Is grading on a scale a flawed method for evaluating student performance?

[Derek Francis]

This is what I would see as an effective grading system.

A - You're exceptional (possibly in the top 20% of students)
B - You're very good
C - You're good (or very good, but inconsistent)
D - You're bad, but you put a lot of effort in
F - You're bad and you put little effort in

 

[Derek Francis]

A curve can sometimes be justified.

In a scenario in which every single student underperforms, it's most likely the professor's fault.

 

[mathwonk]

as a retired professor, i agree with micromass about how hard it is to choose a few specific questions that accurately measure competence or lack of it. Even when we give out a syllabus with a list of specific topics and even specific theorems to know, with references, and I ask students simply to state those same theorems, most cannot do so. Even among phd candidates this occurs, for the reason apparently that the students are unwilling to actually learn what we tell them to, rather they look at old tests and memorize the same questions hoping these will reoccur. So in a sense my test is excellent in that it accurately measures the incompetence of my students, but in practice people stop allowing me to write these tests this way precisely because they do expose the lack of knowledge of the students. in fact our students actually argued that it is too much to expect them to understand the syllabus well enough to answer general questions, and they should instead be tested only in a predictable way that they can more easily prepare for. So to them the purpose of the test in helping them to learn a basic groundwork of material is unreasonable, and they petition simply to make it a hurdle that anyone can surmount. testing well is really hard. try to think of a single question, say in calculus, such that, if a person cannot answer it, they should fail. i assure you, even if you choose differentiating the simplest most familiar functions, you will be surprised at the outcome, much less if you ask the definition of a ,limit, or a derivative.

 

[bluemoonKY]

A curve can sometimes be justified.

In a scenario in which every single student underperforms, it's most likely the professor's fault.

I agree that it's likely (but not definitely) the professor's fault if 100% of students are incompetent. However, I've personally been in classes at community colleges in which probably 90% of the students in the class were incompetent.

If I was the Dean of the community college in which the electrical engineer failed all or almost all of the students in the class, I would definitely think it fishy if a professor failed everyone or almost everyone in a calculus class. What I would do if I were the Dean is ask the professor to submit all the students' final exams back to me. If the students' exams indicated that they were actually incompetent with the course material, then I would personally sit in the professor's classes to investigate if the professor taught the material decently. If the professor taught the material decently enough, then I would ask the professor to personally submit the students' subsequent tests into me to examine personally. If I personally thought that the professor did a decent job of teaching from my observations and all the students still could not competently answer the calculus problems, I would not fire the professor for failing all the students. I would be grateful to the professor for the professor's upholding standards and upholding the reputation of the college.

 

[bluemoonKY]

Even among phd candidates this occurs, for the reason apparently that the students are unwilling to actually learn what we tell them to, rather they look at old tests and memorize the same questions hoping these will reoccur students learn what the professors assigns or fail the class.

The Dean should let the professors decide what the students must learn to pass the class.

So in a sense my test is excellent in that it accurately measures the incompetence of my students, but in practice people stop allowing me to write these tests this way precisely because they do expose the lack of knowledge of the students. in fact our students actually argued that it is too much to expect them to understand the syllabus well enough to answer general questions, and they should instead be tested only in a predictable way that they can more easily prepare for.

You sound like the electrical engineer/math professor my father told me about. In any sane educational institution, the professors or the Dean of the College would decide what the students should be required to know to pass the class, not the students making the decision of what they should have to learn to pass.

try to think of a single question, say in calculus, such that, if a person cannot answer it, they should fail. i assure you, even if you choose differentiating the simplest most familiar functions, you will be surprised at the outcome, much less if you ask the definition of a ,limit, or a derivative.

The students' final grade for the class should be determined on how the students answer a multitude of calculus problems, not just one calculus problem.

 

[Derek Francis]

An unfairly generous curve wouldn't happen in a scenario in which only 10% of the students are competent. In order for a curve to happen, every student must underperform.

If the highest grade is 100, no curve happens. If the highest grade is a 95, only a light curve will happen. If the highest score is an 83, just imagine what the rest of the class got.

It would be rare for a class with zero competent students. Gen ed classes tend to have upwards of 50 people in them. Large sample size. And students in majors tend to be more competent in general.

 

[Derek Francis]

In any sane educational institution, the professors or the Dean of the College would decide what the students should be required to know to pass the class, not the students making the decision of what they should have to learn to pass.

I'm not saying the passing criteria should be whatever the students want it to be.

