Partial Credit

    I was motivated to draft this statement after witnessing the many errors that are prevalent in the many exams I've graded over the years, and then hearing the many requests for re-grades and more partial credit after the fact. I believe that it is crucial that everyone understand the type of partial credit I award and the manner in which mistakes, whether they be algebraic, computational, or conceptual, will affect your score. While this is a statement of my personal grading philosophy, and therefore applicable only to my students, I think any student of mathematics would be well-served by heeding the warnings and advice in this statement. My grading philosophy is remarkably similar to that of many other professors as well.

The difficulties of partial credit

    The first thing to understand about partial credit is that earning it is a privilege. Because of its ubiquity in college math courses, students are perhaps left with the impression that they have an inalienable right to partial credit. Whenever you are tempted by this thought, keep in mind the alternative: all or nothing credit.

    When it comes right down to it, partial credit is really the only fair way to grade mathematics. But it is not without its difficulties.

    One of the hardest tasks for a professor (or TA or other type of grader) is grading exams in a fair and consistent manner. Certain sacrifices have to be made. Due to time constraints it is usually impossible to inspect every line of every question meticulously to find mistakes.

    Additionally, there is a reasonable expectation that once students have reached the mid- to upper-level courses in mathematics, they should have mastered the basics of algebra, trigonometry, and elementary calculus. Of course, we all commit simple errors, especially in a pressure situation such as an exam. This is an understandable and expected behavior, but we must learn to overcome it. In a perfect world, test scores would reflect a student’s knowledge of content and concept, not routine calculation; nevertheless, there is no such thing as a perfect world, and there is no such thing as a math problem that doesn’t require intermediary computations.

    One difficulty is that many of these "basics" to which I refer above are not really that basic at all. Some of the algebraic manipulations we think we understand turn out to be quite subtle and often elude us. Take, for example, a situation in which one takes the square root of both sides of an equation. How many of us forget that we need the positive and  negative square roots? This should be a rudimentary concept, but quite frankly, very few of us had teachers through junior high and high school who focused on such issues. Perhaps it was viewed as too trivial. Perhaps we only encountered problems where the negative square root led to a nonsense solution, so we got into the habit of disregarding it. Whatever the case, we must learn it now. For many of these kind of errors, I recommend a website by a former professor of mine which deals extensively with such "common errors" (The Most Common Errors in Undergraduate Mathematics). Please read it and practice any concepts that are unclear. If a problem's solution fundamentally depends on obtaining both roots, then I can guarantee that very little partial credit will be awarded if one of the roots is ignored.

    So what about "less offensive" mistakes, like accidentally changing a positive sign to negative, or calculating 4+4 on one line, and then writing 7 on the next? Students are shocked when they only receive, say, two points out of five for a tiny error made somewhere in the midst of intense calculations. Again, I remind you that graders are humans with serious time constraints. Allow me to explain the difficulty of this situation for a grader.

    Suppose that I am grading a problem. I look at the top of the paper and see that the student has set up the equation correctly, but when I look at the bottom of the paper, I see the wrong answer. If my goal were to award the maximum possible partial credit, I would first have to follow every computation, line by line, until I spotted an error. That might be easy enough; since I have already computed the answer for myself, I might be able to see quickly where the student went wrong. But this is not enough. I then have to suspend my knowledge of the correct answer and follow through on the calculations for the rest of the page under the assumptions of an incorrect intermediate step. You see, the claim that is often made when students request more credit is that, despite having made a small error, the rest of their work is correct. I, as the grader, then have to verify that indeed the answer obtained is correct given the wrong number inserted partway through the calculation. If more than one error is present, the work in doing this is further increased.

    I cannot as a grader spend all my time figuring out where small errors are made and then following them through the remainder of the calculation. I could, if I spotted them easily, award more partial credit. But if I don’t happen to spot the minor error, then I would have to award minimal partial credit and pass quickly to the next exam. This seems less fair to me because it depends on luck, namely, whether you were lucky enough for me to spot the error quickly enough. So I must be consistent and award minimal partial credit for all such mistakes.

    This is nothing to be upset about. If everyone receives the same partial credit, albeit low, for the same mistakes, no grades really suffer. The averages are lower overall and then that means that the curve is lower overall, and that means that your final grade can only improve. The only case in which partial credit can reasonably be disputed, then, is when two people receive a different number of points for the exact same work. Naturally it is difficult, if not impossible, to be completely accurate, especially over a large number of tests, so these types of inconsistencies will inevitably crop up. However, no professor I know is unwilling to re-grade a problem under such circumstances.

