There are a few schools of thought as it comes to homework that I got to see today, these are inclusive of, but not limited to, the following:
1. The Surprise Attack - Oh you forgot to do it? HAHAHAHA YOU'RE GETTING AN F. Too busy to finish because you were up late working on other homework? BOOM DOUBLE F! Some teachers use homework as an artificial way to add "rigor" to their classroom. "If they ain't workin', they ain't learnin'" is one way to look at class. How could this be helpful? It very well may encourage, albeit punativly, to work on course material outside of class. How could it hurt? Turning the kids off of learning completely because they associate it negatively and in some cases don't do the work out of spite.
2. Extra practice - Take what we learned today and practice it at home. This makes sense. The teacher gives information and the student uses that foundational knowledge to complete a project, questions, or does an activity. This is a more traditional approach, but the extra practice is a good thing. One of the drawbacks is that it's more difficult for the students to ask questions in real time about what they are struggling with. If enough students struggle with the homework then it may force a reteach causing you to "lose a day" in your instruction, but that's worth it if they end up getting it.
3. Foundation first - Students work with the content on their own, discovering the basics of what will be covered in class. This is my favorite. I like for my students to work with the material first and come in ready with questions. If they get the bare bones down before class, then we can spend class time practicing with me there to help. It will also be more fun for many of them because they'll be working instead of sitting and listening to me tell them what they need to know. Drawbacks include kids just not doing it because they know I'm going to explain most of it in class or students being overwhelmed by the new stuff that they quit half way in.
Should we grade it or not?
I use homework as a formative exercise. To help students get better and perform better on assessments. When I do grade it, it's usually for completion.
What about cheating / fake answers for credit?
I look for those things and react accordingly, but the reality is that it only hurts the students to do this to themselves. I don't take things like that personal.
Kids cheat. Did I really blow anyone's mind with that? If you are reading this in class, that kid, the one right over there. The one who you can't see all the way but you know he has his headphones in and keeps pulling out his phone? 67% chance he's cheating right in your eyeballs.
The bigger question is "why?"