What are the pros and cons of having students choose which HW problems are looked at?

Question

I am considering a new HW policy that I haven't seen elsewhere. Each week the student must turn in $n$ problems, of her choosing. There will be no HW score beyond documenting whether the student is turning in honest attempts at these $n$ problems. (my general thought would be to make each HW contain $n$ challenging problems, a few medium problems, and $n-1$ easy problems)

If the student chooses to do so, they circle one of their solutions and the "grader" reviews it and gives the student very detailed feedback, including comments on the correctness and clarity of each step, the writing, and one take home message they should work on. No numerical grade would be assigned here, it is purely for feedback purposes, and it is optional (although there is no reason not to choose this option).

What is my desired outcome

• students do the problems that will allow them to learn the most. Struggling students can work on the easier problems and get feedback on them before moving to harder ones.
• stronger students can turn in the hardest $n$ problems and choose to get feedback on the one they found most challenging (but still had a solution for), without wasting time on the problems they found trivial.
• the lack of a numerical grade tied to the HW (beyond completion) seems to increase learning. See the answer to: Is it worth grading calculus homework?
• by choosing the problems that they will receive feedback on, they are learning how to identify when they aren't sure about a solution and when they are confident in it. Being able to identify you don't know something or when you think your argument is shaky is one of the most important skills in mathematics.
• students will see the value in HW as a learning tool rather than something they need to do just to get a grade.
• students are learning to own and take pride in their struggle. By circling the question they most want feedback on, I think it reinforces them to constantly be trying to learn more and not focus on the things they already know well.

What I want to avoid (possible drawbacks)

• non-struggling students doing easy problems just to quickly turn in the assignment. They end up learning less than in the traditional HW setting and do poorly on the tests because they haven't gotten feedback on problems of the appropriate difficulty level.

The goal is to make HW purely a learning experience that everyone can get the most out of but to also move away from a grade obsessed classroom culture. What are the advantages and disadvantages of this HW policy? and what changed to overcome some of the disadvantages?

Edit: another way of breaking down the questions (instead of easy/hard) is to use standards based grading as in this question Standards-based grading in calculus. So I'd mark a question as

• computation - applying formulas and rules used in this class,
• application - abstracting a concrete problem and figuring out which mathematical tools are most useful to solve it,
• adaptation - changing a formula or rule to fit a new, but similar type of problem
• theory - demonstrating a deep, theoretical, understanding of a concept presented in this class

And I might have a requirement that they must do problems from of at least 3/4 types and not tell them which ones are the hardest.

Answer 1

Question 1: Will you tell your students which are the easy problems and which are the hard ones?

• If yes, you get the vonbrand scenario
• If not, how will the "strong students" know which are the challenging questions to work on?

Question 2: how much mathematical maturity do you expect in these students?

• If you are talking about junior or senior level college students in a subject they are doing their major in (so computer scientists in an algorithm class, or pure math majors in a topology class), then your scheme of having students figure out which questions to ask for help on may work.

• But if you are talking about more introductory or "service" classes, my experience¹ suggests that oftentimes the students do not know what they don't know. They may dismiss a question as easy and solved, and be completely wrong about their solution. The policy of only one question being read for each set may be indirectly punishing the students for something that is not really any fault of their own.

Question 3: Will you supplement the homework assignments with detailed solutions?

• If yes, why bother asking the students to circle a question to be read? They can just compare against the model solution. (And the TA will thank you for it, even if it is the TA's job to write up the solution sheets.)
• If not, how will students learn if they have more than 1 question they don't know how to do?

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Some suggested modifications:

1. Don't make the problem sets too long. Keep $n$ small such that students at least have time to try every problem.

2. Don't tell them which are the hard problems.

3. Grader grades each question turned in on the scale of 0 to 2, where 0 is effectively blank/little effort shown, 2 is mostly correct, and 1 is good try but some mistakes. Grader is not require to comment. At the end, the grader totals up the scores, any sum bigger than $k$ for some $k$ depending on $n$ gets a pass on the homework. Your description of the desired outcome suggests that $k \approx n$ is a good choice.

4. Publish solutions promptly after the due date of the homework. (Ideally written by the person who will be fielding questions about the official solutions.)