How to (and how not to) earn more partial credit

    Over the years we have been taught that when we don’t know the answer, we should just write as much down as we know and hope to get some partial credit out of it. This may be suitable for high school. (In fact, A.P. study guides thrive on this very principle.) In college, though, this is increasingly frowned upon, especially in upper-level courses where proof and technique are emphasized. If a problem requires you to find a derivative and you write down a list of derivative rules because you think they're related, should you receive some partial credit? Only under very rare circumstances. Indeed, even if you write down the definition of the derivative itself, you still have not demonstrated its application to the problem at hand. (Very rarely do we actually use the definition of the derivative to find derivatives.) Therefore, very little or no credit will be awarded for just having the right concepts on the paper somewhere. A minimal requirement for partial credit in a situation like this is that you correctly identify the specific rule that applies. At the very least, this demonstrates an understanding of the rules and equations you memorize. (A better idea is not to memorize rules and equations at all, but to understand how to derive and apply them. . . but I digress.)

    Speaking of things just appearing on the paper somewhere, we often hear the complaint that the correct answer was written on the page in the midst of some calculation somewhere and yet credit was not awarded. My thought is that if you come across the right answer, but then you go on and do other calculations and find some other answer, it is clear that you do not consciously realize you have found the right answer. An enormous part of getting the right answer is realizing that what you find is indeed the answer. To this end, circle your answer when you think you have it and demonstrate on your paper the logical progression of ideas that leads to that answer. Give your answer in the form that the problem requests. Don’t do subsidiary but important calculations on the back of the paper or in the margin. What do I mean by "subsidiary but important"? If you just need to multiply two numbers together to get to the next step in solving an equation, fine. Do it on your calculator or in the margin or on a piece of scratch paper or wherever. But, if you’re plugging in values to get your final answer, show your work in plugging in those values, no matter how mundane it seems. This way the grader can see how you went from an expression with a bunch of variables, to an answer with actual numbers in it.

    Finally, use English. (I don't mean use English as opposed to some other foreign language, although I did grade an exam once with notes in Japanese all over it). By this I mean use words and sentences to describe what you're doing. I know this is a math course, but you are far more likely to receive partial credit if you stop every so often and explain why you are performing the various calculations on the road to the answer. If you are finding the critical points of a function, explain why your method is the right one for finding critical points. Then, when you are finished finding them, explain why you know you aren’t missing any. And do all of this in full English sentences. Trust me, it’s worth the few extra seconds you take to organize your work. I reserve the right to deduct points for sloppy work whether you have the right answer or not. If I don’t know how you got it, why should I believe it? Of course, if you get the right answer, there’s a reasonable assumption that you did it correctly, even if you’re work is sloppy or incomplete. But if you get the wrong answer, don’t expect any partial credit at all.

    After your tests are returned, please assess your work in light of these observations and consider where you think you deserve more partial credit. Try to understand from the above discussion why you didn’t receive that partial credit and do better on the next test. The last thing professors want to do is bicker with students and quibble over one or two points, so in the spirit of maintaining a good relationship between professors and students, please be careful about requesting re-grades. As I said earlier, no professor is unwilling to correct grading errors, but very rarely is it a "mistake" when you don't get as much credit as you think you deserve. To give you an indication, out of the thousands of tests I have graded in the past, I can count on one hand the number of times I've changed someone's score due to a re-grade request. (I'm not counting the times when I have misread a solution and made a mistake, but I try not to let this happen very often either.) You would be better served by identifying the mistakes you made and learning how to do the problem correctly. I'm more than happy to answer your questions if you're unsure of where you made your error, and I am far happier to do it if your request is not prefaced with, "You didn't give me enough partial credit on this problem."

    I leave you with one final remark. A number of years ago, an unmanned vehicle crashed on the surface of Mars and ruined a potentially important scientific mission. Do you think the scientists involved in that mission will be remembered for their many advanced degrees or their many years of experience in NASA? Will they go down in history for the remarkable achievement of designing a functional Mars lander that traveled thousands of miles through space? No. They will be known for the fact that the Mars lander crashed because the two groups of scientists working on the mission were independently using the American and the metric system of measurements, without the other knowing. A "less offensive" error involving a "trivial" difference in units of measurement resulted in the destruction of millions of dollars of space research. I realize that your exams hold no such dire consequences, but it is an important lesson learned.

Sean Raleigh