I am currently grading a problem in an exam on analysis for mathematics students. One subtask is to calculate a certain integral (The resulting term has to be as simple as possible.). For this part I am able to give up to $n$ points in essentially any way I see fit, provided that I do so consistently (the number $n$ is already fixed though).
Of course I am interested in an answer to this question in order to apply these to my specific case, but I also find the question of how to assign points fairly for questions where there are many long paths to a concrete solution interesting in general.
Some facts which may be significant:
- I am not allowed to give fractions of points.
- The exam paper does not contain any information on how the problems will be graded other than how many points are assigned to each subtask of a problem.
- There are significantly more questions in the exam than can reasonably be expected to be solved in the given time; in particular, one does not have to solve all exercises to obtain full marks.
I have thought a great deal about how to assign points in such a situation, but none of the methods seems to be satisfactory. Here are some of these possibilities:
- All or nothing: Assign $n$ points for a perfect answer and none otherwise.
Assessment: Small mistakes are excessively penalised.
The usual approach at my university to grading such questions is to record all the different routes the students have followed to carry out the calculation, then either
- partition every such route into $n$ "even" chunks, and then assign one point for every successfully completed chunk; or
- isolate $n$ "significant steps" (e.g. rewriting the integrand in such a way that it may be easily integrated), and award a point for each such step.
Assessment: This seems rather unfair to me for the following reason: There are infinitely many sequences of steps that one may perform, only some of which lead to a term which is sufficiently simple so that one may deem to have completed the calculation. Now, one only really knows that such a sequence leads to the solution when the sequence has been completed, so if a student simply performs the first few steps of such a sequence, he hasn't done anything qualitatively different from a student who has performed steps in any other sequence.
- In view of the assessment of the previous method, I could solely consider calculations which lead to a definite answer, and then penalise every mistake in the calculation by the subtraction of one point, so that one obtains $\max(0,n - m)$ points, where $m$ is the number of mistakes.
Assessment: If someone writes something trivial like $\int f(x) \, dx = 0$, he will only have made one mistake, and will thus receive $n -1$ points. Also, students aren't aware during the exam that they have to complete their calculations at any cost.
- Again in view of the assessment of method 2., I could again solely consider calculations which lead to a definite answer, determine to which correct route it corresponds the best, and then apply 2.2..
Assessment: Again, students aren't aware during the exam that they have to complete their calculations at any cost.
- Judge how much the final result differs from the correct one (by missing signs, summands etc.), and then assign points for how strongly the calculated result resembles the correct answer.
Assessment: Judging such a resemblance is highly subjective. Also, it is possible that one arrives at a solution which looks similar to the correct one by completely invalid means.
- Again in view of the assessment of method 2., one could award points to any sequence of steps which could be considered to constitute a promising attack on the problem.
Assessment: I would have to determine what constitutes a promising attack.
One final observation: I have not considered this up until now, but I could create a grading scheme where there are more than $n$ things one may write which give points, and the final number of points awarded is $\min(n,m)$, where $m$ is the number of things one has written that give points. This would allow for further, perhaps more nuanced, methods.
(P.s. This is my first question on stackexchange, so I am particularly eager to read any criticism of my question.)