At the top of every paper I intend to grade is the sentence :
"You must show enough work so that I can reproduce your results."
I have found that this phrase solves a lot of practical problems about how much work a student needs to show. It also helps when the students solves a problem in an unexpected way. It allows a student to use technology intelligently as long as I am given enough information so that I can get the same result using the same technology. It enables me to effectivly evaluate the error a student has made and give the appropriate amount of credit for the work.
This leads to another aspect of grading a student's work that developed carefully over the thousands of problems I graded. I give points for correct mathematics. I do not take off points. When a student looks at the number of points earned for a particular problem, the student will see a +10, not a -2. The students got the 10 for doing several things correctly that would have lead to the correct answer. Unfortunately, the student made an arithmetic error when computing part of the answer and so did not earn the 2 points allotted for determining the correct answer. It should be noted that one consequence of this grading policy is that a bald answer without supporting work will get 2 of 12 possible points.
Consequently, it takes a long time to grade a set of tests. The effects, however, make it worth the effort. I have learned a lot of mathematics by following the work of a student who took an unexpected path. But more importantly, grading lots of papers teaches the grader what sort of misunderstandings students have, and that in turn enables the grader to try to find ways to eliminate thsoe errors next time around. At the very least, one learns to warn students about a mistake students often make on a particular kind of problem. At the most, the teacher can modify the problems used to teach the concept so the class will make the mistakes early on, exposing and hopefully eliminating the potential for those mistakes to occur.
Another consequence of grading this way is that the grader learns a lot about the particular habits of mind of the student. This information in turn is very helpful when the student or the parent wishes to know what can be done to improve. The teacher is aware of a lack of organization, or not checking work,or poor computation skills, or careless marking of a diagram or any of the other habits that interfere with success and can communicate those habits to the students and parent.
Comments to the student congratulating a clever move will have more impact than criticism about a bad move. A teacher can take the opportunity to point out specifically to a student what might have resolved the error. The feedback is up-close and personal and has impact.
But perhaps the most important aspect of grading papers this way is that it conveys a sense of value to students. It implicitly tells them that the important part of mathematical work is the process. They will get points for failed attempts if those attempts are appropriate and reasonable. They will get more points for a clever observation than for remembering a few steps from a previous problem. The students will learn that mathematics is about logical reasoning and making connections more than mathematics is about remembering rules and following them carefully. And that is a big deal.