One of the things I dislike most about teaching at the small SLAC I'm at is that students are... um... very responsible and grade-conscious...?... but just as competent as any other 18-year-old around (which means "intermittently so"). I assign projects in class with the goals of grappling with open-ended problems, developing critical thinking skills, and learning to communicate mathematical ideas, and they amaze me. (See previous post on ODEs.) On the other hand, I assign regular homework and they amaze me in a bad way.
There is an astounding variety of great excuses for late work. I thought I'd dealt with that by putting three free drops in the homework grade -- three free "I drank too much! My dog got sick! I'm on a volleyball tournament! I forgot I actually have syphilis!" Three free passes, no information or justification needed! One might hope that this would cut down on the number of desperate emails with the above information. Somehow it does not. After all, my students can do some math, and they know that if they get that homework in late and get it score that free drop will be used on the lowest-scoring homework and they will get a few more fractions of a percent in their grade. Must I go to a system in which you only get the drops if you actually refrain from handing in that homework? (That would be amusing in a game-theoretic kind of way.)
Perhaps I should accept that students come in with a Pavlovian response to assigned homework, which involves the ritual of the late assignment as an essential component. Perhaps the ritual of the late assignment is a bonding act, a way of connecting in a comforting way in this scary world of college. It is certainly a way for these students to reassure themselves that they are responsible and doing the right thing: they are trying very hard to get me to let that assignment count.
Of course their priorities are ridiculously misplaced if we simply look at the numbers. With 30-some written assignments in a semester each worth 25 points and normalized to 1/7 of the final grade, each point on a written assignment is about .0013 in the ultimate reckoning. That fourth missed assignment will take you down .03 points. I look for gaps of 20 points to set my grades. To make up that difference you will need 600 assignments, or one really spectacular success or failure on a midterm. Study, young people, and do practice problems -- don't waste your time trying to convince me to accept your late assignment!