# How to assign homework when answers are freely available or attainable online?

I find that making homework meaningful is becoming increasingly challenging. Let us suppose that I am planning for next semester's first-semester or second-semester calculus course at my university.

In all likelihood, we will be using one of the common calculus texts because the university expects consistency, both from semester to semester and from teacher to teacher. Solutions manuals exist for most of the common calculus texts, so assigning problems from the text has downsides. But this point is slightly moot, because WolframAlpha can do essentially all the problems anyway (albeit with poor exposition). There is also an endless supply of answers awaiting at Math.SE and similar fora.

Unfortunately, writing my own exercises (which I currently do) doesn't prevent or penalize the use of WolframAlpha or online math fora.

One possible solution would be to make homework count much less. But I dislike this idea, as I find that students who do homework are much more competent on average than those who don't; homework should be consequential enough for students to take it seriously.

What are possible, good ways of overcoming the availability of free and proficient homework assistance?

• A radical approach would be some kind of flipped classroom scheme. This eliminates the illicit assistance problem since you, the instructor, will be supervising and assisting them instead. – Potato Mar 13 '14 at 21:28
• Re the last paragraph before the question. The students you teach should be on average adults or very close to it. They are typically allowed to drive cars and to vote and to do all kinds of other things. I think you should also trust them to be able to decide if the homework is important for them or not. Inform them and then let them do as they please. – quid Mar 15 '14 at 18:07
• Where I come from we don't use homework to evaluate students. When I realised this is done in the USA, I was shocked. You're begging for these and other kind of problems. – Git Gud Mar 25 '14 at 14:30
• If you look at textbooks from 100 years ago, many of them have ALL the answers in the book. In other words, we have become more conservative, less liberated. Then again, this was from a time, where it was very much understood that drill was critical to learning math. You should also be interested that books on Amazon reviews used for self study (e.g. Dover books or Schaum's) get great ratings if the solutions are included. This is mostly from self studiers or people looking for supplemental texts. IOW, completely perpendicular to HW grading--from people wanting to check answers and LEARN! – guest Jul 7 '18 at 17:52
• Basically the assigned HW is a sort of fetish of college teachers. But it is not good COACHING, the way a sport is trained. – guest Jul 7 '18 at 17:52

My suggestion is that you stop thinking of homework as an assessment tool, and instead find the true meaning of homework, which is a method for students to gain proficiency with the ideas on their own. The key is to design your other assessment tools so as to motivate the students to see that doing homework is a good idea.

My practice, for example, is to assign a lot of homework, and encourage students to do it, but I don't collect it. Rather, I give an open-book/open-notes quiz every lecture (or every other lecture when it is too much), based on the homework---usually just a single question, similar to but never identical to the homework, aimed at testing their understanding of the main concepts or technique for that lesson. The idea is that students who have actually done the homework will find the quiz straightforward, and those who haven't will have a hard time. I make these quizzes count for about 50% of the course grade, so there is ample motivation for the students to be prepared. And the best way to become prepared, they quickly see, is to do the homework, even though I don't collect it. This arrangement also has the advantage that I can encourage the students to form study groups and work together on the homework; studying together, they rise to a high level. Also, I can freely encourage my students to post any questions they might have about their homework to math.stackexchange, or to ask other students and professors, and there is no "cheating" issue, since I don't collect the homework, and instead they sincerely want to learn whatever it is they are asking about.

