Efficient ways to grade proof-based math courses

Question

My school employs undergradates graders for most of the undergraduate math courses, and this year I have been tasked with grading an intro to proof-based linear algebra course. This is students' second exposure to proofs after the $\epsilon$-$\delta$ intro to analysis sequence, so unfortunately they don't know how to do anything but fill the blanks in an $\epsilon$-$\delta$ template, i.e. they haven't thought through a logical proof really. This makes grading a bit tough since there are around 30 students writing horribly long, incorrect proofs for what could be 2 line proofs.

My question is, how can I grade the homework assignments more efficiently? I've tried the 'one problem at a time' method where I grade one problem on every single sheet and repeat, and I've tried the 'one sheet at a time' where I go linearly and grade every single problem on a single set and do this for each student (looks like Fubini's theorem applies here :p) I'm interested in other graders' routines.

Answer 1

For a small class (with, say, ten students), it's easy to sort the papers from "poorest" to "best" answers. Their scores should thus go from lowest to highest. For larger classes (like yours with 30 students), it may be difficult to compare all of them at the same time. You might want to do the following:

If the students do mostly good work, then I would use a "subtract points" method, where the score is initially a perfect score, then every error gets a corresponding deduction in points.

If the students do mostly poor work, then I would use an "add points" method, where the score is initially zero, then every correct idea gets a corresponding addition in points.

If the students do work that is in between good and poor, then you might need to use a combination of both methods. This can lead to problems: Say the perfect score is 10, and the student made errors worth a total of 6 points, but showed correct ideas worth a total of 6 points. So is the score 4 or 6? For cases like these, I would tend to give the average of the two scores (in this case, 5).

Answer 2

I often use a variation of Joel Reyes Noche's proposition, where for each question I first rank the answer into one of two states: "at best mostly inadequate" and "at least mostly adequate".

For the former, I see which ideas can be worth partial points, if any, and never let these partial points exceed one third of the points for the question. If too blatant error are also present, they may induce deduction and a question with a start of an idea but a false reasoning could get zero. As soon as I see enough blatant errors, I set the mark for the question to zero and pass to the next one.

For the latter, I remove partial points for small errors, usually (but not always) retaining at least two-third of the points the question is worth. In some cases, there are enough errors that I don't want to give more than two third, but enough ideas that I don't want to give less, and I am done.

This is crude, as summing points for independent questions is not really a sound grading scheme. I sometimes have a separate note for quality of writing, so that I can boost students who did few questions but did them well (I prefer to value quality over speed).

If it is a final exam and the student do not see their paper without the grader being present, I refrain to annotate the paper too much as this costs time. If it is a test that I turn back to student, I annotate a lot to help them see what they need to improve.

I also have given once a final exam where the first exercise was a quizz with 6 question giving a multiplying note applied to the rest of the paper. At 2 correct answers or below, the multiplicative coefficient was zero, which speeds up the grading quite a bit. I kept this part very elementary so that I would not mind at all failing hard a student not answering at least half of them.

Answer 3

When giving grades to exams I do the following, assuming there is a large number of exams:

1. Solve the problems, preferably in writing (that can be later shared with the students).
2. Process one question at a time.
3. Read through enough answers so that I am no longer surprised by them.
4. Read all the answers in detail. Write comments and mark errors while reading. Give zero points or full points where obvious.
5. Create a rudimentary grading scheme for the most common types of answers.
6. Go through the answers and grade them according to the scheme. Improve the grading scheme as you go by adding notes such specific examples of what gave how many points, and by writing down how many points you gave to one of the strange answers. Postpone any answers that are difficult to grade.
7. Go through the remaining difficult answers that don't fit nicely into your grading scheme.

The rudimentary grading scheme is specific to a given type of answer (There often are two or three styles of answers to any given question.), though they should of course be comparable in that answers showing equivalent understanding should get an equal number of points. I typically separate a number of key points (steps in a proof, common errors that are not committed) and assign points for each. I don't otherwise remove points from errors, but do set a maximum of 5 points out of 6 if the answer contains blatant nonsense.

For example, if the task was to solve the inequality $|x-1| < 2$, different types of answers might be using the definition of absolute value, solving $-2 < x-1 < 2$, drawing a picture and using the geometric interpretation, and simply ignoring the absolute value and instead solving $x-1 < 2$. The final method of answering is a special case of earlier ones, and thus does not require different scheme for grading.

For the first method, the preliminary grading might be: 2 points for using the definition of absolute value correctly, 2 points for solving both of the inequalities, 2 points for presenting a coherent and correct answer. The geometric approach might be worth at most 3 or 4 points if executed correctly (since it is not rigorous, but does lead to a correct answer and does demonstrate understanding).