# Curving grades without creating competition among students

I've recently taken a new position within a math department at a large university. The department has an official policy that in most lower-level undergraduate classes (let's say anything in the calculus sequence and below), roughly half of the students should receive A's and B's, and the other half should receive C's and below.

I tend to disagree with this policy, but being new to the department and rather inexperienced with the student population, I'd like to honor the policy for the time being. So for the first time in my teaching career, I plan to curve my class to achieve a desired distribution.

One tenet of my teaching philosophy is that I try to be as transparent with students as possible about how they will be assessed, and how their performance will translate into a letter grade. This means that if I am to institute a curve next semester, I want to be able to tell students exactly how the curve will work. Of course, I'm very worried that telling them only half of them can get A's and B's will create a very competitive environment, so students will not want to study with and help each other, so learning will suffer (in fact, I'm almost certain this will happen). This leads to my question:

How can I simultaneously

1. tell the students how I will assign grades,
2. achieve the distribution desired by the department, and
3. still get students to buy into working together?

Seems impossible to me - maybe it is...

One quasi-solution (which potentially strays a bit from condition 2) is to have guaranteed grade cutoffs. For example, if a student achieves an 80% in the class, she is guaranteed at least a B-, even if 75% of the class scored higher. In this scenario, the perceived effect of any curve is that it "can only help and not harm." This could give students at least some sense of being in control of their own destiny and that they are not only being assessed in comparison to their peers.

The problem with this solution is that determining those grade cutoffs can be tricky at the outset. If I make them too high, then at some point in the semester it will become clear that the curve will supersede the cutoffs, and competition is likely to creep back in. If the cutoffs are too low, then I could be locked into awarding an overly generous distribution that would be frowned upon by my department.

What can I do?

• For what it's worth, in my undergraduate experience curved grades were always used in the "only help not harm" design, and I did not get the sense that this created too much competition. The courses where curves came into play were typically huge courses, so there isn't a lot of game theory support to competing rather than collaborating with a specific individual or small study group. The problems on both ends that you reference still stand of course. Dec 16, 2020 at 18:36
• I asked basically this question a year ago; consensus was that it's impossible (no actual solutions given): academia.stackexchange.com/questions/135815/… Dec 16, 2020 at 18:52
• Is there a problem if more than half of the students earn A's and B's? What if every student met all of the learning objectives for the course and aced a final exam? Dec 16, 2020 at 22:01
• Is this a one-shot game or a repeated game? If you are going to teach this class many times, perhaps you can use the scores from the previous time (or times) you gave the class to curve this year's students. You won't avoid the competition if you have to curve the first year, but it helps in the long term. Dec 17, 2020 at 2:33
• OP previously posted this question at matheducators.stackexchange.com/q/19281/77 but didn't mention it.
– JRN
Dec 17, 2020 at 4:58

I think you're onto the right approach with the idea of a curve that only helps.

Based on your description, it sounds like you are in the United States. If so, the simplest way to do this would be to compute twice:

1. Use the standard US metric of A=90%-100%, B=80-89%, C=70-79%, D=60-69%, F<60%.
2. Apply the curve.
3. Give the student the maximum of the two grades.

If you put together examinations that are far too difficult, then you may still have a problem. No matter what, however, a curve can only do so much to save you from this mistake, however, since overly difficult exams will also tend to increase the arbitrariness of assessment.

One other thing that may help in setting the curve is a trick I learned from the professor that I TAed for while in graduate school. When determining the curve, he also included all of the students who dropped the course. In a big undergraduate course, there were always a decent number of these, and they would generally be doing worse than the students who at least stuck it out to the end. The dropping student thus generally absorbed the failing grades at the bottom of the scale (except for a few deserving cases) and took the sting out of the curve for the students working hard but struggling in the class.

• This is valuable advice, but won't actually guarantee a predefined outcome or split in grades. Dec 16, 2020 at 19:33
• Correct, and I consider that a positive feature. I despise the idea of a predefined grading split that could give a C to a student who is getting 90% of all the answers right. I believe that if the instructor isn't asking enough of the students, it is the faculty and not the students who should pay the price for that mistake. Dec 16, 2020 at 19:40
• Yes do not forget to count the dropped students as Fs.I think there are just different philosophies about whether a student who understands the material well and does all the homework etc should reasonably be expected to meet all the learning outcomes (mastery model) and also whether meeting all the learning outcomes itself should be enough for an A or if an A should require going beyond the learning outcomes. It really depends on your ideas about measurement too.
– Elin
Dec 18, 2020 at 20:50

### Achieve the grade targets through iterations of the class across multiple semesters, not by curving the marks.

Simply put, your goal is to produce a particular shape to the marks without treating students unfairly. So, simply teach the course, and assess it at a level of difficulty you think is appropriate.

