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This question emerges from a short chat with André Nicolas in the comments of his answer to this question.

I decided to downvote his answer because I considered that he should have given hints on how to solve the problem rather than providing the OP with a direct answer. He answered

Possibly, but my feeling is that it is difficult enough for OP to understand a direct answer.

If I read him well, his case is that, when the OP is having a hard time with understanding even a direct answer, hints might not be helpful.

My position was rather opposite

My case : it is not always the case that a difficulty to understand even direct answer implies that it it preferable to be presented a direct answer. Being walked through the process of solving the problem may be a better way to eventually understand the final answer rather than proceeding backward and trying to understand a direct answer after it has been presented.

Although, I recognize that this is a difficult question and to which there is no general answer. Still, when answering a question, one must decide between one of the two option (or something in between).

  • Can you think of specific circumstances in which would indicate that a hint will be more helpful than a full answer?

  • Do you that hints tend to benefit only sufficiently "advanced" OP and that it is better to provide struggling OP with complete answers? If so why?

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    $\begingroup$ If you are looking for a somewhat general rule, you will not find it. $\endgroup$
    – Mariano Suárez-Álvarez
    Nov 13, 2013 at 1:19
  • $\begingroup$ I am aware of the fact that no general rule exists. I am more looking for something like rule of thumb... I'll edit the question to make this clearer. $\endgroup$
    – Martin Van der Linden
    Nov 13, 2013 at 1:21
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    $\begingroup$ Rule of thumb: give the kind of answer that you expect to be the most beneficial. $\endgroup$
    – user103402
    Nov 13, 2013 at 1:22
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    $\begingroup$ Students are different. What is best to do is a guess, modulated by experience. A fleeting electronic relationship is different from seeing the person in the office. Descriptions that one uses depend on whether effectively one has a common language with the student. $\endgroup$
    – André Nicolas
    Nov 13, 2013 at 1:27
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    $\begingroup$ @ user103402 : my question precisely aims at rationalizing a little this expectation you talk about -- if possible at all. Either the expectation on which answer will be most beneficial on the random flip a coin, or it based on something. If it is based on something, I think it might be useful to share what this something is about. $\endgroup$
    – Martin Van der Linden
    Nov 13, 2013 at 1:35
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    $\begingroup$ I've found that, despite the endless threads on meta, people ask, answer, and vote however the #%+} they want. I respect both your (@OP) opinion and A.N.'s judgement. Why can't we just leave it at that? $\endgroup$
    – The Chaz 2.0
    Nov 13, 2013 at 2:43
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    $\begingroup$ Your profile says that you're a Ph.D. student, and perhaps you have had to opportunities to teach students (as a TA, or maybe even directly). Not everyone benefits from hints all the time. There are times when you can see from the question that the OP is confused, and hints will not help them. In that case giving a full and pedagogical answer can be a teaching experience for them, that they can use later on. Perhaps their teacher sucks, and they can't manage to learn how to solve these kind of problems. But sometimes one can try to size up the OP and see a full solution is better. $\endgroup$
    – Asaf Karagila Mod
    Nov 13, 2013 at 6:19
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    $\begingroup$ Bluntly, I consider your downvote reprehensible. Downvotes, if used at all — I don’t use them on the main site — should be for mathematical errors that the answerer doesn’t correct in a reasonable period after a comment mentioning the error has been left. (Yes, I know that people will vote as they please.) Moreover, André is one of the best teachers here; while I don’t always agree with it, I trust his judgement far more than most, and certainly far more than that of someone with relatively little teaching experience. $\endgroup$
    – Brian M. Scott
    Nov 13, 2013 at 11:13
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    $\begingroup$ I can speak from my experience as an often-time asker. I am, perhaps somewhat strangely, much happier to receive a hint than a full answer. With a direct answer I have to retrace someone´s thoughts which for me is rather taxing and headache inducing, much more so than having to rederive the thing myself and I don´t even speak about the pride of doing some non-trivial work myself. $\endgroup$
    – Adam
    Nov 13, 2013 at 11:36
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    $\begingroup$ We should distinguish between "full answer" and "explanation about how to solve a problem that happens to provide a full answer to the question asked in the process". I have seen full answers here on MSE that have been so far to the extreme that it was inconceivable it could be of any use to anyone other than those who already knew how to solve the problem or just wanted to copy an answer to fill out their homework sheet. $\endgroup$
    – user14972
    Nov 13, 2013 at 14:30
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    $\begingroup$ @ Brian M. Scott : I must admit that I do not know about the best practice on mathSE concerning downvotes (when they should and should not be used). A comment would probably have been enough. After André pointed out that he had been through a process of trying less direct answer with the OP (which I could not see as he erased the comment), I wanted to change my vote but it was too late (past 30 minutes). $\endgroup$
    – Martin Van der Linden
    Nov 13, 2013 at 14:37

2 Answers 2

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This isn't so much an answer to your question as it is an elaboration on your "something in between." It was too big for a comment.

Sometimes I like to write what I call roadmap solutions, which are more or less halfway between a hint and a full solution. Roadmaps consist of a list of intermediate hints / leading questions guiding the reader through the proof, with spoiler-ed out proofs of each individual step. Example:

Question. Prove that $A=B$.

Answer:

Step 1. First we want to show that $A\subseteq B$.

Take an $a\in A$. Because (something), $a\in B$. Since this holds for all $a\in A$, $A\subseteq B$.

Step 2. To show that $B\subseteq A$, it will be easier to use the contrapositive.

Take an $b\notin A$. Due to reasons, $b\notin B$. So, by contrapositive, $B\subseteq A$.

Step 3. Now we have $A\subseteq B$ and $B\subseteq A$. Does this mean that $A=B$?

Yes.

(done)

This method puts faith in the asker to actually try to work out each hint before proceeding to the next, rather than simply looking at the answers right away and reading it like a full proof. Sometimes they'll do it, sometimes they won't. The hope is that it will at least give them the opportunity to work the proof out step by step by providing contingencies in case they get stuck on one small part, or by providing a path to the proof in case they can't see the forest for the trees right away. Personally, I like learning this way, especially when I'm in unfamiliar territory, so I write answers like this every so often.

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    $\begingroup$ This is really great. I'm going to start trying to answer certain programming questions this way on SO as an experiment. It is sometimes hard to find a middle ground between helping somebody and doing it for them. $\endgroup$
    – Jason C
    Nov 18, 2013 at 7:30
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An argument pro-full answers from this question :

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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    $\begingroup$ "When you have the full solution, you can choose to only take the hint that you need." - I claim that many students find this very difficult in practice. They often read the full solution, think they now know how to do the question, immediately forget the solution, and move on without learning anything. $\endgroup$
    – mdp
    Nov 13, 2013 at 10:36
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    $\begingroup$ One problem is that even students who are interested in educating themselves often don't know the best way to do this (which sounds patronizing, but I see it a lot). Giving a hint is less dangerous - if they need more they can ask for it, but if you give too much you can't take it back. $\endgroup$
    – mdp
    Nov 13, 2013 at 10:41

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