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I was wondering what is your method for giving hints. I find it generally harder than posting full answers (but less tedious, as some trivial points may be omitted), but still I have to write a something like more than half (with respect to details) of a full solution and then reduce it to hints (unless I know the full solution beforehand, e.g. with very simple questions).

Even after that giving a hint is not easy: how to point to right direction and not to give away the full solution? How to not spoil the teaching content behind the question? How to make someone learn rather than "guess the teacher's password"? Finally, from multiple key points, how to pick those suited for the OP? For example, does the OP barely understands the subject and wish only for guide-through, or maybe seeks a deeper understanding and showing a correspondence/analogy to some other theorem is more appropriate?

Any rules of thumb, suggestions, experiences, hints? ;-)

PS The context is math.SE of course, but it is not limited, if someone wishes to share their teaching experience, I would welcome such posts very much, as those too probably apply here.

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    $\begingroup$ A good question. Experience does help, but our mileage varies so much (as does the askers background) that sometimes my hints are duds. Either they give the show away, or help too little. I feel good about a hint, whenever it provokes the OP to ask more questions. Feeling like a Socrates today? Alas, I cannot always come up with a hint that works as well as that. Can't estimate my batting average here. Depends on who's pitching. $\endgroup$
    – Jyrki Lahtonen
    Jul 5, 2013 at 21:15
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    $\begingroup$ I do it on the basis of several decades of experience, and I don’t always guess right. If there’s one thing that’s important to remember, it’s that hints are rarely as helpful as you think they should be. This kind of hitting the nail on the head is rare. This is more typical. And this is not unusual. $\endgroup$
    – Brian M. Scott
    Jul 6, 2013 at 3:14
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    $\begingroup$ You can give a hint, but you can't force them to take the hint and work with it. If the OP wants MSE to do most of the work, then hints are pointless. $\endgroup$
    – Calvin Lin
    Jul 11, 2013 at 4:08
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    $\begingroup$ @CalvinLin I think it is still better than giving the full solution right away. $\endgroup$
    – dtldarek
    Jul 11, 2013 at 7:08
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    $\begingroup$ Speaking from very very limited experience, here's one concrete suggestion that I like: teach by example. Put the OP's original question aside, and have the OP work on a simplified problem of a similar form. Of course, this does have its time and place. $\endgroup$
    – Gyu Eun Lee
    Jul 12, 2013 at 3:10

3 Answers 3

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Even after that giving a hint is not easy: how to point to right direction and not to give away the full solution?

The only thing that helps here is experience. Unless you have just learned about the relevant mathematics recently, it is usually hard to relate with someone who is learning the material for the first time. It helps if you have had some experience teaching a course that covers those material so that you will have some gauge of students' common problems.

How to not spoil the teaching content behind the question?

Socratic style can help a bit: instead of telling them which theorem to use, asking them a directed question so that they will remember that they have learned such-and-such theorem.

How to make someone learn rather than "guess the teacher's password"?

There is almost nothing you can do about this... that is more a function of the learner than the teacher.

Finally, from multiple key points, how to pick those suited for the OP? For example, does the OP barely understands the subject and wish only for guide-through, or maybe seeks a deeper understanding and showing a correspondence/analogy to some other theorem is more appropriate?

This is why many users ask for more context when faced with "homework-type" questions. Without additional information and clarification from the OP, you can only guess at where the difficulty is and what constitutes a good hint for his or her level.

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    $\begingroup$ I don't think that the word "experience" is standing out enough. I suggest using $$\huge\mathbf{Experience.}$$ Because you really can't stress this enough! $\endgroup$
    – Asaf Karagila Mod
    Jul 5, 2013 at 9:19
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    $\begingroup$ I have the problem that most of the askers are not frequent users, and often don't even respond to questions (perhaps taking them as rhetorical, or, more cynically, too much trouble to answer). $\endgroup$
    – Eric Tressler
    Jul 6, 2013 at 10:20
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    $\begingroup$ @Eric: I get a significantly better response to questions embedded in incomplete answers than to questions asked in comments under the original question. $\endgroup$
    – Brian M. Scott
    Jul 8, 2013 at 2:52
  • $\begingroup$ I really appreciate your answer. I feel really silly asking this, but still I have to: are there any schemes/templates of hints that come up frequently? For example, I observed that breaking the problem into smaller (but reasonable) chunks (if possible) often helps more than mentioning appropriate theorem or making an analogy or considering a simplified special case. Is it possible to put just a bit of your experience into words? Anything that could help giving hints more consciously, rather than "feeling out what would help". $\endgroup$
    – dtldarek
    Jul 10, 2013 at 13:20
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    $\begingroup$ @dtldarek For problems that will need breaking up into small parts, I generally don't bother with answering it if the question is one for which I should give hints instead of full answers. That is because the StackExchange platform is not the best for tutoring! I generally limit giving hints to questions for which I can identify one or two crucial intermediate steps that needs to be done, and write something like "You should first show $X = Y$ is true." $\endgroup$
    – Willie Wong
    Jul 10, 2013 at 13:31
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    $\begingroup$ @dtldarek: Probably the two most useful techniques are: (1) restating the problem, usually with some expansion, pointing out exactly what needs to be done; and (2) breaking it into smaller pieces, thereby providing a roadmap without actually doing all of the work. Questions that focus attention on the key point are good, if there is a key point. Sometimes giving a single key step makes a good hint. (The SE platform is not idea for tutoring, but it’s not that bad, either.) $\endgroup$
    – Brian M. Scott
    Jul 11, 2013 at 4:54
  • $\begingroup$ +1 for @BrianM.Scott's "roadmap". The difficulty of course lies in drawing a suitably sparse map. $\endgroup$
    – Willie Wong
    Jul 11, 2013 at 8:10
  • $\begingroup$ @Brian: According to the Geneva conventions, not ideal for torturing either. Although sometimes it seems users like to use it as a torturing device... :-) $\endgroup$
    – Asaf Karagila Mod
    Jul 17, 2013 at 16:33
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One neat thing you can do is use >! spoilers:

$$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-a)^{k+1}}\,dz$$

That way you can basically first write a "real" answer and then hide away the parts that you think are the ones the OP should first try to deduce by themselves. In contrast to reiterating which part the OP still has not understood, this way probably saves a lot of interaction.

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Someone recommended Polya's How to Solve It as a text for learning how to write mathematical proofs, but I think it's a far better text on how to give hints.

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  • $\begingroup$ I have read it (in fact I have it on my shelf and I take a peek every once in a while). $\endgroup$
    – dtldarek
    Jul 17, 2013 at 15:30

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