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For grading exams, I usually assign points to the exercises in such a way they sum to 100, and then I correct the exams by giving partial points based on such assignment.

Lately, I feel this is a poor choice: students which obviously don't get the topic sometimes get acceptable grades (because they sum some points for every exercise), while some students who understand the topic get bad grades (because they didn't do all the exercises, but some exercises they completed very well). Is there any resource (i.e. book or article) discussing this methodology of grading? What are the alternatives?

I think I'd like a grading methodology that analyses the whole of the exam not as the sum of the parts, but first I'd like to read about pros and cons.

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    If a student doesn't do all the exercises, then how do you know whether they understand the material? – JeffE Jun 24 '17 at 21:34
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    The question seems to me to be asking for education literature on grading methodology, of which I am sure there is plenty, though the present answers do not address this. Can the OP clarify? It might also be useful to indicate what type of class this is for. – Kimball Jun 25 '17 at 7:46
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    Whatever you do, you need to make sure it minimises the surprise of the grades. Something looking at the whole thing is very subjective, and you'll get endless attempts of grade rebuttals. – Davidmh Jun 25 '17 at 8:30
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    Note that time management is an important task for a student in an exam. He has to know when to skip an exercise, when to stop working on one and go to the next, he has to be able to estimate how long each exercise will take him. So the students who do only part of each exercise might actually be right in doing so. Of course the exam should be designed in such a way that at least the good students are able to solve everything in time, so if you have no complete exam at all, you might reconsider your time planning (general rule: your TI for the course should be able to do it in 1/2 the time). – Dirk Jun 26 '17 at 7:51
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    @Bemte - better how? Better for who? I've been to 2 colleges, and had at least 30 different oral exams, the most significant deciding factor of my grade has always been the mood of the day of the examiner. That is an atrocity, and a primary reason why almost everyone has moved to strictly objective, easily quantifiable tests. – Davor Jun 26 '17 at 12:29
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If good students who understand the topic well are getting bad grades because they don't do all the problems, you should think very carefully about whether you've made the exam too long.

If this is what is happening (and I suspect it is), there is a very easy way to fix it:

Assign fewer problems.

Then you don't have to worry about coming up with a new kind of grading scheme that looks at the test as a whole, and which the students who get low exam scores will undoubtedly complain about.

If you designed the exam by looking at exams that professors had set in previous years, they probably also made the mistake of putting too many problems on the exam. It's a very easy mistake to make. I've heard the guideline that professors should be able to solve the problems on the exam in one third of the time that you give students, and I'm not at all sure that it shouldn't be less.

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    If there are too many problems for a student to do them all properly in the time allowed, they have to pick how to allocate their time. One strategy is to do a few problems thoroughly. Another is to spend a fixed amount of time on each problem, doing as much as you can in that time. The two types of papers the OP is getting could reflect those strategies, rather than a difference in understanding. – Patricia Shanahan Jun 25 '17 at 13:34
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    @Patricial; If a substantial fraction of students turn in tests with some problems blank because they didn't have time to solve them (rather than because they have no idea how to attack that particular problem), your test is measuring how quick the students are, and not how well they understand the material. You don't want to write tests that measure the quickness of the students, or their grasp of good test-taking strategies. (At least I don't.) – Peter Shor Jun 25 '17 at 13:50
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    I'd strongly echo the suspicion that your exams are too long ... and that, in reaction, the students may take different strategies to attempt to cope, as @PatriciaShanahan speculates, again having little to do with understanding of the material. It's true, there's a mythology in mathematics that quickness is a high virtue, but it that's all we're testing, we should not charge tuition, I think. – paul garrett Jun 25 '17 at 18:38
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Lately, I feel this is a poor choice: students which obviously don't get the topic sometimes get acceptable grades (because they sum some points for every exercise), while some students who understand the topic get bad grades (because they didn't do all the exercises, but some exercises they solved very good).

There are two aspects in this:

The combination of your assignments and your points scale does not yield results proportional to the skill level of participants.

The assignments in the exam may indeed be designed in an unfortunate fashion.

