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Recently a few discussions on homework questions appeared (or re-appeared) on meta.

Originally I wanted to post something like this into one of them. But perhaps it is better to post this as a separate question. (Although it might happen that the site begins to be cluttered with several similar discussions.)

My question is:

  • In your opinion, what are the reasons that some answerers prefer to post full solutions?

If you feel like that, you may also explain whether you find these reasons good/bad.

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    $\begingroup$ I downvoted this question because it seems to be biased towards one side of the argument. $\endgroup$ – Ronald May 24 '12 at 12:33
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    $\begingroup$ @Ronald, where is it written that a question should be unbiased? $\endgroup$ – Gerry Myerson May 24 '12 at 13:10
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    $\begingroup$ @GerryMyerson it is written in several places that it is good etiquette to give reasons for downvoting; I am doing so. $\endgroup$ – Ronald May 24 '12 at 13:38
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    $\begingroup$ @GerryMyerson: I don't know that questions inherently should be unbiased, but if one reads a particular question as being biased, then it is reasonable to vote it up/down not just on good/bad question but on agree/disagree with the perceived leaning of the question (I'm specifically talking about meta questions here). $\endgroup$ – Isaac May 24 '12 at 15:20
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    $\begingroup$ It occurs to me that a relatively small group of people could change the social norm about this pretty quickly just by downvoting over-explicit answers to homework questions whenever they saw them. Hardly anyone does this. $\endgroup$ – MJD May 24 '12 at 15:52
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    $\begingroup$ (This was not meant as a recommendation, only as a description of the state of affairs.) $\endgroup$ – MJD May 24 '12 at 16:08
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    $\begingroup$ @MarkDominus True: I would think that the comments noting that full solutions often receive more upvotes really suggests that the full solutions are broadly considered to be better. $\endgroup$ – Ronald May 24 '12 at 20:06
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    $\begingroup$ I wonder what's wrong with a full proof. Nobody forces you to read it all. You can always skip any part of it. $\endgroup$ – Makoto Kato Jun 19 '12 at 23:59
  • $\begingroup$ @Ronals, but that is not a good reason for downvoting. $\endgroup$ – JMCF125 Feb 10 '14 at 21:44

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"In your opinion, what are the reasons that some answerers prefer to post full solutions?"

One reason that I don't see mentioned yet is that we are a bunch of show-offs. When we can solve a problem, we're very proud of ourselves, and we want everyone to know we can solve it, so we post a full solution. Hints aren't enough - we could be hiding the shameful fact that we don't really know how to fill in the gaps - it has to be the full solution to be convincing.

Well, anyway, that's one reason why I occasionally post full solutions. I try to resist. I think that more often than not, I succeed, and just post hints (and then get annoyed when someone else posts a full solution, and even more annoyed when they don't acknowledge that everything they've posted is implicit in the answer I already put up, and really really annoyed when they get all the upvotes that I deserved to get...where was I? I seem to have gone off-topic...), but I can't always resist the temptation.

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    $\begingroup$ I admire great hints rather than full solutions, such as Bill's, but you're right here! $\endgroup$ – Pedro Tamaroff May 24 '12 at 17:58
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    $\begingroup$ This is a great answer :-) $\endgroup$ – joriki May 29 '12 at 14:27
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    $\begingroup$ I agree with this, but I also think that the layout of the site prevents step by step hinting in a small way. When teaching in person then it is a different matter, I never just give away answers without first dealing with understanding and hints. I haven't been here very long but when answering a question I gather there are three possibilities...post a comment with a hint, post a full solution in the answer box or post a hint in the answer box. It is very difficult to judge which of these to do. Firstly it is hard to judge the level at which the OP is at... $\endgroup$ – fretty Jun 25 '12 at 13:23
  • $\begingroup$ Secondly it is difficult to judge whether they actually care enough about the maths to want to work towards a solution via hints. Thirdly your answers are being read by many people that also know how to answer the question (some in better ways than yours) and so anything you do post has to be impeccable else gets downvoted. It is quite a weird balancing act but I am slowly getting used to it. $\endgroup$ – fretty Jun 25 '12 at 13:24
  • $\begingroup$ But yes, I agree about wanting to show-off your own solution. I will just have to learn to resist as you have. $\endgroup$ – fretty Jun 25 '12 at 13:26
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I have been trying to transition to giving hints rather than full answers, but in the past I have certainly been guilty of posting whole answers where hints would have been more appropriate. The thing is, in my opinion giving good hints is actually harder than giving full answers. Normally I am not well-versed enough in the subject to immediately know my hint will lead to a solution, so I have to work it out myself anyway. So at best, I'm stuck converting a full answer into a hint. But then I also need to figure out what the meat of my argument is, and where the OP is stuck. A good hint needs to convey the crucial idea that the OP needs to bridge a gap in their understanding. That's hard!

