# Should full solutions be given (indiscriminately of the question being homework or not)?

Meta seems to bee filled with questions on this particular subject, and I have seen eloquent and convincing arguments from both sides, but the thing that leaves me ambivalent towards leaving full solutions is the lack of consensus. The majority of the questions related to this topic are on issues of policy, but this question is an issue of "best practices," codified (to the extent possible on MSE) or not.

First let me quote Eric Naslund's answer to this thread, which proposes a policy that opposes full solutions:

I am opposed to such a policy, as I believe it would have a negative long-term impact on the site as a whole.

Here are just a handful of the many problem associated with such a policy:

I can definitely see that line of reasoning, and considering the utility and ubiquity of search engines, I think that full solutions help form a de-facto searchable archive that is much more useful than a bank of hints. If this is combined with the continuing closure of duplicate questions, then one could easily see the residual benefit to the overall community (which is what makes MSE what it is).

However, with this community of mixed ideologies, the inevitability of encountering the idea of the community serving as providing pedagogy must be considered. The argument that full solutions can often be pedagogically harmful to students because of their lack of providing the student with the ability or opportunity to reason through the problem with hints only is also understood.

Personally, I can see reasonable arguments for both sides of this debate, and as both an answerer on MSE and a student I have tried to occupy a middle ground, by avoiding solutions ala Gauss (to use his own words, `no self-respecting architect leaves the scaffolding in place after completing the building') and providing full solutions with the full reasoning behind them, as well as any caveats, special cases, or notational issues that might be relevant. My pedagogical goal (if a high school student may have one) is to allow the person who reads my answer (not particularly the OP) to understand how to reason mathematically in a way that would allow them to tackle problems like the one posted and derivatives (no pun intended) if they were to come across them. This however still contradicts the "hints only" approach to MSE, which (along with the discord in meta) has lead me to asking this question.

edit

I just saw the responses to this question, and I simply thought it might be relevant as an example where giving a hint was pedagogically advantageous.

• A link. Another link. The underlying problem, cheating aside, is that people often don't ask questions about understanding mathematical concepts and theorems, but instead assign their homework to MSE. Conversely, answerers often focus on solving the homework, rather than attempt to convery understanding of mathematical concepts and theorems. – user14972 Nov 6 '13 at 8:35
• One way to "attempt to convey understanding of mathematical concepts and theorems" is to explain those ideas and some of their applications in the course of an answer that might well include a de facto solution to homework. The idea that optimal instruction requires third parties to supersede the judgement of the OP or the answerers (who might well be, and often are, experienced teachers) as to how to conduct the conversation strikes me as arrogant in the extreme. – zyx Nov 6 '13 at 9:27
• @zyx I think the point Hurkyl is making is not that it goes against the interest of the student to post full answers, but that we do not want people to simply assign their homework to MSE. Some students simply do not care about understanding anything and the question is whether we are encouraging such students to come to MSE for ready made solutions and whether this is something we want or not. I agree that some people have a strange paternalism in that regard though. – Michael Greinecker Nov 6 '13 at 10:12
• He was making several points, one of which was to negatively characterize many answerers (that is, volunteer contributors to a knowledge-sharing web site) as part of an "underlying problem", and deciding for them that they are not attempting to convey understanding. – zyx Nov 6 '13 at 10:28
• That aside, as an economist I think you understand better than most that the question of "what we are encouraging" is not settled so easily by declarations of principle or by speculations about motivations and outcomes (that are basically a restatement of personal philosophy, tastes and assumptions). For example, there is really no way to dis-courage anyone from posting, in an environment where asking is anonymous and cost-free but can produce significant benefits even in case of hints. And there are costs to attempts to disincentive various kinds of people or postings. @MichaelGreinecker – zyx Nov 6 '13 at 10:34
• @zyx Sure. I just wanted to clarify that there is a rationale for a policy that is not based on paternalism. And these policies do increase the cost of assigning ones homework to MSE. I'm sure you know my views on the issue by now, but I have no interest in playing advocacy here now. – Michael Greinecker Nov 6 '13 at 11:13
• @zyx Some of the people who "supersede the judgement of the OP or the answerers" also might well be, and often are, experienced teachers. (I don't mean to sound petty, or dismissive of the reasonable point you are making, but I think this observation is necessary for balance.) – mdp Nov 6 '13 at 14:22
• I'd like to refer to a previous answer I wrote on a very similar topic. – davidlowryduda Nov 6 '13 at 15:44
• The only sure fire way to prevent students from cheating, is to have no answers of any kind on MSE. Because one question in one place that is not answering a homework question, could be discovered and read by somebody who is answering a homework question. And vice versa. So well-intentioned attempts at avoiding giving students short answers fall short of their goal, and possibly only serve to hinder those who are not doing homework =). – Kent Fredric Nov 7 '13 at 12:59
• @KentFredric I'm not sure I follow - a well-crafted hint should be helpful to the OP whether they are doing homework or not. (And furthermore, if they are asking a question that could reasonably be homework, then they are probably trying to learn something, and so a hint that leaves them something to do is - I think - highly likely to be more useful in the long run than a complete solution). – mdp Nov 8 '13 at 16:24
• @KentFredric: Reminds me of the Simpson's quote: "you tried your best and you failed miserably. The lesson is, never try." Making progress towards a goal is still progress even if you don't reach it. But, IMO, having a "sure fire way to prevent cheating" is an overly naive goal anyways. But setting that aside, as MattPressland alludes to, generally speaking, the people who are advocating measures against cheating also believe that the measures remain worthwhile even if we completely ignore the issue of cheating. – user14972 Nov 9 '13 at 9:47