However, if there are a lot of anomalous test scores, it's worth examining why this is happening and how it can be mitigated.

 

[Choppy]

One of the issues that I see in this thread is that some people are applying a competency-based evaluation model to a subject that is more academic in nature.

Competency-based evaluation is necessary for certain professional certifications. An example might be a license for a medical doctor to practice medicine or perform a specific surgery. In this context a competency threshold is very important. One either has a sufficient skill set, or one does not.

Academia is somewhat different for a number of reasons. First, I would argue that the consequences are less cut and dry. Once you've passed a first-year calculus class it's not like you now have a license to differentiate. Rather, you should have a foundation in mathematics that serves as a prerequisite for understanding more advanced concepts in mathematics and other sciences. Second, a threshold for what constitutes "competent" or "not competent" is more difficult to establish. As has been argued already, even if a student misses a very basic question, the student can still demonstrate that an adequate understanding of the material has been reached. Finally, in academia, one of the reasons for grading in the first place is to stratify students from each other - this stratification is used for deciding who should get into graduate school or who should get what scholarship or in some cases jobs or internships. From this point of view, grading on a curve tends to make more sense.

 

[Derek Francis]

The problem with grading students on a scale at large is that your grade will more so depend on who your classmates are than your hard work. And that sounds pretty damning.

If you study with geniuses, you get left behind. If you study with idiots, you get a free pass.

 

[mathwonk]

here is a real life example. i told my honors calculus students to learn to state the fundamental theorem of calculus, (of course i gave them the statement) since i would ask it on every test, and if i ever failed to ask it, they could get extra credit by answering it anyway. after 4 tests, still not all of the 8 students could state it correctly. encouraged however that 7 of them could, i asked also a question on the final whose answer is given by understanding what the fundamental theorem says. although almost all students could finally state that theorem only one of them understood the sentence they had memorized and correctly answered the question.

to be more specific, in part the fundamental theorem says that every continuous function has an antiderivative. so on the final i gave a specific continuous function and asked if it has an antiderivative, and since it was a complicated function whose antiderivative is not easily guessed, 7 of them said, in contradiction to the theorem they had just stated, that it did not have one. thus 7 out of 8 students stated on one question that every continuous function has an antiderivative, and on the next question stated that no, that particular continuous function (say cos(x^2)), does not. i did not fail any of them but they were very indignant that being "honors students" they did not all get A's.

in teaching we try to get people to learn certain things, and we also try to get them to understand those things enough to use them, and then we try to measure our success at these tasks. it ain't easy but it helps if the student actually wants to learn rather than just get a certain grade.

 

[mathwonk]

bluemoonKY i was just trying to simplify it for you. so try to think of 10 questions such that anyone who cannot answer 5 of them should fail, if you prefer. i have actually had a top teaching professor in my department suggest that we might try failing anyone who cannot differentiate correctly, say the functions, exp(x), sin(x), ln(x), and x^3. this suggestion was not adopted, presumably as too harsh. you should recall also that i said that i had told people in advance what the questions were that they should be able to answer. do you think it too strict to fail someone who cannot correctly answer a question with a single sentence answer, that they have been told to expect, and have been asked it before several times, and given the answer to it repeatedly? (like state the FTC?) The point is that grading accurately is very difficult. i have had colleagues who would give a zero to a student on a calculus problem, for answering the question "find two real numbers whose sum is 100 and whose product is maximal", by saying simply 50 and 50, without showing work. i vigorously argued for at least partial credit since the answer was right and no instructions to show work were given. if you want to see work, i say ask for it or ask a less trivial question. there is little agreement on these matters. But I still feel if the teacher says, "no matter what else you learn, be sure to be able to state the FTC on the final", and you show up without knowing that, you are making a poor impression on your grader, especially if you have never ever come to office hours for help.

 

[Derek Francis]

This might be more logistically demanding than it's worth, but what about a comprehensive grading system for every major course?

Some tests scaled, some tests non-scaled, some tests standardized.

 

[vela]

you should recall also that i said that i had told people in advance what the questions were that they should be able to answer. do you think it too strict to fail someone who cannot correctly answer a question with a single sentence answer, that they have been told to expect, and have been asked it before several times, and given the answer to it repeatedly? (like state the FTC?)

I don't see the point of asking students to regurgitate the FTC on an exam. What were you hoping to reveal other than the students could (or apparently could not) follow instructions? As you found out, being able to state the theorem and actually understand what it says are two completely different things.