5. Have someone (grader? you?) field any and all specific questions about the solutions, after they are published. (But note, not necessarily after the students get their work back from the grader.) (And note, don't answer questions of the type: "Can you show me?" Make sure that they read the solutions and formulate a clear question first.)

Why?

1. By not telling the students which are the easy problems, they will most likely, on average, have to try working on a few of the hard problems before they discover it is beyond their understanding or ability. This will at least show them what they don't know. (And knowing that is a good thing.)

2. By grading them on a scale of 0,1, 2, you (and the grader) are given the opportunity to indicate to them that something they thought they know is perhaps something they don't know very well. (Clears up misconceptions.)

3. By publishing the solutions and letting the students work through them before asking questions, and with the possible additional feedback from the grading, the students actually develop the mathematical maturity to appreciate, without your help, what they know and what they don't know.

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¹ FWIW, in my previous job homework is not graded, and I discuss the students' works with them in a small group (two or three students at a time) setting. In my current institution homework is also not graded, while we have weekly exercise sessions during which the students can work on the homework and ask for help from TAs.

Answer 2

Knowing (a little) of how student's brains work, you'll get $n - 1$ "easy" problem attempts + 1 "medium" problem attempt each time, period.

You have to force their hand a bit. Perhaps give extra credit for turning in harder problems or some such. Or make it a requirement that $x$ number of hard problems get turned in for getting a homework grade (where $x$ is, say, half the number of programmed homework sets).

You want to give weaker students extra oportunities/homework. But you have to consider that students that are weak in your class are probably also struggling elsewhere, so just piling more work on them won't help.

Answer 3

This is more appropriate for a comment, but I lack sufficient rep to post it as such.

Your concern that non-struggling students will do easy problems in order to quickly finish the assignment is valid. One way around this that I have seen work well is to assign each problem a number of points based on difficulty and require a fixed number of points worth of problems be turned in for each homework set.

The advantage is that you can tune your point distribution each week, based on the previous week's results, until you have students consistently working on problems appropriate to their level.

Answer 4

You asked expansively for pros and cons, so here's some thoughts.

I think you're mixing a couple different issues:

(1) Compelled homework, which has pluses of getting them to do it and minuses of the work/bother in monitoring it. Personally, I'm against it, but you seem to be making this a tacit assumption in favor, so fine.

(2) Your particular scheme:

Which has a serious minus of being so complicated my eyes glazed reading it...this may affect the students who feel gamed, also may make it harder to administer. IOW not KISS.

In addition, your second bullet is relatively low value. Not zero, but near zero. This is because strong students will knock out easy problems fast. And most classes don't give an amount of homework that is seriously derailing the strong students. I mean, it's not like telling someone to do 1000 pushups per day.

The feedback (bullet 4 and part bullet 2) feels like a complicated kludge. Why have this circling system versus just having published solutions? Or office hours? Or tutorials? Or in class questions? The circle feedback is a lot of work you are committing yourself to, to write these little letters back. And for that matter what if students have questions on several (need to pick one, more gaming)? Why do only strong students get feedback? Tutorials or math center or office hours or published solutions or questions in class or combinations of these are all more efficient in the real world.

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My practical advice is just do a standard slate with equal split of ~4 easy (plugs, small calcs) ~4 medium (worked solutions, multi-stpe problems) and 0-2 hard (word problem, derivation, two techniques needed, etc.) for the nightly drill. Most books are in order anyway and make clear the A/B/C problems. The simple solution is the kids will do the problems in order. The weaker kids will get through the easy ones (slowly, but will), founder on the mediums, and not seriously attempt the hards. The strong kids will just pound it all out, and the 4 easy ones won't cost them much time, might even help them achieve more automaticity.

You should also mention to the kids that nothing stops them from doing more drill. They can do the entire book of drill problems. That you think the 10 problems a night are enough to pass, but if they are worried about an A versus a B or a B versus a C, to do more.

I would also tell them that test questions (what they care about) will be VERY similar to homework problems. Stamp the ground and make a horse sound. And then follow through. (Why would you examine them on things they don't drill? Why assign drill that doesn't help them with exams?) And of course you should have a similar split of easy/medium and very few hard on exams...this allows students to find their level...and doesn't skunk the weak students. And have them in difficulty (not coverage) order. Other than broader, multi-lecture coverage, is similar to their drill sessions (except with some moderate time/performance pressure).