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    $\begingroup$ "Giving good hints is actually harder than giving full answers." Very very true and I find this difficult as well. $\endgroup$ – user17762 May 24 '12 at 11:22
  • $\begingroup$ I wanted to upvote because of what Marvis highlighted, but I also wanted to downvote because you use "guilty of posting whole answers" which is a tremendously negative context. So I'll just not vote for now... $\endgroup$ – Asaf Karagila May 24 '12 at 12:38
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    $\begingroup$ @AsafKaragila I didn't mean to frame posting whole answers as a bad thing. I included that statement because I wanted to be frank that I cannot without some degree of hypocrisy speak in favor of hints over answers. $\endgroup$ – Alex Becker May 24 '12 at 13:36
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I can think of a few reasons:

  • Homework and homework-like questions are low-hanging fruit - an easier way to earn the reputation. Full solution usually gets more upvotes.

Although we should not put too much worth in reputation, I personally remember that I worked to get 2K to be able to edit answers and questions. (Unless the threshold was different back than.) And I also worked towards 10K to be able to see deleted questions.

Both these features are useful - so I could pretend that my main motivation was that I would be more useful for the site if I have access to them. But another reason is that getting reputation points feels good, just like any other kind of reward.

  • If I put a hint, I feel kind of obligation to get back to the question and see whether the OP did some progress from the hint. Instead of going to the same question several times, writing full solution is more comfortable.

I used to do this earlier - I've checked quite often whether there's something new at that question or whether I was pinged. I don't do that anymore as this approach takes too much of my time. Doing something like this would be the closest thing to dialogue between teacher and student, where teacher sees what are the problems and can direct the student towards solution.

However, even if I posted some comment or answer with a hint and I don't come back to the question, there are certainly other people who can help the OP. And we have also chat, which is more interactive.

  • I can also understand the point made by JDH, see here, that giving just hints may leave to half-answers or incorrect answers.

I have seen some cases, where hints were not correct - the poster of hint probably missed some step in the solution. But in such cases we have many other experienced users here, probably some of them will notice this and point out the problem.

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I often try to give hints, but there are times that I post full solutions, and when that happens I'd say my reasoning is usually one or more of the following:

  • I'm in a rush (or I just want to post first) and don't have time to be clever and think of an appropriate hint (where to stop, what to say to spur them on in the right direction, etc.).

  • I feel that the answer I have in mind is so short that I don't think there exists a useful hint that isn't pretty much the whole thing anyway.

  • Alternately, I feel that the answer I have in mind is complex enough that I can't think of a reasonable-length hint that I'm confident would suffice.

  • I'm feeling shaky about my grasp of the material myself, and I want to write out my solution in full to make sure it's correct.

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I sometimes give hints, and sometimes full solutions.

The reason is that sometimes I am unsure of my ability to write a complete solution, but I know what is the correct way of approaching the problem (and I usually prefer wasting my time doing something else, and not filling the details myself).

Other times, however, I feel that I can give a very good account of all the details. I think this can be very helpful and instructive. Not all people who come here to ask help in their homework are looking for final solutions, some want to learn but their local environment is not well-suited or they might have teachers which are poor educators and it's hard to follow them.

I prefer to think that people who write properly formulated questions really want to learn, but could not do it outside this site. Of course that when someone merely spits a question from the questions sheet (or worse, e.g. "hurry up guys!") I will avoid giving any help at all.

However, when I want to help someone, I want them to fully understand what I meant. Hints can be hit-miss sometimes, but giving a complete solution to a problem means that you have the opportunity to explain the relationship between the propositions and the reasoning you apply. I like doing that.

Even when students come to me in office hours just to ask how to solve one specific question, I often end up giving them a bunch of theory too. During the fall semester when I taught math/comp. sci. students they liked it; now when I teach engineering students they like it less.

Of course when there are users which I see constantly asking questions from homework, I may prefer to switch to hints again because I fear they might not be trying hard enough to chew on their own.


