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It seems like the internet has made it very difficult to prevent cheating -- simply looking up the solutions to textbook problems -- on undergraduate math homework. So there is a danger that grading math homework assignments could be unfair to the students who do not cheat.

What is the best way to handle this? One system is to check homework for completion only and have a weekly ten minute quiz in class that consists of one or two problems taken directly from homework. Is that a good way to go?

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    $\begingroup$ For that matter, much standard graduate-level stuff is on-line... as it should be! And I am not in favor of commanding students to pretend to ignore the internet, etc. And it's not just about math. "Friction" of transfer of information has been vastly reduced, which changes the "physics" of schooling and other stuff. No easy answer... except that, truly, the "laws of nature" have been changed in this regard. $\endgroup$ – paul garrett Jul 19 '18 at 21:43
  • $\begingroup$ Who is asking this question? A student or a professor? The answers given may be opposite. And, of course, this question seems to be asking for opinions, which would be off-topic. $\endgroup$ – Gerald Edgar Jul 20 '18 at 0:10
  • $\begingroup$ The asker is the professor or anyone who wants to create an optimal learning experience for the students. I'm not sure that the answer is purely a matter of opinion, as some policies will lead to better learning outcomes than other policies, and in principle the difference in outcomes could be measured. It would be helpful to know what has worked well for other people. $\endgroup$ – eternalGoldenBraid Jul 20 '18 at 0:23
  • $\begingroup$ Online homework systems that randomize the questions and give multiple attempts are great practice. If a student goes through the hassle to “cheat” without learning anything on the homework, then they deserve to do poorly on the exams. $\endgroup$ – Aeryk Jul 22 '18 at 2:50
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I should preface this by saying that I am working in the US, at a large public university that serves a population that largely consists of first-in-family college attendees, low income students, and many students who speak a language other than English at home. I have previously taught in low-income public high schools with similar demographic profiles.

Homework gives students an opportunity to practice performing the kinds of computations that they are expected to learn in a course, or to build familiarity with the definitions and theorems that they are expected to understand by the end of a course. I find it incredibly difficult to believe that a student can master material (at any level) without the kind of practice that is provided by homework assignments. Since the goal of homework is to give students an opportunity to practice the skills that they will ultimately be tested on, I am incredibly sympathetic to the argument that homework should not be graded, as the students that actually complete the homework will master the material and do well on exams, while students who don't complete the homework will probably fail those exams.

Unfortunately, students (especially undergraduates, and even more especially high school students) are highly motivated by "points" and often don't value anything that isn't worth any "points" in a class. For example, one quarter I had two precalc classes, one right after the other. In one section, I took attendance every day and factored that into a "participation" grade. In the other section, I took attendance, but did not factor that into the grade at all. The classes were otherwise as identical as they could be. In the class where I gave credit for attendance, I had an average of about 90% daily attendance, while the other class averaged out to about 60%.

As such, I think that it is necessary to give students some kind of grade for completing their homework. Moreover, I think that such a grade should be directly tied to the homework (rather than to quizzes based on homework problems; I think that students often fail to see the connection). That being said, I think that the key work here is completion. I typically don't knock myself out grading every problem, and will often grade homework in a completely binary manner (1 point for turning in a substantially completed assignment, 0 points for not turning in anything or turning in something that is incomplete). Alternatively, if homework can be done online (my institution uses either WebWork or WebAssign for most lower division classes), I tend to take the score given by the computer, and "round up" to the next letter grade.

The goal is to give students an incentive, in the form of "points", to complete their homework. I am trying to exploit my students' innumeracy (they'll work quite hard on their homework for a tiny fraction of their overall grade) and the Skinner box-like reinforcement that comes from giving students "points" in order to get them to do something that is good for them (i.e. finish the homework). At the end of the day, the homework grade shouldn't have that much effect on their overall grade (exams or projects will be much more important).

In short, yes I grade homework, but (1) I focus on completion over accuracy and (2) I don't make the homework grade a huge component of the overall course grade. In answer to the secondary question ("Should I give homework quizzes?"), I think that is a fine strategy, but I would employ it in addition to assigning some kind of homework grade (with, perhaps, more weight given to the quizzes) and I would be careful not to spend too much lecture time on quizzes.