For instance, I have found it to be problematic if different points weigh very differently. In particular, if in each assignment, some points count for effort, while others count for thinking, students that totally lack any understanding of the topic might still get through reasonably well by gathering all the effort-points.

There is no general rule of how to achieve a uniform weight there (because it depends strongly on the particular topic and tasks), but try to check your points scale against "personas" with different strengths.

Results of your systematic grading scheme contradict your gut feeling about a student. Learn to distrust your gut feeling.

I have experienced plenty of cases where students that I thought had written mostly nonsense ended up with not that bad a result, and other students that I thought were "almost perfect" were actually just in the middle field at best.

Whenever this occurs, after summing up the points from each assignment, I am surprised by these results, and I sometimes re-check some of the graded assignments, just to be sure. I usually do not find any mistakes in the grading, and if this has taught me anything, it is that using a fixed1 grading scheme that breaks down assignment into pre-defined steps or solution parts, each of which yields a point is a good choice.

Without it, my grades might be totally off, significantly distorted by things like my general impression of the student, the regularity of the pattern by which correct statements appear in the student's exam among incorrect ones, the ratio of correctness versus completeness of answers, etc. In other words, the grade would be biased by individual factors that I was not planning to evaluate at all.


1: Well, more or less. You can never quite predict all the out-of-the-world solutions for some of the simplest tasks that some students find.

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    I particularly appreciate the 2nd part of this answer. – Daniel R. Collins Jun 24 '17 at 22:37
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In maths at Oxford they used to square scores on exam questions before summing them, to encourage complete answers to fewer questions rather than partial answers to many.

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    Could you give an example of how this works out in practice? I can't quite visualise the effect. – Weckar E. Jun 26 '17 at 6:49
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    A single score of five with two scores of one is worth as much as three scores of three, if the square-first method applies, rather than being significantly less in the direct-addition method. A four and a zero is worth more than a two and a two. A single perfect ten is worth much more than ten trivial observations that score one each. @WeckarE. – Nij Jun 26 '17 at 9:38
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Continuing an aspect of @PeterShor's answer above, and @PatriciaShanahan's comment, (that too long exams test quickness, which is probably not tightly related to subject matter master, and that you may also be in-effect testing students' choices of coping strategies for too-long exams...):

A specific way to have a too-long exam test something perhaps tangential to the subject matter, is to in-effect require students to do a lot of new thinking in real time (as opposed to more-or-less recollection of their thinking during the presumably longer period of study prior to the exam). This is obviously somewhat related to raw quickness, but also related to some sort of sense of "composure", much as being able to extemporize while speaking in public... which is a good thing, but may not be a criterion for underlying sense, etc.

Yes, in mathematics, for example, there is a tradition (I think somewhat misguided) to pretend to test peoples' "problem-solving ability" on exams, by coming up with "interesting" questions. My objection is that this does not well test capacity to deal with routine situations (which could be "remembered" rather than figured out in real time), and not only tests quickness and composure, but can contribute to stress and distraction from the routine questions. For example, students may not be able to distinguish the two, due to self-doubts, etc., thus sinking too much time into non-routine questions.

Both at undergrad and graduate levels, I am first interested in whether students can do the routine, cliched, universally-useful things... not whether they can improvise in real time, etc. The latter is not really essential to be a good mathematician, for example.

In reference to the original question: exams are probably too long. Also, probably require too much thinking, rather than recollection from (presumed) previous preparation.

  • There is a difference between testing students' problem-solving ability, testing students' ability to memorize and plug numbers into formulas, and testing whether students actually understand the material you're trying to teach. Ideally, you want exams that accomplish this last criterion. – Peter Shor Aug 5 '17 at 18:47
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If you want to favour finishing some problems over nibbling at every problem, you can simply change the way you "give partial points" to each problem to be more convex.

If we take the sale from @MadJack,

  • no answer or very, very wrong,
  • got started, made some progress, but took a wrong turn somewhere,
  • half-way there,
  • minor error,
  • perfect

Then instead of ranking those (for an easy example, out of 8) as 0, 2, 4, 6, 8, you could do 0, 1, 2, 4, 8.