The point is that grading accurately is very difficult. i have had colleagues who would give a zero to a student on a calculus problem, for answering the question "find two real numbers whose sum is 100 and whose product is maximal", by saying simply 50 and 50, without showing work. i vigorously argued for at least partial credit since the answer was right and no instructions to show work were given. if you want to see work, i say ask for it or ask a less trivial question. there is little agreement on these matters.

I'd give the students zeros who just wrote down the 50 and 50. The point of a test isn't simply for students to write down the right answers. It's for students to demonstrate that they know what they're doing. If they can't or don't explain their reasoning, there's no reason to conclude that they did anything more than make a lucky guess or memorized the answer from the homework.

 

[mathwonk]

in testing you always want to know what you are testing by a certain question. e.g. if i get my students to admit that every continuous function has an antiderivative and then they don't think that sin(x^2) has one, maybe they don't know that sin(x^2) is continuous. so on the next try maybe i'll add in an intermediary question asking whether sin(x^2) is continuous, to see just where they are going off the track.

or maybe all i gave them was a technical version of the FTC, like " if f is any continuous function then the integral of f from a to x (written in symbols instead of words) is differentiable and has derivative equal to f(x)". but maybe they don't understand the integral sign, and don't realize that integrating from a to x really is a function of x. i.e. maybe they don't know what an abstract function is, and think that a function has to look like one of the usual suspects like sin, or x^n, so maybe next time i'll put the FTC into words like "and this of course means that every continuous function is the derivative of its (signed) area function, even if we don't know a simpler expression for that function than the integral sign itself".

or maybe i'll ask the question as an example, like "can you give an example of a continuous function that does not have an antiderivative?" anyway, you start out asking it the way your professors asked it and books ask it, and over time when nobody seems to get it, you try to learn how to communicate better with your own students. but sometimes you feel some of them are not helping themselves or you much.

 

[mathwonk]

"I don't see the point of asking students to regurgitate the FTC on an exam. What were you hoping to reveal other than the students could (or apparently could not) follow instructions? As you found out, being able to state the theorem and actually understand what it says are two completely different things."

well i have to admit it was news to me that there are people so brain dead that they memorize a statement they have been told is important without thinking at all about what it means. but after i learned that, i began to ask it in more informative ways. it takes a while to realize that some of your students are actively trying not learn anything.

 

[mathwonk]

"I'd give the students zeros who just wrote down the 50 and 50. The point of a test isn't simply for students to write down the right answers. It's for students to demonstrate that they know what they're doing."

In my opinion if you make these, very reasonable, assumptions then you have to tell them to the students before penalizing them for violating them. you cannot expect your students to know what you think the purpose of the test is unless you say so. i think that answering any question correctly indicates at least some intuition. so on my tests i always say, "for full credit, give a full explanation for every problem in this section, don't just write down the answer". they are not mind readers.

anybody who even graphs the relevant function y = x(100-x), of that maximization problem sees that it is finding the high point on an upside down parabola that hits the x-axis at 0 and 100, so the max is obviously in the middle at x=50, and only a person with very weak math intuition would use calculus to do the problem.

in my view it is a sign of intelligence to use only the appropriate level of technique for each problem. tests are timed affairs, and over answering easy problems takes time you may need on hard ones. but in any case the rules of procedure should always be clearly stated.

this reminds me of a test where the prof asked us to prove a certain map of the sphere to itself was not null homotopic. my friend developed the theory of degree of a map and calculated the exact degree. i on the other hand merely showed the degree was not zero, without computing it exactly, and deduced the map was not null homotopic. i thought my roommate was over answering the question and wasting time, even showing he did not appreciate what was really needed, but maybe some people would give him more points for the unnecessary parts of his answer since they did show a thorough grasp of the subject.

 

[vela]

it takes a while to realize that some of your students are actively trying not learn anything.

I think you've summed up why teaching is not so straightforward in that simple statement. ;) Your best laid plans for instruction can go horribly awry when students can't be bothered to think about the material for more than a few seconds.

In my opinion if you make these, very reasonable, assumptions then you have to tell them to the students before penalizing them for violating them. you cannot expect your students to know what you think the purpose of the test is unless you say so.

I agree that you have to tell students what your expectations are. In fact, I explain this point about showing work on the syllabus; it's one of the points on the syllabus I go over in class; it's a point I bring up before they turn in their assignments; it's a point I bring up after giving them 0s for not showing work, and it's part of the instructions on every exam. Also, in my experience as a student, you're constantly told to show your work in physics and math courses. A student claiming they didn't know they had to show work is being either incredibly dense or disingenuous.

they are not mind readers.