Now comes a good point to think about it. If we want people to learn mathematics, and we really just want them to learn it. I was told (and I agreed since then) by one of my teachers that the only reason they require us to submit homework is that they want us to work on the problems, if they knew we would do that there would have been no submission of homework. However other courses already ask us to submit homework so we would do those first and not give any time to that specific course. Therefore all homework is for submission.

So we want people to chew on the problems chosen for the homework sheets, because those are often the problems which require the student to exercise the definitions and the theorems and are usually chosen for a specific level of education. But the problem, again, with posting full answers is that we want them to chew on the problems on their own, I agree that this is the only way to advance in your studies.

Alas, if someone is reading on their own. The book is well fitting to their level, but they come with questions from the book, and they come and ask and ask and ask. It would seem that they are not trying to solve the problems hard, if they don't see a solution right away - they come to ask for help. This is bad because when you struggle with the problem you learn a lot more.

Now ask yourself, would you have answered in full the questions of someone who was self-studying a certain topic? Is this any different than a homework sheet? Yes. It is different because of honor code.

But my goal is to educate not to be a moral compass, if someone wants to learn - I will do my best to help them learn.

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    $\begingroup$ Now that is a proper rant. $\endgroup$ – Asaf Karagila May 24 '12 at 7:12
  • $\begingroup$ @Peter: I meant final, not full. Thanks for reading, though! $\endgroup$ – Asaf Karagila May 24 '12 at 18:14
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    $\begingroup$ "Full" means it solves the problem wholely, and "final solution" means it is the last possible one. (And it doesn't remind me of good times, if you have to ask.) $\endgroup$ – Pedro Tamaroff May 24 '12 at 19:52
  • $\begingroup$ @Peter: I mean that some people really just want a final answer, with a limited solution to accompany that. Full solutions are not necessarily harmful, but wanting just a final answer only is harmful in most cases. $\endgroup$ – Asaf Karagila May 24 '12 at 19:55
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    $\begingroup$ @Asaf : Don't let some engineering students ruin your view of all of them! Things make more sense when they're placed in their theoretical background. Moreover, background knowledge is serendipitous. $\endgroup$ – bright-star Mar 3 '13 at 8:52
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    $\begingroup$ @Trevor: Theoretical background makes sense if and only if the student wants to understand it. Otherwise it becomes a shackle which inevitably holds him back from getting that passing grade she/he wants so badly. $\endgroup$ – Asaf Karagila Mar 3 '13 at 9:01
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    $\begingroup$ Passing grades can be dangerous later when you need that actual understanding you skipped, especially for real work. $\endgroup$ – bright-star Mar 3 '13 at 9:14
  • $\begingroup$ @Trevor: True that. But it's not that any of my student learned any calculus from me. If anything, it was the other way around. :-) $\endgroup$ – Asaf Karagila Mar 3 '13 at 9:18
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The main reason I sometimes post full solution is that it is fun to nail it down completely and write out all the arguments. Most of us would accept that when we write out anything, we would like it to be complete and in fact thats how we do our homework.

Also, often I do not check if the question is tagged homework or not (unless the OP explicitly mentions that in the question). If I think the question is interesting (or) I can answer the question, I try to answer it.

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Hint: full answers are seen as more complete, and get more upvotes.

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    $\begingroup$ Your hint hasn't received many upvotes. Consider expanding it to a full answer. $\endgroup$ – Austin Mohr Jun 15 '12 at 16:50
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I don't know whether I am "guilty" of prefering full solutions to hints (or whether this is bad or not). Like I said in a recent comment:

Sometimes I give complete solutions; sometimes I don't. It usually depends on a complex combination of (i) the question; (ii) the effort the student has displayed; (iii) the difficulty of the problem; (iv) how much the problem might be suitable for a more general discussion than a specific one. I can't say I have an algorithm, more a heuristic, very similar to the one I use in person with students when they come to office hours: sometimes I give them hints, sometimes I walk them through solutions.

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I find it educationally better to provide full solutions.

If you are learning, it's better to receive a full solution - if you only need a hint, that's great. When you have the full solution, you can choose to only take the hint that you need.

However, if you receive a hint: this may or may not be the hint that you needed to progress. In a medium such as StackExchange, it is difficult or impossible to know if a particular hint actually matches the questioner's difficulty.

If a student is here with an interest to educate themself, they will not be harmed by receiving a full solution - as they can apply this understanding to challenge themselves on novel questions.

I actually find it important to ask the opposite question - why do people think it's better to provide a hint?!