Then, if you're off your desired grade distribution after the semester's over, you know how to adjust it the next time you're teaching. If too many students get high grades, you can increase the difficulty of the assessment next semester. If too many students get low marks, you can decrease the difficulty next semester.

This process can be repeated until your grade distribution is at about the level desired by the university. If one of the other professors criticizes you for being off the desired mark threshold, simply tell them that you're new and you're still in the process of calibrating the difficulty of the course. It's an iterative process, and you're probably not going to get it right the first time.

• This is the right approach to grade curves. Grade curves are expected with a large enough sample of students. With a normal class size, you can expect sampling errors. To get the curve right, you need to look at student grades across multiple years. If this is your first year teaching, then you can't do that yet. If the department wants half of the students to get a C or lower every single semester in every class, then they are curving wrong. Since you're in the Math department, the Statistics teacher should be able to back you up. Dec 18, 2020 at 14:58
• There's also going to be random variation.
– Elin
Dec 18, 2020 at 20:51
• @Elin Sure, but with a class of hundreds of students, you'd expect it to tend towards the normal distribution. Dec 19, 2020 at 8:19

If a policy like this works to the detriment of an individual student, giving them a grade lower than what they fairly earned, then it is immoral. Such policies should be changed and the faculty should get together and come up with better policies.

However, interpreted in a certain way, there shouldn't be a problem, though, again, it takes buy in from faculty and administration. If the "half good - half bad" policy is interpreted as a goal to be achieved over time then it can be benign. If a course is offered in a term and the grades are "off" then the difficulty of the course should be adjusted for future offerings, with the already-earned grades standing as assigned.

Let me give two extreme cases. Suppose you wind up in a given term with a bunch of brilliant students. They are all overachievers and get near perfect marks on all graded work. How do you proceed? Flip coins?

The other extreme case, which I've actually seen happen, is that a group of students think they are overworked and "go on strike" and refuse to turn in any work at all. Again, how do you make this "curve" work? I was once the new faculty member who faced this situation when I worked a bunch of MS students harder than they were used to. I had to tell them that it was possible for everyone to fail in order to get them going again. I needed a discussion with the Dean to justify this, but she agreed and it worked out. The students got to work and raised their standards.

The grading scheme needs to be fair on its face to every individual. They should have a good idea along the way how they will likely come out at the end and what they still need to do to achieve their personal goal in the course. If the grading scheme doesn't do that then it is immoral and needs to be fixed. Student life shouldn't be a battle against unfair odds.

I find your instincts here to be good and appropriate.

However, if your class grades deviate from the norm, be prepared to answer why. You need to justify your actions, of course, and you may need to adjust the student tasks to make a desired overall outcome more likely. But a rigid scheme will eventually be unfair to someone.

"Fix" the policy, not the (individual) grades. Adjust the difficulty of the course over time to match expectations.

Let me add a couple of notes for balance.

A department wide grading scheme that leaves all courses of "approximately" the same difficulty can actually serve students well. If the standards are high, then students, when they join a course, know how hard they are expected to work and that it will be pretty hard. It also prevents some "gaming" by students who seek an easy path by choosing courses or professors. There is no problem with this until it is misapplied to a single section/course/professor.

When it is applied to the course of a new professor, without thought or understanding, it is also unfair to that person who may need some time to become adjusted to the students and their expectations.

Finally, I'll note that it sometimes becomes necessary to provide a "gate" to a major as the university may not have the resources to handle a too-large number of advanced students. CS sometimes has to deal with this when the field becomes "hot" for some external reason, driving too many students to want to major in it. These gates can take many forms, such as not being able to declare the major until the end of the second year and requiring certain grades in foundation courses to do so. This is necessary, if unfortunate. But the system still needs to be fair to individuals and not contain hidden traps.

### Achieve a grade distribution by having easy, intermediate and hard questions

Suppose you had an exam with ten questions, all worth equal points:

• 1 very easy question
• 2 easy questions
• 4 intermediate questions - stuff you should be able to answer if you understood the course "well enough"
• 2 tough questions
• 1 very hard question

For a class of typical students, you would expect the results to be bell-shaped. So this setup allows you to claim that you satisfy departmental policy. But it's also more fair and objective than the post-test shifting that you feel like you're being pushed into.