Neither am I. I can only go by what they write down. If they don't explain what they did, how am I supposed to assess their understanding?

 

[mathwonk]

ah yes, the syllabus. once i handed out the syllabus with an explicit requirement at the bottom saying: "if you read this, email me immediately". after one week i had received exactly one email. apparently no one actually looks at the syllabus in my world.

 

[Ben Niehoff]

I'd give the students zeros who just wrote down the 50 and 50. The point of a test isn't simply for students to write down the right answers. It's for students to demonstrate that they know what they're doing. If they can't or don't explain their reasoning, there's no reason to conclude that they did anything more than make a lucky guess or memorized the answer from the homework.

I agree that you have to tell students what your expectations are. In fact, I explain this point about showing work on the syllabus; it's one of the points on the syllabus I go over in class; it's a point I bring up before they turn in their assignments; it's a point I bring up after giving them 0s for not showing work, and it's part of the instructions on every exam. Also, in my experience as a student, you're constantly told to show your work in physics and math courses. A student claiming they didn't know they had to show work is being either incredibly dense or disingenuous.

Then you would certainly have earned my ire if I were your student. In what you write here, you've repeatedly assumed that everyone has the same understanding of what "show work" means. Sometimes a question is so obvious that one cannot envision breaking it into simpler steps. If you were asked "86+27=?" on a calculus exam, would it ever cross your mind that the examiner intended you to write out something like

 1
 86
+27
---
113

even if the instructions did say "show work"? On a calculus exam?

We have focused mostly in this thread on the fact that some questions which are obvious to the teacher are sometimes surprisingly un-obvious to the students. But let's not forget that the opposite also occurs: That a question the teacher thought was of the appropriate difficulty (especially taking into account the previous fact) might turn out to be so bleedingly obvious to some students, that it seems silly to "show work".

And as I mentioned, people also have a different understanding of what "show work" means. We can all agree it means "Break the solution of the problem into atomic steps, and write out those steps". But what are the appropriate atomic steps? What counted as atomic in arithmetic shouldn't be necessary to write out in calculus, or even in algebra. But even in the same level, different students will have a different appreciation of what is a "unit of problem solving", and if your grading system relies giving points for seeing specific steps written out, then you are unfairly penalizing students who can think for themselves (and you're also expecting people to read your mind!).

My philosophy, on the 50+50 question, would be this: Is the answer right? If so, and there is no work, do I have reason to believe the student was cheating? If not, then full points. Maybe the student thought the question was more obvious than I did. Maybe the student had an algebra class in high school where the teacher constantly repeated something like "The product of numbers with a fixed sum is highest when those numbers are least different", and has internalized that fact as atomic. (That is not nearly as silly as you might think...a lot of math teaching focuses too much on learning things by rote.)

If you think my above example is too unlikely to appear on a calculus exam, then consider this sort of example, on an algebra exam, that better illustrates what I'm talking about:

Solve for x:
$$36 x + 41 = 113$$
Would you penalize a student who just wrote $x=2$? If not, then what would you expect? Would it be good enough for them to write "It works when I plug it in"? If that doesn't count, would it be ok to write

$$36 \times 2 + 41 = 72 + 41 = 113$$
? Or would you insist on using a method that doesn't rely on guessing the answer*? Would it be enough to write

$$x = \frac{72}{36} = 2$$
? Or would you need to see

$$x = \frac{113 - 41}{36} = \frac{72}{36} = 2$$
? Or would you insist on

$$\begin{align*} 36 x &= 113 - 41 \\ 36 x &= 72 \\ x &= \frac{72}{36} \\ x &= 2 \end{align*}$$
? Should the student also have to write out long division to find 72/36? Should the student also have to write out long subtraction for 113-41? What counts as "show work" and what doesn't? Consider also that the exam is timed, and most students will be looking for ways to be as efficient as possible.

If your answer is "I expect the students to regurgitate the exact sequence of steps that I used on the board when I solved similar problems in class", then you are teaching recipes, not mathematics.

* Oh man, guessing the answer and showing that it works is such a ridiculously valuable skill in mathematics, I really hope you don't kill off students' developing intuitions by penalizing such answers.

 

[Ben Niehoff]

A slight addendum:

Part of the problem here is the wording of questions like "Solve for x". It sounds like "Hey, this guy needs to know what x is, and wants me to tell him. I can see that x is 2. There you go!"