• I like your aswer, but I think it is a bit utopistic. I think that many students are honest and will see the advantage, but there are also more than a few who will not do the homework (as long as it is not collected). What if at the end most students would fail the course since they were "cheating on themselves"? – Markus Klein Mar 14 '14 at 9:20
• Another drawback of this is that students need not (and most likely won’t) write down answers in a way that is readable by others. Not is learning to communicate mathematics this way important for its own sake in my opinion; it also helps to find errors or alternative ways in your solutions and learn things by applying them once more and from a slightly different perspective. – Wrzlprmft Mar 14 '14 at 10:02
• Markus, with frequent quizzes students get the same kind of feedback that they get with frequently collected homework assignments, when they see their quiz grades. @Wrzlprmft, I emphasize writing and communication on my quizzes, insisting that answers are written essay-style in complete sentences, etc., even in a calculus class. – JDH Mar 14 '14 at 11:42
• @ChrisCunningham, I find that giving so much credit to the quizzes is exactly what motivates the students to do a lot of homework on their own. I have heard students describe the situation to others as a relentless pressure to be on top of the material. – JDH Mar 14 '14 at 22:30
• @ramanujan_dirac Your remarks don't really make any sense to me, and I think you may have misunderstood the proposal. If a student knows the concepts and ideas, then they will do well on my quizzes, and it is not possible to "mug up" the answers. – JDH Jun 8 '14 at 18:26

The underlying issue is something that is hinted at in JDH's answer, but not explicitly stated. So I'd like to state it.

The key is to decide what your homework is for. The big words are Formative and Summative. In short, formative assessment is designed to help you learn, summative assessment is designed to figure out what you've learnt.

The distinction is not clear cut, and I'm no expert so I won't pretend to be. In truth, all assessment is a mixture since unless we know what we have learnt, how can we figure out how to learn it better? And if we don't use the assessment of what we've learnt to find out more, what was the point in measuring it?

Nevertheless, we can broadly assign assessment to one of the two classes by its primary use. I suspect that the majority opinion would be that homeworks are for formative assessment. We only grade them to encourage students to do them, reasoning that if we don't give the students credit for doing them then they won't choose to do so of their own free will (but note quid's comment on the question!).

Unfortunately, this sends out the wrong message. By counting them as part of the final grade, the students view the homeworks as summative: a measure of what they have learnt, and therefore view them as something to achieve rather than as something to reflect upon.

Also unfortunately, the default assumption on students' part seems to be that all assessment is summative. This is because the majority are not aware of these nuances, and are more aware of the occasions where they have undergone summative assessment rather than formative (the latter being, by its nature, less stressful and more subtle, and so less noticeable).

Therefore it is not enough to simply say "Do the homework for your own good". You will need to set up a culture of reflection. Here are some broad guidelines. Again, I'm not an expert and have not implemented (all of) these myself.

1. If you must include a homework part in the overall assessment, make the bar as low as possible. Essentially, if they try the homework they should get the credit.

2. Use feedback from the homeworks in your teaching. Use the homeworks to find out what people found difficult and then go over that again in class time. Make it so that the students see the link between the information that they give you through the homeworks and the help that they get afterwards.

In my course, the rule is that they need 8 out of 12 homeworks "approved" to be able to take the final exam. Other than that, the homeworks don't count. Nevertheless, because we don't really do (2), a lot of students simply do the first 8 and then stop.

• This answers deserves more upvotes than it has. – Mark Fantini Sep 25 '14 at 0:51

I agree that this is a problem in teaching calculus and other classes where solutions manuals and online forums are readily available and I don't know that there exists any one thing that you can do. No matter what, a student can always find "help" somewhere.

First some remarks

• If a student gets hold of a solutions manual, then it is of course easy for the student to copy down the solution. But the student still faces the problem of having to reword the solution since you (most likely) also have access to the solutions manual. When you see a solution that is a copy of the one in the solution, the student has cheated and you can follow your school's policy on what to do. And so, having a solutions manual still does require some level of effort on the students part. (Granted, there are problems where it is hard to write down a solution that is different from the one in the solutions manual).

• As you note, you can solve the problem of solutions manuals by writing your own problems. This, as you note, still doesn't eliminate the student seeking help elsewhere. But again, seeking help online or with a tutor requires some effort.

So by saying this, I guess that I am trying to point out that it might not be a huge problem. It is my opinion that as long as the students actually learn the material, I don't care if they studied solutions in a manual or got help online.

Anyway, to answer your question, what do I believe that you can do?

• As you also note, making the homework part of the final grade smaller of course takes care of it, but I agree that this goes against the desire for students to spend time outside of the classroom.

