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

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.

• I understand that there is already a best answer chosen, but I too have considered this approach. My variant would go like this: There are 3 sections of homework, the "easy" (rote) problems, the "medium" problems, and the "tougher" problems. Choose two of the three sets. The catch is, there is a higher quantity of "easy" problems than "medium" problems, and more "medium" problems than "hard" problems. This means students capable of doing the hard problems will likely aim to do fewer problems, while those who need the easier problems will get the practice. I'd love feedback. – Opal E Jul 29 '14 at 5:51

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?

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.

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 to question 3: - yes - the point of the circled question is mainly for the student to get feedback on their presentation and attention to logical details> Iv'e noticed that students when comparing the solution to their's often think their solution is identical (when it is not) and if their solution is different, they have no idea whether it is correct or not (so that is the point of the circle + having the solutions). – WetlabStudent Jun 13 '14 at 1:29
• But: if the student already is uncertain about the validity of their solution, they would already be motivated to study the solution closely. The big problem you want to be tackling is in fact the questions which the students are confident they know how to do yet in fact they don't. – Willie Wong Jun 13 '14 at 7:50
• hmm, that is actually my current grading scheme, so I guess I shouldn't modify it – WetlabStudent Jun 13 '14 at 13:09
• @MHH the problem is that if they think the problems marked will be part of the final grade, they'll mark those they know are right, while you want to check those where they aren't sure... – vonbrand Jun 14 '14 at 0:41
• @vonbrand but the marked ones aren't part of the final grade under this new scheme, which I haven't deployed yet. – WetlabStudent Jun 14 '14 at 4:43

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.

• An alternative could be: harder problems count for two simple problems. – Taladris Jun 12 '14 at 9:43
• A variant I used to do is that the problems they choose have to add up to be more than some X. Assuming that the low numbered options won't add up to X, even if they do all of them, this forces them to choose some harder ones. – ncr Jun 13 '14 at 2:19

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.