This setup requires a somewhat rare piece of knowledge: what exactly is easy, intermediate and hard? This is obviously relative to the skill of "typical" students. Because an individual class can just happen to be quite good or bad, you need to iterate over several classes to settle on a good understanding of what a typical class is capable of.

Practically, in your first term, you don't want to be wildly off the mark with estimating where the middle of the class is on a difficulty scale, because that will undermine your standing in the department when negotiating the setup for next term. So you need early data to help tune your final exam. But any early test you do is probably going to skew to the left of the right. So the prudent thing to do is to make the early test relatively low-stakes, don't let it count for a large percentage of the overall grade.

Also, practice a poker face. If the results of the early test are bad, tell the students this was the "wakeup call" and that they need to step up. But you might yourself decide to scale down the difficulty on the final exam a bit. Conversely, if the results of the early test are very good, tell them this was just a getting to know you test and that the final exam will be harder. Making this early test low stakes is what allows you to do this trial.

Of course, you can also use the knowledge of the people around you for testing.

• Ask other lecturers to rate the difficulty of your questions, based on what they know about the questions
• Have your TAs playtest the exams. Assuming TAs need to have done well on a course in order to be allowed to TA it, this gives you an idea of the upper bound of what you can expect from students. If even the TAs have trouble with your test, it's probably too hard.
• Getting lower end of the curve testers can be harder, because these are probably less motivated to just out of the blue re-take a test they didn't do well the first time. However, if you have any students who are re-taking the course after failing it before, having a discussion with them about what they found hard and need help with, can give you some of this insight.

### Iterate to get the difficulty right

After your first time teaching this course, you can iterate. Any intermediate questions that everyone got right should probably be considered easy, and intermediate questions that less than about two-thirds got right should probably be reclassified as harder (or, you change your lectures to focus more on those topics).

Unlike typical "competitive" bell curve grading, this system doesn't punish students in a class for doing well together. If you have an exceptional group of students, they can all get good grades. But, depending on how many years you look at when tuning your test for next year, the next year's students are going to face a tougher test. So the competition element moves away from students in the same year, to students in successive years.

Now, you get an interesting situation. If this is year Y, and next year is Y+1, then the students in this year have nothing to gain by "going on strike"; class Y might make things easier for class Y+1, but that's cold comfort to class Y. Students in class Y-1 have nothing to lose from sharing useful materials with class Y, since they already have their results. Even students from Y-1 retaking the class have nothing to lose. Only students expecting to fail the class of Y and retake it in Y+1 have a reason not to help, but they probably don't have much ability to. Likewise, the upcoming class Y+1 doesn't have any real leverage to hinder class Y to make their own test easier next year.

Overall, the system has a mild bias upwards.

• In my experience this generally works reasonably well, except that I consistently find a second peak in the grade distribution at 0%, corresponding to students who simply don't show up for exams or really make no effort at all. Dec 17, 2020 at 15:11
• @BrianBorchers: Yep, in some courses, bimodal distributions are common.
– J W
Dec 18, 2020 at 13:35

Quite obviously, the mutual satisfaction of those criteria is impossible. Adjusting student marks to a curve puts students into competition ---i.e., a higher mark by one student lowers the marks of the other students and vice versa. That is a mathematical corollary of scaling.

May I suggest an alternative way of looking at this. Since you are new, one of the things you use your first iteration of the course for is to calibrate the difficulty level of the material and assessments adequately. Scaling of marks can actually help here because it allows you to adjust misjudgment of this level --- if you made the course too hard you scale up and if you made it too easy you scale down. Once you have taught the course once, and you can see the raw marks on your assessments, you will then be able to recalibrate the difficulty level of the course so that it accords roughly with the desired mark outcome in the department policy. At this point, you ought to be able to proceed without further scaling in future semesters, allowing some deviations from the desired outcomes from time to time. There may be some oscillation in the quality of the students from time to time, and so some departure from the desired outcome is to be expected.

It sounds like the official policy of your department allows some wiggle-room here, and the department does not expect that all courses will exhibit the desired pattern of marks. I recommend that you impose scaling the first time you teach the course, and then in future iterations you should try to calibrate the difficulty level of the assessment to get outcomes that are roughly where your department is aiming. At that point you will be in a strong position to argue against strict application of the policy to your course, in view of the efforts already made to calibrate the difficulty level of the assessment to the desired outcome.