• This is radical, but I suggest implementing oral exams or presentations and letting them be part of the final grade. This, of course poses some practical problems that might be addressed in another question. This could serve as a motivation for the students to understand the problems that they are solving better.

• Instead of having oral exams, you could also simply tell the students that they have to present a problem from the homework on occasion. Maybe you could make every 30 min. every Friday "present a problem Friday". You could determine in advance who should do it, or you could pick someone at random in class. That way there is an incentive to understanding your homework. This, of course, can be a bit controversial, but in many countries this is actually practiced and so it might be worth considering.

• I agree strongly with the oral exams part. Something similar I do in my calculus courses is to have "short answer" questions on the exam / homework, where the student must explain in their own words some topic like integration, or some theorem, or even some abstract concept ("Why do some limits not exist?"). If/when a student submits something I strongly believe is not their own words, I discuss it with them outside of class, probing to see how much they truly understand the question. – Steve D Mar 13 '14 at 22:38
• @SteveD: I agree with what you said! – Thomas Mar 13 '14 at 22:39
• Daily one-question quizzes, with the homework grade being the geometric mean of the quiz and homework grades. – Kevin O'Bryant Mar 14 '14 at 0:45
• @Kevin Why geometric mean? – Chris Cunningham Mar 14 '14 at 13:16
• The geometric mean is small if either number is small. If one copies the homework (gets 95%), and then bombs the quiz (gets 5%), the usual mean is almost passing, while the geometric mean is harsher. – Kevin O'Bryant Mar 16 '14 at 0:08

Here's another idea: Give a completely different type of homework assignment.

For example, this year, collect all of your homework as usual. Look through the homework for examples of solutions that are clearly done by students, and which have mistakes in them. Choose representative samples of the student homework which have evidence of common misconceptions in them. Typeset them in LaTeX to make them anonymous.

The homework assignment then becomes; for these three example problems, find the mistake in reasoning in this example homework, and explain what you think the reasoning is that led to this mistake.

This has the benefit of directly giving students counter-evidence to the likely mistakes and misconceptions they have, and being an assignment that cannot be Googled (remember you have LOTS of examples of homework assignments, you can easily change them up for the following year).

You cannot prevent them from finding solutions, but there are some tricks to keep the difficulty of the homework and the make it more diffucult for those students to google solutions:

• If students have to proof theorems, don't name them in the question. If they have to perform an algorithm (like "Perform three steps of the Euler iteration"), don't name the algorithm, just write down the definition.
• You can ask the students to proof something in a specific way. For those who do not "google" the answer, this is even easier for them. Often, this is also a good way to use results of the course and not a standard solution from a book not using results from the lecture. For example, I like the derivation of Neumann series via Banach's fixed point theorem. Instead of asking to show that the following sum is a inverse of (I-A), you can ask to show that this is invertable using Banach's fixed point theorem and to define a appropriate fixed point function.
• (This happend once to me by accident. I don't know if this is a good thing to do on purpose.) I asked to students to proof a not so known theorem and by grading it, everyone was having the same solution which was wrong (A part was missing, something was proven by contradiction, but the contraction was wrong). I saw that in one book a gave as reference there was this proof with all the mistakes in it. All the students copying from that got a wrong answer.
• You can try to reformulate the result. For example, only let the student proof a special case. Or add some additional case.

Some ideas I've developed over the years:

• Give the answer (for example: sin(x)/2) and have the student find two or three questions (ex."what is the function whose derivative is") with this answer, of course before studying integral calculus.
• "Audrey thinks that the integral of a product of functions is the product of the integrals of the functions. Write her a letter giving two counterexamples (one simple, one more complicated) to show that this is false." If the subject is the definite integral, one of the counterexamples should also have a graphic explanation.
• The same idea can be used for derivatives, trigonometric and algebraic identities.
• When teaching functions of several variables: "Find 5 examples of functions of 2 variables from different sciences. You can look anywhere you want, but you've got to identify your variables." Usually, one or two A= 2*Pi*r would appear as an answer. When giving back the assignment, it was a good time to discuss the difference between variables and constants.
• $A = 2 \pi r$?? – Daniel McLaury Jul 15 '15 at 23:30

What about just giving kids the answers and asking them to rediscover them? It would be harder to check, though...

