Grading by assigning points to exercises

Question

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.

Answer 1

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.

Answer 2

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.

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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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

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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.

Answer 6

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.

Answer 7

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.

Answer 8

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.