Lastly, in regard to the desire to blunt the competitive instincts of your students, one major thing to try to get across to them is that their learning and knowledge matter more than their grades. If students behave in a way that enhances their learning process (e.g., by working together, helping each other, etc.) at the expense of a couple of marks due to scaling, this will stand them in a better position for their future success. If they believe that then they will be more likely to focus on the learning process and act in a cooperative manner.

• "their learning and knowledge matter more than their grades" is true, but with a large caveat: Learning and knowledge don't help students reach GPA cutoffs for scholarships. For example, the NSERC grants require a minimum of 3.7/4.3. Get a 3.69 and your application goes straight in the trash regardless of what you've actually learned. Dec 17, 2020 at 15:41
• I agree with @JSLavertu. Grade cutoffs for scholarship programs have created a very strong incentive for schools to inflate grades (what used to be a C is now a B or even an A). First, schools brag about placement in prestigious programs, so making the students seem more competitive is in the school's interest. Second, with a scholarship on the line, some students will become desperate, so inflation relieves some of the pressure on the students. As long as people judge students by their grades, the grades will matter. Dec 18, 2020 at 15:06

This situation is likely not as bad as you think, especially if you have a large class (I strongly question the wisdom of applying a curve in a sparsely enrolled class, but even then the situation may not be as bad as you or the students expect).

Let me demonstrate by way of an example. Let us assume 100 students, with the top quartile receiving A's, the second quartile receiving B's, the third quartile receiving C's, and the lowest quartile D's and F's.

First, note that any two students in the same "grade level" can study together with impunity. If Awesome Adam beats out Amazing Alice, who cares? They both get A's.

But what about students in different grade levels? What if Billy B. might want to study with Charlie C., but is afraid that raising Charlie's grade might hurt Billy? What is the likelihood that, if Billy and Charlie work together, Billy gets "bumped down" a grade for his assistance?

Almost none, as it turns out. First, note that no matter how much Charlie might improve his score, Billy can only ever be hurt by Charlie if Charlie actually does better than him. This is already fairly unlikely, at least in my experience, as very often students who study together are rarely so close in skill and understanding that the less prepared student will outperform the better.

It could happen, though. Suppose that against all odds, Charlie actually outperforms Billy. Will Billy be hurt?

Probably not. Note that Billy is only "bumped down" by Charlie if Billy was already in last place among all B-grade students. That is, Billy can only be hurt by Charlie both if Charlie outperforms him and Billy is already in last place among the B-students. Otherwise both Billy and Charlie will receive B's in the class.

Only four students in the entire class, the students at the very bottom of the A, B, C, and D grade levels, can possibly be hurt by partnering with a lower-level student who subsequently outperforms them. What about those four students, though? Should they avoid partnering with somebody for fear of a later upset?

Again, probably not. Finally, note that tutoring is rarely simply a one-way street. Although Charlie might learn more by working with Billy, chances are that Billy will also learn at least a little from studying with Charlie. If this translates into even just one extra question correct for Billy on his exam, he's probably better off than he would have been otherwise. Billy only has to beat one other B-grade student no matter what score Charlie gets. If Billy can beat Bottom-Barrel Bob, it doesn't matter what score Charlie gets -- Billy is still getting a B. And who knows, maybe by studying with Charlie, Billy might be able to beat out Amusingly Average Andrew. That would be a good situation for both Billy and Charlie.

tl;dr: A small number of students working together are not actually competitors in a large class. They are competing against other students in the class who are not part of their study group.

There are already good answers here but let me tackle this from a different perspective. Notes: this answer pertains mainly to big multisection courses that are required for many different majors. I also assume the course is not purposefully designed as a weed out course.

## You're might be overthinking this!

Preface:

For several years I taught the lab section of a intro level math class that most students in the university need in order to graduate. Thousands of students took the class each semester, and I taught 10 or so sections with roughly 35 students over that time.

The imagined situation:

If I heard of the policy you described while teaching my first class, I would have been very concerned as well. I think everyone who hears it imagines the same scenario: what if everyone does really well? Do I have to give bad grades simply because the quota says so?

The reality:

Individuals are unpredictable but group outcomes tend to be consistent. My grade distribution from class to class and year to year had very little variance. The overall grades over all sections were also very consistent year to year, which in my opinion is a desirable outcome. Big discrepancies from year to year are much more likely to be the fault of the instructor, instructional material, or differences in exam difficulty / grading than they are to be differences accountable to the students.