(In light of some down-votes, allow me to clarify:)

I wouldn't recommend this as an always thing to do on homework, but I've seen teachers give their students answers to homework when they want their kids to focus on the process, not the actual numerical solution. And this can be real problem solving. Instead of "What's the answer?" the question becomes "How is this the solution?" If the problem is difficult enough, it'll still be challenging.

• Can you explain why you think just giving the answers is a good idea? (I give daily homework from the odd-numbered problems in the back of the book. That allows students to check their work. These are exercises. Anything that I would consider real problem-solving, I wouldn't give answers for. But I'd like to hear your reasoning.) – Sue VanHattum Mar 15 '14 at 15:02
• I wouldn't recommend this as an always thing to do on homework, but I've seen teachers give their students answers to homework when they want their kids to focus on the process, not the actual numerical solution. And this can be real problem solving. Instead of "What's the answer?" the question becomes "How is this the solution?" If the problem is difficult enough, it'll still be challenging. – Michael Pershan Mar 16 '14 at 15:55
• If you edit your answer to include what you put in the comment, maybe the downvotes will be reconsidered. I think your answer is a good one. – Sue VanHattum Mar 16 '14 at 17:04

This problem was already solved at my university when I was an undergrad in the mid 80s. The assignments were fixed, in that they were the same for all sections, and unchanged along the years. They represented a training platform/self-assessment benchmark to know if you were ready for the tests. The marks came from the tests.

This had the additional advantage that it helped the professors to be more uniform in their teaching, because they were all preparing the students to tackle the same practice questions.

• Love it. Very powerful. And good attention on the result in terms of training. Will really annoy the "I'm Robin Williams in Dead Poets Society" types, though. – guest Aug 30 at 19:40
• Indeed. Another good thing was that classes had "lab time", meant for the students to work on the assigned questions, with abundant student TAs available for getting help. – Martin Argerami Aug 30 at 20:06

Unfortunately, writing my own exercises (which I currently do) doesn't prevent or penalize the use of WolframAlpha or online math fora.

Wolfram Alpha will do great if you have a long list of problems that are all posed in the same formulaic manner, and that all consist of nothing but calculation. 1. $d(x^7)/dx=?$ 2. $d(x^7+x^6)/dx=?$ etc.

It's a whole different story if you make the problems you're assigning not so formulaic -- even trivially so.

1. Evaluate $d(r^j)/dr$, where $j$ is a contstant.

To you and me, this is obviously a trivial variation on problem 1, but to a student who is trying to use WA without understanding anything, it's a complete mystery, and they won't even know how to put it in a format that WA can accept.

WA can't do word problems. WA can't do proofs. Ask your students to check whether the answer has the right units, or to check that the answer makes sense in a special case, or to describe why the answer increases when one of the parameters decreases.

The basic point is that WA can't think. Ask your students to think. A computer can't do that.

• +1, although I'm not too comfortable with your statement that "WA can't think." (It depends on what you mean by think.) – Joel Reyes Noche Sep 29 '14 at 1:03
• Have you tried using Wolfram Alpha? I just typed in 'd(x^r)/dx' and it told me the answer. WA is explicitly designed to accept as broad a range of inputs as possible. If it can't quite tell what you want, it will aim to tell you all the possible things you might have meant. I heard Mr. Wolfram say that's the point. See en.wikipedia.org/wiki/… – Jessica B Mar 13 '15 at 9:44
• @JessicaB: Are you saying that you typed in $d(x^r)/dx$ and WA told you $rx^{r-1}$? Did you have some way of telling it that $r$ was a constant? If not, then WA is giving the wrong answer in the case where $r$ is not a constant. – Ben Crowell Jul 12 '15 at 13:44
• @BenCrowell Yes I am; try it yourself. WA makes its own assumptions about what is a constant and what is a variable, based on, I guess, convention. If you want more, I'm sure it will offer to do so if you pay. There are nice ads at the side of the page that imply that paying them will increase your grade. – Jessica B Jul 13 '15 at 7:03
• @BenCrowell Actually, I was wrong. If you tell it the parameters for the function $r$, it will work with it as a general function (without requiring money first). – Jessica B Jul 13 '15 at 8:35