It might be clearer to write the problem statement as "Show how to find x". But you would have to accept any response that does that, even if it doesn't follow the methods you taught. It should be totally valid to answer "x=2 because it works and the answer is unique".

 

[Choppy]

I don't know, I think a directive to explicitly show one's work is a reasonable request.

I agree that there is a certain amount of subjectivity to it. And a lot can depend on the level of the course. In Ben's algebra example I would have different expectations for an eighth grade class where basic algebra is being introduced compared to a first year calculus class where the students should have lots of experience with that kind of manipulation and may skip some more obvious steps to save time on an exam.

The guiding principle I have as an instructor is whether I can follow the pattern of thought from what is written down. I tend to offer the benefit of the doubt to students when I'm on the fence about something. But sometimes the student has just skipped too much to award full points.

 

[micromass]

It should be totally valid to answer "x=2 because it works and the answer is unique".

I would totally accept this answer if you also give a reason why the answer is unique.

 

[Choppy]

I had to memmorize the first paragraph of the declaration of independence in 6th grade. Do I remember it now? No. Did I never need to remember it for anything beyond getting a quiz grade? No.

The actual text may not have been all that useful for you* but I suspect that wasn't the real point of the exercise. The point was likely an exercise in memorization itself. This is a very important skill to have, or at least the exercise is an important experience to have because it can help a student to gauge how difficult it will be to commit different things to memory later on when they are more critically important.

*I think there's also a strong argument to be made that not just the main ideas, but the specific wording in the Declaration of Independence is very important, particularly if you happen to be American.

 

[vela]

If your answer is "I expect the students to regurgitate the exact sequence of steps that I used on the board when I solved similar problems in class", then you are teaching recipes, not mathematics.

Choppy described my philosophy perfectly. I expect students to be able to articulate their thought process. If they simply write down an answer without explaining how they arrived at it, I don't know what their thought process was, so they don't get credit. As I said in an earlier post, they need to demonstrate they know what they're doing rather than simply write down an answer. If they figure out a clever way to arrive at the answer, great, as long as it's correct.

Being able to communicate their ideas to others is a skill that students need to develop. What level of detail they need to go into is a matter of knowing their audience, which, in this case, is the grader. I don't think that just because students are taking a physics or math class, this is a facet of their education that should be ignored.

 

[mathwonk]

i once had a linear algebra test in which i was asked to find a maximal orthonormal subset for the pairing <f,g> = f(1)g(1)+f(0)g(0), defined on the space of polynomials of degree ≤ 2. My complete answer: {x, 1-x}. I initially got a zero, since the prof expected me to have to apply the gram schmidt orthonormalization process to the standard basis {1,x,x^2}, and thus obtain a much less elegant answer. when i pointed out my answer was indeed correct, he reluctantly gave me full points, raising my grade from D to B. I offer this as another example that what is obvious to one person is less so to another.

I was able to articulate my reasoning, but normally I only do this when asked to do so. In this case it is obvious that one wants functions that equal 1 and 0, respectively at 1 and 0, or at 0 and 1. they are obvious. moreover since only two terms are involved in the pairing one should not expect three orthonormal functions to exist. (this was a fuzzy feeling i had, due to my ingrained math intuition, which it seems was correct.)

I admit that being lazy, as well as arrogant, I always exploited the inability of professors to find problems that really required knowing the theory to be solved. I.e. I prided myself on being able to solve problems from scratch without knowing the material. Thus I was actively engaged in the sort of recalcitrant behavior I deplored as a teacher, of refusing to learn material I felt I personally had no need for, due to my above average ability (in my own benighted opinion).

The foolishness of this attitude only slowly made inroads in my attitude, until I eventually became the epitome of a hard working student when I realized I was a bush leaguer compared to the real stars of my subject and needed every advantage I could acquire. Unfortunately it was somewhat too late by the time I began to study, in my late 20's, to catch up to the smarter harder working members of the profession. Still it was fun, and left me with a lifelong commitment to convince other lazybones students to give themselves more of a chance to succeed.

 

[mathwonk]

in relation to my last post, i remark that your grading criteria may also differ if you are a research mathematician rather than not. i.e. a research mathematician may be looking for signs that the student can solve problems creatively rather than just memorize canned material. this may result in assigning points for an original solution even when it is clear the student has not learned the given topic. the idea is that creative people are more valuable in research than others and we want to encourage them. of course as i implied later, even creative people are sometimes benefited by knowing what has gone before. so we ask ourselves, are we trying to measure how much of the stuff in this course X has learned, or how much can X accomplish in the subject if given free rein? I admit to a bit more of the latter attitude due to my own experience as (at least semi-) creative goof off.

