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 regrades 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 wellserved 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
upperlevel 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 regrade 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 upperlevel 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 regrades. 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 regrade
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