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    $\begingroup$ There are many pedagogical arguments against providing full answers, but the main one used here is that it undermines the concept of homework to do someone's problems for them. $\endgroup$ – Alex Becker May 24 '12 at 11:49
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    $\begingroup$ If there are "many pedagogical arguments", then please give them. You can read mine above. $\endgroup$ – Ronald May 24 '12 at 11:51
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    $\begingroup$ I do not agree with your concept of homework - the concept of homework is to improve and consolidate the student's understanding. A good student should understand that answering the homework questions is not their ultimate goal in life: the student's clear understanding is more important than the teacher receiving the correct work to mark. $\endgroup$ – Ronald May 24 '12 at 11:52
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    $\begingroup$ Of course answering a homework question is not the ultimate goal in life. But homework is not assigned for no reason. The entire point of most math homework assignments is for the student to figure out how to solve such problems. If the point were for them to see the answers, it would be a handout, not a problem set. $\endgroup$ – Alex Becker May 24 '12 at 11:54
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    $\begingroup$ I would prefer to take responsibility to educate someone properly, than to take responsibility to police homework. $\endgroup$ – Ronald May 24 '12 at 11:57
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    $\begingroup$ The main argument though is that you are not properly educating someone by doing their homework for them, but rather undermining their education. And there is the other people in the class to consider, who aren't cheating on their homework. $\endgroup$ – Alex Becker May 24 '12 at 11:58
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    $\begingroup$ OK, but I don't agree with that argument. How does helping someone with one question of homework "undermine education"? It is the fundamental mission of education to impart knowledge and understanding. I agree that the ultimate goal is for a student to be able to answer questions independently - but this does not necessarily mean homework questions MUST be done independently. If they cannot be done independently, then extra help is an essential part of the learning process. I would say that refusing to offer full help undermines education. That's my argument. $\endgroup$ – Ronald May 24 '12 at 12:06
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    $\begingroup$ Very few people here are upset about providing answers to "one question of homework". The issue is people who repeatedly post entire assignments, which happens with some regularity. $\endgroup$ – Alex Becker May 24 '12 at 12:09
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    $\begingroup$ The main place "hint" vs "full solution" comes up is "should we give hints or full solutions to people who ask lots of HW questions". $\endgroup$ – Alex Becker May 24 '12 at 12:25
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    $\begingroup$ One argument in favour of giving hints is that students may gain more by working out the problem themselves using a hint than by looking at someone else's full solution. As a student I preferred to be given a hint, even a big one if necessary, as I found it more rewarding and a better learning experience to work out details myself. $\endgroup$ – Matthew Towers May 24 '12 at 13:16
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    $\begingroup$ I agree that a hint is useful to some students in some circumstances. However, I think that the students who have come to post here to get an answer are probably not in that position, and it is a mistake to presume that they are. (If you know you would prefer a hint, then you can ask for a hint) $\endgroup$ – Ronald May 24 '12 at 13:41
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    $\begingroup$ My argument against full solutions would be that one learns to solve problems by solving problems, not by reading other people's solutions of problems. $\endgroup$ – MJD May 24 '12 at 15:11
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    $\begingroup$ strongly disagree... if that's always true, why does this site exist at all? $\endgroup$ – Ronald May 24 '12 at 20:01
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    $\begingroup$ To answer non-homework questions and to provide hints for homework questions. $\endgroup$ – MJD May 25 '12 at 21:52
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    $\begingroup$ OK. I am not clear what the distinction is. It's simply saying we don't want to treat high school-level mathematicians in the same way as we would treat everyone else. I find this attitude disrespectful to learners of our subject. Sigh. $\endgroup$ – Ronald May 27 '12 at 20:53
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For practice with mathematical writing, and to be sure that the "hint" is actually correct and sufficient, if carried through.

For example, I "answered" a question here with a short hint-like statement. It seemed obvious to me but it turned out that there was a subtlety to the question and what I said amounted to circular logic.

It's dangerous to get in the habit of thinking you know something, without actually following through with the details.

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    $\begingroup$ That is a good argument to work out the details yourself, but not necessarily to post them! There are many examples of things one should check carefully on one's own, even though there is no reason to publish those details. (Unfortunately, many people -- myself included when I was younger, if I am being honest -- often misconstrue this practice as an opportunity to skip the behind-the-scenes work.) $\endgroup$ – Michael Joyce Feb 24 '13 at 17:03

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