I have tried so many different things in my classes regarding homework and I have finally found strategies and policies that I am currently comfortable with. I am a strong believer in homework. I was always a good student myself when I was in high school, but I really only did my homework because it was expected of me and I didn't want to disappoint. It wasn't until I got to college, where homework was optional, that I finally understood how important it was to my understanding to get some practice with different concepts, mostly because at first I wasn't doing the assigned homework and therefore wasn't doing as well on tests :(
In my classroom I grade homework based solely on completion because I want it to be based on effort. Students can earn a check plus(100), check(80), check minus(60), or zero on each homework assignment. In order to earn a check plus students must have attempted every problem they were assigned. At the beginning of the year, I usually get a lot of flack about this because if a student doesn't do even one problem, they earn a check instead of a check plus and they don't agree with losing 20 points because of one problem. My answer to this is to ask them if they deserve a 100 if they didn't do the entire assignment and they usually agree that they don't. They usually still whine (which is when I just point to my "Be a winner, not a whiner" poster!! haha), but quickly learn that they must put effort into each problem.
My faaaaaaaaaavorite part of my homework policy is how I handle students that don't quite understand the assignment, but are still trying and therefore deserve a check plus (to me it really is all about the effort!). I explain to my students at the beginning of the year that their homework assignments are meant to help them learn and that I still want it to be of value even if they were struggling. If they don't understand a part or parts of their homework assignment I ask them to write a meaningful question for each problem they struggled with (none of this "I didn't get it" or "I don't know" stuff is allowed...1) because those aren't questions and 2) because in order to help them "get it" I need to know what about "it" they don't get). This year I am going to use an idea I saw for smart questions on pinterest this summer.
I really really love this because it gets them thinking about what parts of the concept/topic they aren't understanding and serves as a reminder for them in class when we're going over the assignment what they were struggling with. Now during class, I (or one of their peers) can help to answer their questions as if I (or they) had been sitting next to them while they were doing their homework and doesn't make the assignment a complete waste for them. This also allows me to differentiate the true "I didn't get the homework" and the "I didn't get the homework" that really means "I didn't do my homework".
Other key pieces to my homework policy:
1) I do not allow homework to be handed in late. Immediately after checking their homework we go over it as a class. I explain to the kids that I'm not going to accept an assignment after they have just been given all of the answers during homework review. Sometimes this makes me feel bad for the students that would truly complete the assignment late, but I have yet to come up with a way to be able to tell who completes their assignment during homework review and who does it at a different time.
2) Once a student has missed three homework assignments in a quarter (consecutive or not) I contact home. I think this is so important because many times a parent isn't even aware that their child isn't completing their homework so I like to keep them in the loop and they are usually very grateful. I keep track of missing assignments by highlighting each missed assignment with a different color. The first zero is yellow, the second is orange, and the third is pink (my department members make fun of me because I'm so crazy about color coding things).