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    $\begingroup$ In Spain, where I teach, at the university level, in introductory mathematics courses, homework is essentially never given in any formal sense. There are no resources available to correct it in a way that guarantees any kind of fairness. Students are given problem sheets - these are usually distributed weekly or as an entire package at the beginning of the semester - and they are expected to do these problems in order to learn, but no one controls that do so. Quizzes did not traditionally form part of the evaluation either, although now some kind of in class exercise solution is more common. $\endgroup$ – Dan Fox Jul 20 '18 at 8:20
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    $\begingroup$ On a different note, focusing on completion over accuracy seems to me a mistake, on the grounds that it is better to learn to do one thing well than to learn to do two things badly. $\endgroup$ – Dan Fox Jul 20 '18 at 8:21
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    $\begingroup$ @DanFox It comes down to understanding what the point of assigning and grading the homework is. As I said above, giving any points to homework is a technique for encouraging students to actually engage with and complete the homework. Maybe the culture in Spain is different, but in the US I have found that if I don't give "credit" for homework, it doesn't get done, and more students fail. Moreover, I think that it is more fair to award credit for completion than for accuracy, because giving credit to accuracy unfairly advantages the students who cheat. $\endgroup$ – Xander Henderson Jul 20 '18 at 12:47
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    $\begingroup$ My basic thesis is "Students who complete the homework are more likely to learn the material, and therefore more likely to do well on exams." I have found that if I give no credit for homework, students don't do it, and fail. If I give minimal credit, more students actually do the work, and perform better on exams. I don't have a ton of data on this, but I have enough experience to be well convinced of this phenomenon. $\endgroup$ – Xander Henderson Jul 20 '18 at 12:54
  • $\begingroup$ It's standard in Spain to provide lists of problems for students to work. What is not so common is to collect them and grade them. In part, university students are expected to have (or to acquire) the maturity to do the problems on their own. While it is my experience that giving a nominal grade does work as motivation, the labor of correcting dozens (or hundreds) of exercises is not something most professors find well compensated by the results, and the more so when the reaction of most students is simply to copy in teams, to get whatever few points are available. $\endgroup$ – Dan Fox Jul 30 '18 at 9:51
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Great answer is given here What's the value of homework and should it be graded

Homework should be given only if the teacher is prepared to give students feedback on their performance.

Additionally, even cheaters can be forced to understand the solution they have found in internet and to adapt it on the problems they were given.

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You complain about access on the internet to solutions to textbook problems. Here is a way around that: write your own homework problems.

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    $\begingroup$ That's a reasonable first step. However, I wrote a graded homework set that used a unique term I created (apparently not found anywhere on the internet). A year after giving the assignment, I was curious to see if it was used anywhere else, and Googling it returned just one result: An image of my assignment and the entire solution posted on Chegg. I'm now of the opinion that the only time I really know my students are working for themselves is when they do it in my presence. $\endgroup$ – Nick C Jul 30 '18 at 19:39
  • $\begingroup$ On Chegg did you find your solution or one written by a student? $\endgroup$ – KCd Jul 30 '18 at 19:50
  • $\begingroup$ I believe the solution was posted there by some expert, but I decided not to view the whole thing as a subscription costs $14.95 per month. $\endgroup$ – Nick C Jul 30 '18 at 20:03
  • $\begingroup$ Supposedly Chegg will give instructors investigating academic misconduct free access, though my experience was that their hold line is long and I decided it was simpler to just pay. $\endgroup$ – Noah Snyder Aug 30 '18 at 13:34
  • $\begingroup$ @NoahSnyder you paid $14.95, or whatever the monthly fee is, how many times to check into possible student misconduct? $\endgroup$ – KCd Aug 30 '18 at 14:19
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Nope. Nor should HS math stuff. Just do kids to the board or grade each other's papers for no recorded grade. and do frequent quiz and testing.

Graded homework is a waste of instructor time. And then in college (especially higher end ug or grad school), hw tends to become too "project-y". Loses the key pedagogy of drill.)

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