Last month, my colleagues who teach 9th grade conceptual physics started using nearly-daily quizzes in their classes. Great move. The students initially rebelled a bit, but they started preparing better for class each day when they saw they would be held accountable for the course material. (Daily quizzes are best established at the beginning of the year so as to avoid the resistance to change. But these work any time.)
The general approach was to have an in-class discussion, demonstration, or presentation; assign some problems for homework; and take a brief multiple-choice quiz the next day about the previous day's class and the homework.
A major student complaint sounded something like "We did the homework, but then we had the quiz before you graded the homework and gave it back. How were we supposed to know whether we were doing the homework right? It's not fair that we have to take the quiz before we get the homework back."
Well, one of the teachers accepted that argument, and postponed his quizzes until at least a day after he had returned the homework. I ask, will the quiz scores be substantially improved by this postponement? I say, "no."
There is nothing wrong with giving a quiz based on homework problems that are due the same day.
A principal battle that I fight about homework is to establish a "correctness" rather than "completion" mindset. A decade of schooling has convenced my students that homework is a mindless chore that must be done for the sake of doing it. I think of homework problems as practice, as ways to remember and reinforce problem solving methods learned in class. Without active participation in the problem solving process, homework is useless. Whether the answer to a homework problem is right or wrong is immaterial -- what matters is the student's engagement.
A quiz can check that engagement. Did the student figure out the most important step in the problem? Can the student solve the problem again with the numbers changed? Can the student explain why the answer makes sense, or doesn't make sense? If not, then the homework problem didn't have the desired effect.
This follow-up quiz must be given right away. By waiting, you're telling the class that the homework isn't really that important... it's okay to give a half-arsed effort because I'll just tell you the answer if you wait a day. And, if the student didn't check his answers with friends or ask the teacher about a difficult idea on the night he was supposed to do the problem, what is the likelihood that he will follow up after the problems are graded? What are you doing every day in class in the meantime, while you wait one night to grade the problems, and another night for students (in principle) to look at their graded work and study? You can move along to new or more complex material immediately if you insist that students pay attention to their problems the very first time they're assigned. If there's substantial misunderstanding, the quiz can provoke a good class discussion about the problem in a way that "Anyone have any questions?" cannot.
Homework is not merely busywork. Regular grading of the homework, along with quizzes and other creative accountability methods, promote the appropriate attention to detail on daily work. Without such attention, you might as well not even assign problems for all the good they'll do. :-)