In which I continue to reflect on this past semester's teaching experience, for future reference:
Late policies: I used to be of the school of thought uninterested in dealing with assignments turned in late. Papers and take-home exams were due when they were due, no exceptions permitted. I've come to think this is not the best policy, for two reasons. First, there are always exceptions: most schools have policies that require exceptions to be made in some exceptional circumstances, and, even if not, there is the occasional situation in which an exemption seems appropriate. It also seems, from my perspective, needlessly vindictive: a paper one day late (or one week late) is not going to throw off my intended grading schedule: when the number of assignments to grade is small is hardly makes a meaningful difference, and when the number is large an additional one is marginal.
Instead, I've come around to the idea that one of the purposes of college education is to introduce students to the management of their own responsibility. For my freshmen, this meant informing them of their responsibilities over the semester and also that I would not be keeping track of whether they had met all those responsibilities. For the upper-level students, this meant giving them a 10%/day penalty for late work. With the freshmen, this meant that the person who did not write enough response papers over the semester received a 0 for the one missed. With the upper-level, this meant that the paper that came in three days late could get a maximum score of 70.
The penalty, when it comes, should be deserved, and accurately reflect to the student the stakes involved. As I discovered, late papers tended to fall into two categories: those that were less than a day late and those that were more than four. The less-than-a-day-late papers tended to be around the class average in terms of quality; those that were much later tended to be lower than the average. So the policy, I think, allows otherwise conscientious students to accept a minor penalty in order to deal with last-minute circumstances, and provides for others an additional, objective reason to accept a low grade: one can argue over how a paper is to be scored; one cannot so easily argue that a 70 is not a reasonable grade for a paper turned in three days late, if that's what the syllabus says.
As a result of all of this, I am inclined to keep this policy barring some new reason for changing it.