In my perception, we have the following approximate categories of questions:

• Homework(-like) without effort;
• Homework(-like) with effort;
• Seeking understanding;
• Research-quality questions.

(Allow me to not bother with the inevitable grey areas ('What is effort?', 'This question is homework but asks for the underlying mechanism').)

• Hints;
• "Some gaps left for you"-type full answers;
• "All details covered"-type full answers;
• Full answers with pedagogical merit;

where there is effectively a continuous scale spanning the first three bullets.

Of these, there are only a few problematic combinations (ordered from most to least opposed):

• "No effort" question + "All details covered" answer;
• "No effort" question + "Gaps left" answer;
• "Effort" question + "All details covered" answer;
• "No effort" question + "Hint" answer;
• "Understanding" question + "Hint" answer.

Inevitably there are exceptions to this generic scheme, but for the most part, this will sum it up quite nicely.

In particular:

It's always acceptable to provide a high-quality post of pedagogical merit (e.g. addressing the general issue instead of the particular, with pointers to important insights). Such posts are an asset to MSE.

Now you ask, "How do I know if my post is of that type?" To which I say: You just know it when you're writing it.

Unfortunately, as it stands now, such posts are relatively rare -- not least because they require great insight in the subject, as well as a lot of time.

So I'd like to ask all of you, the next time you write an answer, to consider if you can add to it in such a way that it becomes more than merely the solution to the question at hand.

In a similar vein to the suggestion to have an MSE blog and the list of generalizations of common questions, we could even post generic questions with the very purpose of accumulating such answers.

• A big thanks for stating a distinction between an answer with formally sufficient details and the next pedagogical step beyond that. A lot of the "didactic" school of rhetoric on meta around the concept (to paraphrase somewhat sarcastically) of helping students by not answering their questions, revolves around the idea that a "model answer" of the all-details-covered type, is a bad response in many cases. But one solution to the alleged pedagogical deficiency of such answers, can be to add more information into them, rather than subtracting it to form hints, partial answers, or silence. – zyx Nov 6 '13 at 9:58
• @zyx I agree that adding more information to a model answer can be helpful - I would love to see more answers that explain how a solution can be thought up, and usually this involves explaining the full solution. But this seems to happen very infrequently, and I think one way of getting it to happen more is to encourage people to ask questions in a way which makes such answers more likely. (E.g. instead of "what is the solution to this problem?", ask "how would I come up with a solution to this problem, given that I know the following?"). – mdp Nov 6 '13 at 14:25
• I like the distinctions made in the above post, and I would like to think answers like mine qualify as pedogogically useful, but I think that I should actively try to reach this high standard on my answers. – Juan Sebastian Lozano Nov 6 '13 at 16:35
• You missed the rarely occurring, but worth more than the rest combination seeking understanding + pedagogical answer. – Asaf Karagila Nov 6 '13 at 16:41
• Unfortunately, the category in which a question falls isn’t always recognized. This question is a case in point. Anyone who’s at all familiar with the OP’s posts should know that it was asked for understanding: he’s been asking sophisticated topology questions for a long time, some of them research level, others asking for critiques of his own arguments or examples. (I’m increasingly inclined to think that in general one probably ought not to pass judgement on a mathematical question that one can’t answer.) – Brian M. Scott Nov 6 '13 at 17:07
• @BrianM.Scott It's true; no scheme can ever hope for flawless application. A scheme's strength lies in keeping the fraction of mistakes minimal, and the ease of correcting those. As to your parenthetical remark, I would concur with the more lenient "... that one can't meaningfully think about." I'm often immediately bubbling with ideas upon seeing a question, but this does not imply that (I know) I'm also able to answer. I might have hearsay knowledge of solving techniques without being able to use them myself. I would be comfortable basing my assessment on things like that. – Lord_Farin Nov 6 '13 at 20:22
• @Juan, my comment was not addressed to you... – Asaf Karagila Nov 7 '13 at 9:22
• @Asaf I'm still at a loss as to what you meant. Of 16 possible combinations, I listed some that are problematic. I stated that pedagogical answers are an asset. I didn't list all 16, and I didn't intend to... – Lord_Farin Nov 7 '13 at 9:31
• I guess I missed that "few problemaic" part... :-) – Asaf Karagila Nov 7 '13 at 9:37
• @Lord_Farin: (On passing judgement.) We probably don’t disagree much in practice: my in general and probably were intended to allow some weakening of can’t answer in some cases. – Brian M. Scott Nov 7 '13 at 9:52
• I agree with the sentiment of Brian's parenthetical comment. I also agree with Lord Farin's last paragraph. A radical proposal: to increase the answerers commitment to their posts, what if we introduce a daily maximum to the number of answers in addition to the number of daily questions? I probably would reconsider posting some of my hints-only answers, if there were a daily maximum of, say 3 answers. Also the FGITW would need to put more thought to their answers? – Jyrki Lahtonen Nov 8 '13 at 19:57
• @Jyrki That's a very interesting idea. I think it merits a separate discussion. – Lord_Farin Nov 8 '13 at 22:04
• @Jyrki: Limiting answers seems a bit impractical when we’re up to about $500$ questions a day. It would also reduce the proportion of answers provided by the most experienced answerers. (And for whatever it’s worth, it would also ensure my departure: the site would offer far too little to hold my interest, and the limitation would be impossibly frustrating.) – Brian M. Scott Nov 10 '13 at 9:13