This is exponential, so maybe an extreme example, but any convex function would do to shift the reward from equally attempting any problem to finishing problems.

Alternately you could reward the introductory "parts" with 1 point and the final ones with 3 points, so the grades would be 0, 1, 2, 5, 8.

To find the right function for you, it depends on your preferences (whether to force integer scores, how to weight different levels of completeness of the exercise, etc). I guess you should experiment a little maybe on the previous tests you had to see how it would modify the grade based on what you do, as well.

The main take-away is that the rewarded strategy for a convex scale is to finish exercises.


A different example of this is what one of my teachers used to do: grade 7 questions out of 5, starting from 5 and getting -1 per unanswered or wrongly answered question (until you reach 0, no negative grades).

If you look at it by adding points instead of subtracting them, the (cumulated) points that each successive (correctly) answered question grants are: 0, 0, 1, 2, 3, 4, 5, which is indeed convex. If this is applied to an exercise in a test, the answering strategy becomes to finish every exercise you start, but only attempt those where you feel you would get more than 2 questions right.

This specific scale is a little harsh however - so I wouldn't really recommend it - and was meant for questions on definitions and such that everybody was expected to know.

  • The Putnam competition does this. Most grades are 0, 1, 2, 8, 9, or 10. So it essentially divides the solutions into "solved it, possibly with minor errors" and "not solved it". – Peter Shor Jun 26 '17 at 10:54
  • Well in that case I'd do every exercise until 8 points and stop there. Increasing rewards is the key if you want people to finish the exercises. 0, 1, 3, 6, 10 for example, you get +1, +2, +3, +4: more points for dotting the i's in an exercise than doing half of another one. – Cimbali Jun 26 '17 at 11:37
  • Right. If the input of grading levels is not linear in effort or difficulty that makes it a little hard to reason about. :) I guess you could also grade linearly but with unequal steps, though that does sound discouraging if you have to do 60% of the exercise in one go to get either 0 or 6 points. – Cimbali Jun 26 '17 at 11:54
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I usually assign points to the exercises in such a way they sum to 100.

In addition to the other answers, you may want to consider whether points summing to 100 is really necessary. How many points is it per problem? Are there really that many discrete steps on which to assess? If not, then this may possibly bias you to giving too much partial credit (like maybe half-points) for any random scribbling, which inflates the scores for the under-proficient students that you seem to be observing.

Consider: The New York State Regents mathematics examinations for high school have short-answer questions (Parts II, III, and IV) each of which are worth either 2, 3, or 4 points (respectively). I think the same is done for PARCC tests used nationally with the Common Core curricula. The tests do not add up to 100 points each.

I follow the same basic protocol in my community-college tests. Short-answer questions up to college algebra are worth 3 points each. In sophomore statistics I may have larger problems, up to 6 points each. The tests may sum to 20 or 30 points (which gets scaled to a percent automatically in the online learning management system).

I write out a complete solution to every problem before giving a test, analyze the length, adjust the problem if necessary, and decide on what section of each problem earns a point (for example: college algebra problems tend to run about 6 lines each, so I'm awarding 1 point for each 2 lines of algebra); i.e., effectively a grading rubric. That also makes for a simple, clear, and fast decision process when grading papers. It may also possibly result in fewer partial-credit points (e.g., compared to some time before I did this, early in my teaching career). If scores are overall too low the first time I do this in a new class, then I may give a one-time linear scaling to all the papers and adjust problem length or difficulty in the next semester.

See also this excellent argument for a 5-point grading system by Joel David Hamkins.