You might be interested in the article Algebra Homework: A Sandwich! by D. Bruce Jackson in the March 2014 issue of Mathematics Teacher. Despite the title, it's not really specific to algebra and it provides one way to address exactly the problem you're talking about. Basically, the idea is to make a homework problem a 3-step process. First, do the problem, then check the answer, and if it's wrong do it again. The students effectively self-grade, from the perspective of getting the correct answers.

I don't remember what the article suggests in terms of how to translate that into a homework grade. Personally, I would want to do some spot-checking to make sure the student is really checking all of their answers (and the penalty for an uncorrected wrong answer would be harsh) and checking that the work that leads to a correct answer is correct.

In my last couple years of teaching I did not grade or count as any credit towards a grade homework completed as an assignment. I did collect it and mark it as COMPLETED ON TIME (COT). If a student had completed on time all homework assignments they were given the option of retaking the test after an additional review assignment was completed. At the end of the grading period if all homework is COT they were given a perk to be negotiated such as dropping a quiz or test grade or bending the grading scale slightly. The availability of homework help should not be a detriment to doing the homework. If the student does the homework they should do better on tests and assessments of any kind. If they cheat themselves during the homework phase and do poorly on assessments the question becomes-"why didn't you know you were having trouble when you did the homework?"

The question as posed asks "What are possible, good ways of overcoming the availability of free and proficient homework assistance?” I find myself wondering what the purpose of the homework is. Since this was not clear to me from the question, I am going to take as given that the purpose of homework (as the rest of the activities undertaken by students and teacher related to the course) is to promote learning.

If so, I think the specific exercises included are not just “the wrong ones” (in that their answers are easily locatable), but the wrong kind. So what is the right kind? First of all, there should be (as Taalman calls it) a "Problem Zero" (http://www.maa.org/problem-zero), where students are asked to articulate the key ideas. This is likely to elicit more thinking on their part, require more work, and gives the instructor more to go on to ensure more learning in the future.

Second, there are many simple adjustments possible to the exercises you have. One is the “give them the answers and ask them to justify” that Michael Pershan proposed above (it seems with limited acclamation). Specific examples include asking them to explain whether their method would work in another case, or articulate the limits of how far it generalizes; comparing worked solutions (which can be harvested from prior student work) to identify either connections or contrasts in reasoning or diagnose errors; or writing their own problems and explaining how they are related.

Since the answers are already findable, clearly the student generating the answers (again) does not have some inherent value. The desired output of the course is for them to learn.

• +1 for "Since the answers are already findable, clearly the student generating the answers (again) does not have some inherent value. The desired output of the course is for them to learn." – JPBurke Sep 24 '14 at 6:44

I actually did research on this (unpublished, but I collected hundreds of questionnaires) and students overwhelmingly claimed to use W|A to check their answers, to see the steps to a solution, etc. In the case of websites like physicsforums or crazyproject where people may be able to copy entire proofs, the students I surveyed again claimed to be engaged in some kind of learning behaviour. I don't see how reading a proof on the internet is any different from having your professor explain it to you in class. In both cases there's a transfer of information from an authority to the student, and the student gains new knowledge. It would be crazy for students to come into a calculus course expecting to develop all of the ideas and methods on their own.

Students have always been able to trivialize math assignments, and it's up to them as self-interested actors to exercise discipline and do what's best for their grade in the long-term. Copy-pasting a proof from the internet and not attempting to understand it (the absolute worst case scenario) won't help them on the final exam.