I do not enter zeros and late work receives full credit in all my courses except Calculus AB. I believe my job as a teacher is to have students learn the material and prep them for college & Career. Allowing them to take a zero does not prepare them for the world or for college. (I also believe in rework, retests, etc - but we will save that for another post.)
But not doing homework is not consequence-free....
I treat homework just like a job in the real world. If a student does not have it done at the beginning of class I simply have them stay that day after school and work some math problems with me (usually something equivalent to the homework). Since we don't grade and rarely review any homework during class time it works fine.
Now some of the arguments I typically hear:
From college professors -- that I am not preparing them for college because they think late work is acceptable. First, no -- the students know it is unacceptable; I preach that no where in the world is not doing work acceptable -- school is their job but it is different then a real job - they don't get paid and I cannot fire them.
And professors and HS teachers have different jobs! College is the opportunity to learn, in High School the learning is required. But since AP Calculus is a college course I do enter zeros but in no other courses.
From other teachers -- that this policy is not like the real world and I am not prepping them. This always makes me chuckle because for 12 years I was an engineer and manager in corporations and mid-size companies. And if you think of homework as a report, or a quote or something like that -- those things are often late in the world. School is a job - but it is not a real job (again cannot fire). What we teach them is most important and consequences need to be what the student finds not pleasant. Zeros are often not unpleasant, hanging with Mr. A after school, well you get the point. [One thing I will concede is it does make more work for me -- but after a couple of weeks it is just one or two students that seem to miss.]
If I enter zeros I allow the students to skip required practice and knowledge -- and if my homework is appropriate and reasonable (I do my best with this) then it is important.
I have a policy, that I call a Homework FAQ sheet, that I send home with students on day 1 signed by them and their parents that lets me keep them after school. (Homework FAQ).
Finally I use judgement with this, it is a policy that requires being reasonable and fair. If it was run equally accross students I would probably lose my job. But it works, six years ago I had about 20% zeros, now I have less than one percent (sometimes I enter a zero at quarter end if the grade stays the same, the student still completes the homework -- helps with grade deadlines).
I really believe our job is to educate, these students are just kids looking for the path of least resistance. Our job is to make sure the path of least resistance requires them to learn enough to be successful after High School.