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    I don't really see the difference between assigning partial credit of 3/10 on a problem, and partial credit of 1/3 on a problem. I'd be much more likely to give half-credit to an incomplete answer on a problem worth 2 points than on one worth 10 points. – Peter Shor Jun 25 '17 at 13:45
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    @PeterShor: In my (and I think colleagues') experience with a 10-pt problem, mentally we tend to look for some opportunity to award any of the 10 point levels, e.g., some small mistake getting 9 points (instead of down to 7 or whatever); perhaps that tends to be a new-instructor mistake. Also, it's known that it's easier to mentally add up (at the end) a sequence of 1's, 2's, and 3's, compared to a bunch of 3's, 7's, and 10's. – Daniel R. Collins Jun 25 '17 at 14:37
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    One advantage with 10-point problems is that you can award 9 points for an arithmetic mistake, 7 points for a minor conceptual error, 4 points for a major conceptual error, and 2 points for a couple of paragraphs which mention some relevant keywords but make no progress whatsoever towards solving the problem. – Peter Shor Jun 25 '17 at 14:39
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    perhaps that tends to be a new-instructor mistake — I do the same thing, but I don't think it's a mistake. Only perfect answers should get perfect scores, but small mistakes should have small effects. Only answers that show NO understanding should get zeros, but small successes should have small effects. (And I don't even try to add up the points at the end; I let Moodle/Gradescope/Excel/whatever do that.) – JeffE Jun 25 '17 at 19:01
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    @DanielR.Collins Well, I think that if the exam is ABOUT arithmetic, then a regular arithmetic mistake would be considered a conceptual error (7 points for a minor conceptual error, 4 points for a major conceptual error), while you can consider awarding 9 points for "trivial" mistakes (e.g. the student forgot to copy that minus sign in the next line, or copied a 1 instead of a 7). YMMV – xDaizu Jun 26 '17 at 11:35
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My CG professor in college had similar worries about grading, especially because many students would develop a subpar understanding of either theory, algebra, or low-level programming, in that they would be particularly weak in one of those topics, which eventually hindered their progress in the course.

His approach was breaking down every question into three grades, one for each of the criteria he wanted to evaluate, and then applying geometric mean among them. It punishes incomplete answers more harshly while also being a bit more forgiving of small mistakes. Of course, this was transparently disclosed at the very first class, so we knew what we were getting into and could opt out if we wanted to.

He also made his exams shorter in size, but deeper in detail, which is something that you should take a look at, regardless of grading issues. If students who clearly don't get the topic end up doing more questions than good students, this seems to stem from poor time management. If completeness of solutions is more important to your subject than their speed, then you should allow them more time to display the results of their learning.

There's another issue you need to solve, now. If I was your student, I wouldn't feel comfortable focusing on your course. If I can get a better grade by just giving incomplete answers, there's not much point in making your subject a priority.

  • In one of my graduate math classes, the professor used the Euclidean norm (square root of sum of squares) of our homework scores to assign grades, to deliberately reward excellent work on a few questions over pretty good work on everything. (The problem(?) with geometric mean is that if even one component score is a zero, your composite score is also a zero, no matter how well you do on everything else.) – JeffE Jun 25 '17 at 19:10
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    @JeffE It is true, and it was intended to guide students into making sure they had a decent understanding of the whole material. In the end, our scores weren't significantly affected. About using Euclidean, it doesn't seem like pointing in the direction OP wanted, who, I believe, still wishes to grade good work overall higher than focused excellent work. – Ramon Melo Jun 25 '17 at 19:31
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some students who understand the topic get bad grades (because they didn't do all the exercises, but some exercises they completed very well)

As others have addressed well in their answers, you can opt to ask fewer questions to mitigate this issue.

students which obviously don't get the topic sometimes get acceptable grades (because they sum some points for every exercise)

To counter this issue, I assign relatively higher weights to more challenging portions of the exam. Here is my typical exam grading flow:

  • Divide up my exams into grade-able chunks;
  • Weight each chunk such that the more challenging chunks are worth more, and the more trivial stuff is weighted less;
  • Grade each chunk on a { 0, 2, 5, 8, 10 } scale (10 is "perfect," 8 is "minor error," 5 is "half-way there," 2 is "got started, made some progress, but took a wrong turn somewhere," and 0 is "no answer" or "very, very wrong");
  • Combine the chunk scores with their associated weights to get an exam score out of some arbitrary total (e.g., 100 is a popular choice).

Students who don't really know what they are doing will do well on the trivial stuff, but they don't do well overall since the parts they knew how to do wasn't weighted very highly.

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