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

[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.