 

[mathwonk]

i guess after reading all these, i am led again to emphasize that we all have different priorities in grading and it is only fair to share those with the students so they can aim for them. at Harvard in 1960, they oriented us partly by teaching what to expect in grading at that school. it was eye opening to me. on one example, student #1 responded to a reading comprehension question by quoting all the salient facts from the paragraph. student 2 rambled a bit it seemed to me and was less comprehensive in his recitation. so i gave an A to #1 and a B to #2. then they revealed the harvard grades, #1 had not actually paid any attention to answering the specific question asked and had merely regurgitated the facts, so he got a C. #2 had done a better job of grappling with the actual question posed in light of the facts given so got a better grade, a B. I was nonplussed at the high level of grading, i.e. a B was considered good, as well as the requirement to actually read understand and address the given question, i.e. a student who knew every fact could be on the verge of a D!

i still stumbled on my first essay paper in philosophy when the prof asked us merely to "summarize the argument in Plato's Republic". When I did so, I got one of those "C-, lucky it wasn't a D" grades. When the 38 out of 40 of us who got C's, complained that we had done exactly what we were asked to do, we were told "at Harvard, it is never enough to just summarize the facts. You must always give some reasoned interpretation of them as well. The section man is bored reading the papers otherwise."

I have never again experienced an environment where grading was as capricious and high handed as there, but never have experienced a place either where performance was so high level. Unfortunately for my students, or perhaps not, that experience may have made me a bit harder grader later in life than most other people. I was educated to believe that grading should be a tool to help one improve ones skill level. I.e. every student, no matter how strong, should be encouraged to leave the course on a higher level than that on which she/he entered.

 

[Ben Niehoff]

Ironically, the most capricious grading system I have experienced came from the math department at my undergrad. Final grades (A, B, C, etc.) would be assigned by the following process:

1. The final exam would be graded without a curve.

2. From the final exam grading, one would keep a tally of how many earned A's, B's, etc.

3. The students would then be ranked according to the total points earned in the course (including the final exam, plus homeworks and midterm exams).

4. Starting at the top of the ranking list, grades would be assigned in the number that had been tallied from the final exam. So if 3 students earned an A on the final, then the 3 top-ranking students would get an A in the course; the next n students would be a B, etc.

The premise was to create an extra incentive for everyone to do well on the final exam. So if 98 out of 100 people got an A on the final, then 98 out of 100 people would get an A in the course.

In real life, it usually had the opposite effect. The class size was around 30, so not a good statistical sample to begin with. And usually only a few people would get an A on the final. I had a friend who earned a 94% in his math class and was given a C as his final grade, because he was ranked 5th and there were only 2 A's and 2 B's on the final exam.

 

[mathwonk]

In calculus I used to grade roughly as follows: "your grade will be no lower than that given by this formula: 15% HW, 60% test average, 25% final exam". But I counted the HW only if it helped raise the grade. I.e. I used 75% test average if that gave a higher grade. Also in counting the test average, I threw out the lowest score among 4 tests. Also I just used the grade earned on the final exam if that were higher than the weighted average. I.e. I calculated three separate grades for every student, either 100% final exam, or 25% final + 75% average of best 3 tests, or the 3 part formula above including HW, and gave them the highest of those three. Oh yes, in calculating the HW grade I threw out several low scores. I was still considered one of the toughest graders. Even with this (to me) rather generous formula, sometimes there were few or no A's. There were also few office visits until literally the day of the test, and few questions in class of any kind. Near the end of my career, I hypothesized that most students were too timid to come to my office and began to schedule problem sessions in our regular classroom in the afternoons, carefully scheduling several at different times so everyone could attend one. This helped some but doubled my own workload to compensate for refusal of students to take advantage of office hours. In more advanced classes like abstract algebra my grades came closer to a curve system, essentially by giving higher grades than earned so as to have more honor grades. I would group the scores into natural groups and give higher grades to the higher groups, but without any limit on how many of each grade. So a curve just meant if the highest score was 80, then 70-80 was probably an A. And a good individual answer could lift a grade above that indicated by the overall score, since if the goal is to teach proof, rather than a specific syllabus, you try to give credit for even one good proof. I never used a curve to reduce the number of high grades, as that was not a problem.