This is an opinion, but perhaps worth sharing:
When I assign that I expect students (particularly lower division students, such as those in calculus or precalculus classes) to do on their own time, I assume that they are going to collaborate, ask the Google, and otherwise use all of the resources that are available to them. Demanding that they do the work on their own is a fool's errand, and can be rather paternalistic. As such, I don't like to make homework a huge proportion of the final grade in a class, as I want a student's grade to reflect their individual ability and learning, rather than their ability to find the answers on the internet. I would prefer to base a student's grade on work that they do in class, under better controlled circumstances, e.g. exams or presentations (with moderately intense questioning).
That being said, most students cannot learn and internalize new material without having had a good deal of practice. Homework provides that practice. As such, homework is a vital part of class, and cannot be ignored. In an ideal world, I would assign homework, and my students would be mature and well-motivated enough to recognize that working through the problems is to their benefit. In this ideal world, it would be entirely unnecessary to grade (or even collect) this work—the effort that the students put into it would be reflected in their exam scores. Since we don't live in an ideal world, and not all students are mature or well-motivated enough to do work on their own without some kind of Pavlovian reward, I typically give some minimal credit for homework.
Returning to my ideal world, once I have assigned homework, I would collect it, and give feedback to all of my students in detail. However, if I have a large class, there are not enough hours in the day to give that kind of feedback. As such, there are a few ways that I can assign work (this is not meant to be exhaustive, but should hit the major points):
- Use a computer driven system, such as MyMathLab, WebWork, or ALEKS, then using the scores returned by the system for the homework component of my gradebook. Such a system has the advantage of requiring very little input from me (once the class is set up), and gives students instantaneous feedback. On the downside, these systems are only good for low-level drill-and-kill style questions, and tend to do a poor job of forcing students to use higher order thinking. I think that the questions also have a tendency to break down problems too much, and give students the idea that there are different problem "types" that, once mastered, give them mastery over the material.
- Assign homework with answers (but not solutions) given, and grade that on the basis completion. Here, the students know if they have obtained the correct answer (since the answers are given), and my workload is tractable, since I only have to check off the assignments. I can also give deeper questions in the homework. The disadvantage is that the students get very little feedback, unless they choose to come and discuss the problems during office hours. I have also observed that students seem not to value work that is graded on the basis of completion—they do the minimum necessary to get the points, then move on. This hurts them, come exam time.
- Assign problems with no answers given, then grade either entirely on the basis of completion, or pick a few problems on which to give feedback and give completion credit for the rest. From the point of view of giving good feedback, this is likely the best that can be done for a large class. Again, office hours are good for addressing problems. On the downside, students are only getting feedback on a few problems, and may have serious misconceptions that are never addressed. This can be somewhat ameliorated by providing solutions once the assignments are handed in.
- As you suggest, assign work but do not collect or grade it. You can then assess their homework performance on the basis of short quizzes (I wouldn't give them answers in advance, as this defeats the assessment). My strategy is to give them 5–10 minutes to answer one or two questions taken more-or-less directly from the homework. Sometimes I even let them keep their homework out to answer these questions (if they did it right, it is a simple copy-pasta). I then grade these quizzes fairly harshly, with the justification being that they had plenty of time to discusses troublesome problems during office hours.
Note that all of this is for a lower division mathematics class, where there is a often a significant requirement that students be able to perform basic computations, which may or may not require deeper understanding, but which does require a fairly large number of problems for practice (perhaps not quite drill-and-kill, but close). As courses get more abstract and the students become more mature (both mathematically and as self-motivated learners), a greater emphasis can be placed on written work (e.g. nicely written or typed work in a proofs-based course).