That future irresponsible OP's can take advantage of a complete answer is a reality, but also, a very weak argument in favor of leaving homework answers at "hint level" for ever. We should care more about all the rest. Leeches will never go away -this is not reason enough to underfeed the truly and honestly needy.

I have been with this site for just over a month and was not even able to participate in much of this period.
Before I bring up my issue, I should say that I am a big fan of this website and am already starting to recognize the acronyms of some of its users and their styles (HINT givers and those telling the student to show s/he's made an effort, and more).

To my point:
We need to try making the solution understandable to the person who asked the question?

Too many times I think of the (poor) student who wanted to get an answer to the question he was asked to do for homework (even if asking for a "free ride") and then he gets a reply quoting Euler that s/he does not even know how to pronounce. Aren't we alienating this student from math and from ever asking a math related question in public?
He or she come in with a question they don't know the answer to and they get replies that are more difficult to understand by 3 orders of magnitude!

I find myself, many times, trying to add, after the learned replies, a reply that I think the question provider would understand.

All math lovers, me included, love the neat solution with attention to every detail. We (again, me included) "edit" anything that does not seem perfect in the question or in any reply. Is there another website where so much of other people's work is being edited? It's ok if you feel confident enough about what you wrote, but its difficult to learn to accept when you are less experienced.

Lets treat the homework questions with the understanding that the form of the answer and the language of our comments may be more important than their completeness in order to keep students close to math.

• I am a high school student, and as such, if I were to be presented with two solutions to a simple homework problem I would appreciate both answers, even if I only understood at a surface level or even if I was only able to recognize the field of math which the problem used. I remember two or so years ago when I saw the derivation of the volume of a sphere using integration of solids of rotation, it gave me a better grasp of what the definite integral did and its power, even though I didn't understand the solution fully. I think both answers have a place here at MSE, and I would up vote both. – Juan Sebastian Lozano Nov 13 '13 at 3:20
• Relevant: #11419. – Lord_Farin Nov 13 '13 at 8:43
• @Juan Sebastian Lozano Muñoz: You are not the type of student I was referring to, so you are not a good example. If you are reading this forum and you remember from 2 years back "the derivation of the volume of a sphere using integration of solids of rotation" then you are not the student to be alienated. – DannyDan Nov 13 '13 at 11:27
• I am curious how this essay is connected to the question posed. Perhaps instead of waxing poetic about users "quoting Euler", and how dutifully we edit "anything that does not seem perfect", you could have given some indication of your opinion on the matter at hand. – user642796 Nov 13 '13 at 16:22
• @Arthur Fischer: I think this is the exact matter. When looking at homework issues, as opposed to other questions, the people who reply should take into consideration, besides the completeness of the reply and whether they provide a hint or a full answer, the readability of the reply and whether they think that the person who asked the question can benefit from the answer. You could, of course, decide to stay high and dry and not take any of this into consideration when you formulate the answer, but I would disagree with that. – DannyDan Nov 13 '13 at 20:01