Grading advice
One of the painful duties of graduate students, at my school, as in most schools, is to grade undergraduate tests and homework. We do not really teach our grad students how to do this, assuming that they will learn, as we learned, by having our tests graded when we were undergraduates. The problem is, at least at my school, that graduate students, as we professors, are not like our undergraduate students. Most of our undergrad have no intention of ever becoming mathematicians or computer scientists. They, for the most part, take math classes because they are forced to. Hence, their attitude to the material is much different from ours. They do not learn the same way and we should grade them differently. Here is some advice on grading undergrad tests, especially valid for remedial classes and classes aimed at engineers, business or other students who take math classes only because they are forced to take math classes.
Grading on a test should be an accurate reflection of the student's understanding of the material. Say, for the sake of argument, that you grade a question worth 10 points. Then students who do everything perfectly should get 10. If it is not perfect it should not receive 10. A student who makes a minor conceptual mistake should get 9. An error in the proper utilization of mathematical formalism is worse (an = between non-equal elements; or a <==> where an equal should be is cause for deductions.) A minor arithmetic mistake should not affect much and the student could still get 10.
Undergrads survive on partial credit. A major mistake should be reflected in fewer points but, unless there is nothing correct in the student's answer, he/she should not get zero. Strive to find what is good, what is wrong with the answers you read and grade accordingly.
There is more to an answer than correct/incorrect. Even though we know that, deep down, an answer is correct or not, the grading is meant to help the student learn. In the end, typically, you should not be grading on a binary scale (10 or 0); you should be giving out grades in the whole range [0, 10], underlining either what is right or what is wrong in the answer. This helps the student learn and helps us distinguish between two students. (Of course, yes, there are exceptional questions where a binary outcome is acceptable.)
A wrong step does not invalidate all that follows. Be especially vigilant of questions with multiple parts. If a students makes a mistake on one part, give full credit for answers of the following parts that are consistent with the erroneous part.
Presentation counts. A complete mess, even if you think you can extract a correct answer from it, is not a perfect answer. One of the most important outcomes of a mathematics course should be clear thinking. And clear thinking is mirrored in clear writing. A mess is not worth much.
Cross out blank answers, somehow. If a student leaves a question blank or mostly blank, put a big X (or Z or whatever, but be consistent) to indicate that the place was blank. I will let you figure out why this is useful.
Be consistent! Essentially the same answer, from two different students should receive essentially the same grade (modulo 1 or 2 for presentation).