have limited experience teaching at this level), so I'd be inclined to give slightly less than half credit on this exercise. The perhaps more difficult half of the exercise is botched and the final answer is wrong (the student should be encouraged to evaluate and compare the initial and final form for some simple values of $x$ such as $0$ and $1$ as a way of checking the correctness of the response).

The precise point values to assign depend on the grading scale and system in use. This feels like a failing answer, so if 1/2 is minimum passing, I'd view this as roughly 1/3 (so in the US, where 7/10 is minimum passing, maybe it would be scored apparently higher), but saying anything precise about numerical values requires knowing well the local context, grading system, and standard practices.

• Ok This is the second answer saying something like that 7/10 is minimum passing. This is utterly contrary to my experience. We would consistently have 60% be a C and when the teacher was a little more demanding (average from my point of view coming from Europe) we would have as low 35% be the bottom cut off point for C. Are community colleges so different from research unis in grading? Also how come they are tougher? I would have expected the opposite.
– DRF
Jul 4 '17 at 11:25
• @DRF: The point is simply that precise point values are contingent on the grading scale used. The situation is even more complicated in the US because of the practice of disassociating point values from the final assigned grade, which is (sometimes) determined a posteriori, based on the distribution of assigned point values (what is called "grading on the curve"). Usually in the US a C grade nominally corresponds to something like the range 70-79 out of 100, but of course it is impossible to state any general rule about education in the US. Jul 4 '17 at 12:38

On a 10 point scale, I'd roughly break up the work as 4 points for correctly distributing the minus sign, 4 points for combining like terms correctly, and 2 points for correctly performing the basic arithmetic inherent in these steps. Since this student correctly performed the latter two tasks, but not the first, I would award 6/10 points.

I've seen it go both ways. "The answer is wrong. No credit" at one extreme. Or as little as 2/10 off for failing to distribute the -1 before the parenthesis.

When students complain to me that their teacher was unfair, having given zero for this, I explain (via a story from my own HS experience) that a doctor can potentially kill a patient for the misplaced decimal.

I also remind them SATs are scored either right or wrong for each answer.

Joel David Hamkins recently gave a sufficiently excellent argument for a 5-point grading system that I'd like to quote it here for the record. (By implication I think this would argue for something like a 4/5 or 80% credit on the short algebra question above.)

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.

I suppose this all depends on what type of expectations are set up for the class and the type of instructions that are given for the exam. Personally, I've never cared for grading problems individually devoid of the performance on the rest of the exam. And this is the type of expectation I set up for the class.

If the exam questions are to be graded independently on a problem-by-problem basis, then go ahead nick a few points [3 out of 10, let's say] because presumably there are numerous other errors [combining like terms, eg] that can be made and they are all worth something.

On the other hand, if there were an exam wide rubric where one of the things to check for was understanding how the parentheses are handled in the presence of a minus sign, then it's a matter of seeing what the performance on the rest of the exam was like. If the student does not demonstrate an understanding of how to reconcile $-(ax^m - bx^n - c)$ (or similar structures) on other problems, then they should get a few points taken off [however many were attributed to the skill / concept / mechanic] for the entire exam [or proportionally per problem]. However, if this was the only mistake of this type on the entire exam, I would lean towards a minimal deduction, perhaps even zero depending on how the rest of the exam looks.

In a nutshell, I try to be qualitatively cognizant of the topic-wise correlation inherent in exam problems. I tend to not want to repeatedly penalize a student for the same mistake throughout the exam, especially if the exam is testing on a diverse set of topics [unlike, say a basic arithmetic exam].