I'm teaching a graduate course in mathematics next semester. I'm planning to have a midterm and a final exam. But I'm thinking about having weekly (or once-every-two-weeks) in-class quizzes instead of homework assignments. My plan (for each week) is to post five problems for the students and one week later randomly select one of the problems for the students to solve during a 10-minute closed-book quiz in class.
I've done this type of thing before in first-year calculus and linear algebra classes without issues. Some students reported that they liked the quizzes because it forced them to keep up and internalize the material more than homework does. I agree with them on that point. I also like quizzes over homework because it is more difficult to cheat and that there is less grading to do.
Is doing quizzes instead of homework a good idea or a bad idea in a graduate course? What are the pros and cons?
Edit: It's a standard first-year lecture-style graduate course. Think complex analysis or abstract algebra or measure theory. I expect 10-20 students, mostly graduate students, but a few senior undergraduates.
Edit: A valid answer could be neither. I'm open to hearing the case for that.