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Due to unknown reason I was assigned as a grader this semester for a certain proof writing class in my universtiy. The class size is 65 and students took it usually comes from a computer science background. The result is horrible as I often graded them with marks of $2,3$ out of $20$.

The class average is usually $8$ out of $20$ not considering students who do not hand in homework. And if I account students who did not hand in homework with students scored $5$ or less, then usually this constitutes to $25$ or more students. So right now I do not really know what to do.

The reason they are performing really badly is because most of them has no proof writing background. As a result when asked to "show" some result is right, most of them just write hunchs crossing their mind and did not bother to justify their claims. The simplest example might be the professor showed for a graph of order $n$, size $m$, if $m\ge 3n-6$ then it is not planar. When the homework was hand in, I found the majority of the students under cooperation took this as a criterion to judge whether the graph is planar. Then they omitted obvious non-planar cases.

I am wondering what I can do. Grading takes me a lot of time and it is hard to spend any more time in writing out why they are wrong, or having regular office hours. But on the other hand I doubt if they really learned anything in the class in an organized way given their performance in the homework. So some help must be needed. I feel the situation to be quite puzzling because at here there are no review sessions, no discussion groups, and most students cram in homework in a few hours without much studying. I do not know what I can do to improve their performance.

It might be that because I am from a foreign country and my view towards US education is biased. Maybe students are expected to be this bad. But I did my undergraduate studies at US, and most of my classmates know how to write correct mathematical proofs despite they may have trouble in mastering advanced mathematical material or doing research. It might be that most of the students are CS majors, and as non-math major students they do not know how to prove something other than checking a few cases. But during the class the professor covered truth table, etc in the first few weeks. So I thought the basics are there.

I thought about inflating the grades on my part, but was dissuaded by fellow post-docs suggesting they should get what they deserved. However I had been an undergraduate and I know low grade hurts people. What can I do? I asked them to seek help in mathstackexchange, but I doubt if they are even willing to learn latex since they usually do not even bother to read textbook carefully.

My grading policy is as folllows:

4 for completely correct solution with an acceptable proof.

3 for completely correct solution with a flawed proof, and in some case without a proof but showed substantial work.

2 for correct or nearly correct solution without a proof or noticeable work

1 for half correct solutions, or ideas not quite correct but makes some sense.

0 for totally incorrect solutions, or confused with problem definition, wrong problem, wrong strategy, wrong use of the theorem, complete nonsense, etc.

update: I really do not know what to do. The student accused me for not explaining grading policy clearly. She thought finishing a math problem with a proof needs to be explained; She also seemed to be much more interested in grades instead of the course content, as she claimed the 10% homework grade is a lot. The message below was met with the response that "However, it is not my job to share those views. I think it would be better if you email your grading guidelines and any other relevant announcements to the professor to send out to the class." I felt at lost with her utilitarian attitude.

Also I finally got the professor's response as I forward the conversation to him. Instead of giving any suggestions on what we can do to improve student's understanding of mathematics, he boldly suggested:

"I don't think you need to spend so much time writing to a student in the class. A few sentences should suffice. Save your time for grading"

I think I have to give up dreaming I may change something as a grader.

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Please don't inflate their grades. Give them what they deserve, perhaps draft up complete solutions so that they see what correct work looks like. Most importantly, ask the professor. Finally, I would like to say that I think this may be more on topic at academia.se. –  mixedmath Mar 12 '13 at 6:23
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@mixedmath: I would hope the professor can help, but all he did is to hand in me the instructor's solution manual and ask me to finish grading as soon as possible. For the students he asked them to work 25 hours a week instead of 14 hours, not to use electronic devices, etc. I do not really know if this helps. –  Bombyx mori Mar 12 '13 at 6:30
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$65$ students in a proof-writing class? That’s obscene. I always wrote copious comments when I graded that kind of material $-$ the feedback is far more important than the score $-$ and I could barely keep up with a class a third that size turning in homework perhaps three times in two weeks. I think that you’re doing the right thing and should continue, with the addition suggested by @mixedmath of writing up model solutions. –  Brian M. Scott Mar 12 '13 at 15:11
    
@BrianM.Scott: Grading itself takes me about 20+ hours every week. I do not really have time to type three clean solution sheets for their homework problems. Neither do I have time to give them detailed feedback. If I am teaching the class the situation would be different, but I am not. –  Bombyx mori Mar 12 '13 at 17:24
    
I may be biased here but 20+ hours of grading sounds rather obscene to me. It sounds like you are the only grader for the class and 65 students is quite a lot for one person. You should maybe ask the department to get at least one other grader. Your contract probably says try expect 9-10 hours tops. Again, I speak only from my experiences. –  Alex R. Mar 13 '13 at 4:03
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1 Answer

I wrote the following to the students. I do not really know if it helps.

To be honest giving low grades is not a psychologically pleasant experience, and I assume neither is that of receiving one.

I understand that you were unhappy that I had been rather strict in grading on rigorous proof writing, because the majority of the students in the class did not know how to write standard proofs in the first place. However, the important thing you learned in the class is not the statement of the theorems or the homework problems you solved, but the skill of thinking logically in an organized way so that you can convince yourself as well others that your opinion is correct or at least worth considering.

I think this is part of the ideals of higher education, that everyone regardless of initial background is provided with enough intellectual support such that he or she can learn the essential philosophy of any subject. Maybe this is not so easily to be carried out in real life - when I was an undergraduate I often felt being challenged in photography, art history, anthropology, and sociology. But my experience in learning the "essential philosophy" of these subjects helped me immeasurably once I graduated from college. In mathematics, the principle of giving logical, complete proof is one of such essential philosophies.

The problems in the homework problems sets are all relatively easy, yet can be time consuming and tricky to write out the solutions. So I hope students may regard this as part of their education goals at here instead of treating it as purely part of the required grading process, even if they choose it simply to satisfy distribution or graduation requirements in the beginning. One of the best way to train yourself is to read the textbook carefully and emulate the author's writing style in his proofs of the theorems and various examples. If you can appreciate how he/she avoided making mistakes, you will be better at eliminating your own mistakes as well. To do that you may attempt to prove all significant statements in the textbook by yourself from the definition and see how it differs from the author's.

I understand that students have to prioritize and homework usually has a low priority; when I was in college I used to take 7 graduate level classes in one semester and barely had time to finish all the assignments. But I cannot inflate the grades because the net effect is the same and no one really benefited from it in the end. Also, in the real world your performance will not evaluated by a grader anymore. This is the message I wish you can share with others in the class. I regret I do not have time to set up regular office hours to help out your difficulties.

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This is a good message. I really liked its content. –  Rustyn Mar 14 '13 at 5:17
    
I don't really know as I have Aspegers syndrome. But I hope it might help. –  Bombyx mori Mar 14 '13 at 5:47
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