Essentially, as class size increases the variations in your outcomes should tend to decrease. This is going to be especially true for intro level math classes: these classes are taken by a wide range of students whose demographics (age, major, etc) are also going to be stable year to year. In other words, the freshman class in year x is going to look a lot like the freshman class of x+1. Additionally, the material in these classes tends to be highly standardized at this point. Nearly all calc 1 classes cover the same things regardless of where or when you take it. The evaluation material, particularly tests, also tends to be similar in difficulty year to year (or at least it should!). The material is also highly objective in nature, so variances in grading 'harshness' will be small (compared to, for example, grading essays).

Looking back, every class I taught shook out similarly, there were 2 or 3 students who got nearly perfect scores, 2 or 3 that failed completely, a handful of D's, lots of B's and C's, and a handful of A's. About half being A/B and half C/D/F sounds about right.

On a final note, sizable deviations away from previous distributions were how we determined something was off -- perhaps a bad test question or cheating took place. Consistency in grade distribution from semester to semester is a desirable outcome as it indicates that the course's difficulty is also consistent. This is most likely what the policy is getting at: if your grades significantly deviate from the distribution empirically derived from previous classes, something is probably wrong and it's probably not that you have class of all geniuses or dunces.

### Preamble

There's been, so far, a lot of answers and comments here containing variations on three themes:

• Calibrate your preset grading so it matches "the curve" after several iterations.
• Only apply "the curve" to help, not harm, students' grades.
• "Curving" grades is somehow immoral/unethical, so don't do it.

Does any of that answer really your question? Maybe, maybe not. (You've not yet "accepted" any.)

First, talk to your departmental colleagues -- the "grading committee" (if such a thing exists) and your fellow teachers. Find out how strictly is this "curving" policy enforced? Ask how they deal with the dilemma. You might just get some good advice.

If, however, it is well-established practice in your department to "curve" the grades then

• It will be easy for you to do too, and you'll be doing your job.
• The students will already be very aware of it, and presumably, have come to live with it.

If it is strict departmental practice and you choose not to follow suit, and inflate your students' grades, your colleagues will likely notice and may take action. So too will the students in general notice that yours are "easy" courses. Maybe students will flock to your courses. Maybe that will increase your workload.

At every university I've seen a "curve" is when the highest performing student is taken as the new 100% point and everyone else graded based on their score relative to that.

I have never seen any legitimate institution use the term to describe arbitrarily robbing students of a legitimately earned academic standing. As stated by others this is a deeply unethical policy, and one which I'm surprised hasn't been sued out of existence in any developed country.

Either your students followed your directions and got the answers to your assignments and tests correct or they do not. If they got the answer correct and followed your instructions properly then they deserve full credit.

Period.

Anything else is completely unethical and frankly a staggering liability for both you and your department. What happens when you're left with multiple people equally qualified for a given standing and have to pick who loses out? How do you do that? Do you flip a coin?

What's going to happen to you when the student who was artificially given a lower grade than they objectively earned alleges racism, sexism, or some other prejudice was involved in the decision?

The only ethically and legally responsible thing to do is to refuse and grade your students honestly and fairly on their academic performance. Tell your department that if they are going to demand students be arbitrarily and capriciously penalized for literally no reason whatsoever then they can have the guts do it themselves, and to take the responsibility for that decision when someone files under Title IX and starts a sit-in in the dean's office.

If there isn't enough room in the department for everyone to progress the only legitimate course of action, assuming growth isn't an option, is being honest about that and setting up either an actual legitimate and fair selection process or a waitlist.

Also as a side note to the suggestion of "hacking" this result by tuning the difficulty of the course: When one of my undergraduates failed a test that student was at fault. When half of them failed or got poor marks that was my fault for failing to properly teach them the material.

The idea that large numbers of students failing classes is somehow a good thing needs to be left in the dark ages where it belongs along with deliberately depriving medical residents of sleep.

• A staggering liability? That's a bit bold. I'd challenge you to cite any case of a successful lawsuit over any impartially applied grading policy at a university. Dec 18, 2020 at 1:47
• A quibble: taking the best-performing student as a renormalized 100% would occasionally be not what you'd want, if/when that student is an extreme outlier. Something more like "the lowest grade among the top 5%"... Dec 18, 2020 at 20:39
• @KevinArlin My citation would be that we now live in a day and age where a professor can be suspended over accusations of racism when they quote and actively condemn a historical document from centuries past which contains racism.
– D3SL
Dec 20, 2020 at 5:26
• If by "no citation" you mean a reference to some of the many incidents which have made major nationwide news multiple times as an example of the current state of affairs on the average campus then sure. This is a 500 character comment, not a peer reviewed article.
– D3SL
Dec 28, 2020 at 10:17