This is going to look a lot like Jeffery Thompson's answer, with some slight modifications (I was working on this answer when his appeared).
First off, I agree entirely that you have to decide what you want students to get out of the homework first, then decide how you are going to grade those assignments. As I said in the answer to which you linked, I think that primary goal of homework in a lower division class is to give students a chance to practice the kinds of problems that I think are important, as this will help them to learn and master the material. Ideally, the students would be mature and engaged, and would do the homework and come to office hours to discuss problems even if no credit were given for homework.
That being said, the kinds of students who are typically in a college-level precalculus or introductory calculus class are often not mature enough to do the work without the promise of a reward. Thus, I collect the homework and assign marks, because if I don't, the students probably won't bother. This is why I give any grade at all to homework—I'm playing psychological mind games with my students.
The next question is one of time; specifically, how much time do I have to grade? In a large lower division class (100+ students), there simply isn't going to be enough time grade all of the work in detail and give meaningful feedback. I would love to comment on all of the work, but my wife would probably kill me if I spent that much time in the office. Thus I will typically choose to grade on the basis of "completion," where completion basically means that the student wrote their name on the top of the page and wrote down solutions (note: not answers, but solutions which show work) to all of the questions. Typically, I'll make each assignment worth 1 point, and either give a 1 or 0. Either it is done or not. In your examples:
- Answers, but no work: I would give no points. The answers were given. It is up to the student to do the work.
- Illegible work: I would give points on the first couple of assignments, and make the comment that it is illegible, then stop giving credit after two or three assignments have been returned.
- Wrong problems: I would give no credit. If the student doesn't do the assigned problems, then they didn't do the work, and deserve no credit.
- Poor quality or incorrect work: I would give credit, but might ask the student to come to office hours. That being said, I might not even notice that the answers are wrong, since the whole point of grading for completion is that I don't have to check their work.
- Poor techniques: I would treat this the same as 4.
- Identical work to another student: Both students get full credit. Ideally, they were working together and therefore have the same answers. More realistically, one student copied off of the other, and was therefore "cheating." However, I assume that students will (to put it generously) "collaborate" on anything outside of class—I have no control over that, and don't want to arbitrate those kinds of disputes.
That being said, I rather like Jeffery Thompson's rubric, and my start adopting something like that (0 points for blanks, 1 point for the attempt, 2 points for correct solutions).
Note that all of the above applies to the kind of homework assignments that I call "Computational Practice." These kinds of assignments (similar to Jeffery Thompson's "Homework") are generally very computational in nature, and not terribly deep. They are meant to get students to master algorithms or computational tricks. These are not far from "drill-and-kill" (though I seem to recall from my ed certification program that there is some fancy jargon for this... "deliberate practice," maybe? that has the right ring to it...), and typically involve a large number (15–20) of problems that should all take very little time (2–3 minutes each).
I would like to specifically address this point:
In fact, it seems to send a confusing and perilous signal to give students a passing grade ("you're doing okay") on work that is inherently flawed like this.
I have had concerns about that, but I try very hard to make it clear to my students what the point of the work is. An example of the exact statement on my syllabus (from a stats class last quarter) is
Homework will be assigned daily. Assignments will be due on the Monday after they are assigned, at the beginning of class. All assignments will be posted on Blackboard. The goal of the homework assignments is to give you a chance to get some practice with the kinds of problems that might appear on an exam and to build your understanding of the concepts discussed in class. However, I do not have the time to give in-depth written feedback to every student, so your work be graded on the basis of completion. Answers to odd numbered exercises are in the back of the text, and I am available during office hours or via email to discuss any problems that you might have.
I then repeat something like that every time a hand back homework: "Remember, I am just grading your homework on the basis of completion. You should not assume that, because you got credit for the work, you are in good shape in this class. The exams will be much more important. If you are having any difficulty, please see me during office hours." This, coupled with the fact that homework is worth only 5–10% of the final grade, seems to get the point across.
As an addendum, I also tend to give longer "Written Assignments," but the goal of these if different, and they are worth a larger portion of the student's grade. The assignments typically consist of two or three more involved problems, and the solutions are graded in detail, with an eye toward good mathematical writing (this is inspired by the "writing across the curriculum" mandate that was all the rage when I taught high school—the goal is to (approximately) learn how to write in mathematician-ese).
I typically give seven or eight of these written assignments in a 10 week quarter, though it depends on the size of the class—it is reasonable to grade something on the order of 50 such assignments in a week, spending 2-3 minute per; in larger classes, I would assign fewer written assignments. The assignments are given out, and students have a week to turn in a rough draft. I then offer comments, and students have another week to turn in a final draft (though, to be honest, the final drafts have a "floating" deadline). I then award points as follows:
- one point for turning in a rough draft on time
- one point for turning in a final draft on time
- one point for coming to office hours and discussing corrections to the rough draft (these three points represent the "completion" portion of the grade)
- three points for writing mathematically correct solutions
- three points for writing in legible, grammatical mathematical English, including the proper use of notation
- one "bonus" point for TeX-ing up their work.
There are fewer of these, and each contains fewer problems. Even with a large number of students, it is possible to grade a couple of stacks of these each week (particularly since much of the feedback is offloaded to office hours, which is allocated time already).