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I've been running into a lot of situations like this:

OP asks how to find a solution to an elementary equation (in the sense of being written in elementary functions), but without an answer that can really be given in an exact form, something like solving $3^{x} + \frac{x}{5} = 2.7^{x}$, something that cannot be solved explicitly, but can be solved numerically. Somebody responds by suggesting that one finagle it a bit, and then appeal to Banach's fixed point theorem. This is of course a sound method, and will work (assuming such finagling is possible), but it doesn't strike me as very appropriate, because the question strikes me as a homework-type problem, or at least one that was encountered in or inspired by a class. I see this problem and think, "This cat's going through a pre-calc, college algebra course," so it seems sensible that a good answer would use the tools the student has at her disposal, and though I cannot speak for everybody when I say this, it was my understanding that something like Banach's fixed point theorem would not be encountered until a good bit later in a math education (for better or worse). Of course, the OP could look up "Banach fixed point theorem" and see what it says, maybe even understand the proof, but it would not be of much aid in the actual class.

It reminds me somewhat of a calculus class I took, and we were given a test covering Taylor expansions, and we'd been told to derive the Taylor expansion for $\exp$. I said that we could do this by understanding $\exp$ as the unique solution to the differential equation given by $f' = f, f(0) = 1$. I got no credit, and when I asked him about it after the fact, he asked (and probably rightly so) how I knew those conditions had only one solution, to which I mumbled something along the lines of, "Because you said so once." Obviously I could not answer his inquiry satisfactorily, and so I did not get credit on the exam. But even if I had given a complete proof that my answer worked, going through to show that any two solutions to $f' = f$ could only differ by a multiplicative constant, it still would've missed the point, because the goal of the question was to test my familiarity with Taylor expansions, and my method didn't do it.

Similarly, using Banach to answer a pre-calc question sounds not only like overkill, but moreover sounds like taking a foreign and undiscussed result/method that the student may very likely not really grasp, and missing the goal of working some other more elementary technique, and so to say, "Appeal to Banach" might be a sound and workable method, it doesn't meet the OP's needs.

So my question is, does MSE etiquette have a mechanism for this? Is there a way to determine the appropriate amount of "technology" for a problem? Is there a way to express to another answer that you think their answer is inappropriate for the OP and their needs? Perhaps should there be?

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    $\begingroup$ In cases like this, where the background of the questioner changes the level of the answer, it would be good to ask the OP for context: where they encountered the problem, what tools they have at their disposal (this might involve the course in which they encountered the problem). Of course, there is nothing wrong with an answer that is too basic or too advanced. Answers are not only for OP, but for anyone else who might find the question interesting. $\endgroup$ – robjohn Jul 1 '15 at 4:21
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    $\begingroup$ @robjohn - I suppose the way I think of it could be captured by a rather absurd inflation of the idea: Suppose you were in a middle school geometry course and some kid started telling you about Hilbert spaces, when all you needed was some simple plane geometry statement. $\endgroup$ – AJY Jul 1 '15 at 4:24
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    $\begingroup$ it would be good to ask the OP for context but beware that some users don't like it when you ask for context. $\endgroup$ – Joel Reyes Noche Jul 1 '15 at 4:45
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    $\begingroup$ My understanding is the SE sites are repository of knowledge in the form of question/answer pairs. Helping the questioner is a beneficial side effect. If the elementary question + advanced answer together can shine some light on the underlying problem and provide insight to other readers. I don't see any problem of providing an advanced answer. Of course, it is preferable the answerer warn the questioner about the the nature of hir answer and limit the amount of confusion. $\endgroup$ – achille hui Jul 1 '15 at 5:30
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    $\begingroup$ @AJY: the problem with being on the net is that you can be talking to middle school student as easily as a college professor and not know it unless they tell you. $\endgroup$ – robjohn Jul 1 '15 at 7:27
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    $\begingroup$ @JoelReyesNoche: they might like it less if their question were closed due to lack of context. $\endgroup$ – robjohn Jul 1 '15 at 7:30
  • $\begingroup$ @robjohn, I know. But when I said "some users," I didn't mean the OP. $\endgroup$ – Joel Reyes Noche Jul 1 '15 at 8:19
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    $\begingroup$ @achillehui this is quiet true. I feel though that under certain circumstances too advanced answers can degrade the quality of the site as a knowledge repository. I think the answers should try to target persons that might reasonably have the question asked to begin with. $\endgroup$ – quid Jul 1 '15 at 9:13
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    $\begingroup$ @JoelReyesNoche I think it is kind you made this warning, but at the same time we should try not to abandon doing what we think, and decided as a community, is beneficial for the site only because of some minority that tries to enforce their point of view, often taking a cavalier attitude towards the rules of the site. $\endgroup$ – quid Jul 1 '15 at 9:26
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    $\begingroup$ @quid Personal preferences for questions are by no means "site rules", no matter how many times you say it. $\endgroup$ – Bill Dubuque Jul 1 '15 at 13:23
  • $\begingroup$ I understand the philosophy that the site is also meant as a place where sojourners of the Interwebz can find answers to their own question, but there'd be no reason for me to ask in the first place if you couldn't provide something that helps me. $\endgroup$ – AJY Jul 1 '15 at 15:38
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    $\begingroup$ @BillDubuque you are perfectly correct. And, I never said otherwise. This does not exclude some users having a cavalier attitude towards the rules of the site, though. The plausibility of that claim is exemplified by some current and past suspensions. $\endgroup$ – quid Jul 1 '15 at 23:53
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    $\begingroup$ Related (10k, but I'd like it undeleted): meta.math.stackexchange.com/q/20486/23353 $\endgroup$ – apnorton Jul 2 '15 at 18:32
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    $\begingroup$ If I think that I can write an answer significantly more likely to help the asker than any yet posted, I simply do. I don’t worry about the existing ones, unless they’re actually wrong: they may help someone else someday (and I say that as one who is much more interested in helping the actual asker than in the archival function of the site). I just use my judgement and prefer to err on the low side. Parenthetically, I note that I would have given full marks for a complete version of your answer, and partial credit for an incomplete version: it’s my job to design questions that do what I want. $\endgroup$ – Brian M. Scott Jul 2 '15 at 22:05
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    $\begingroup$ @BrianM.Scott: I fully agree with your viewpoint. I have tried to express similar point in my answer. $\endgroup$ – Paramanand Singh Jul 4 '15 at 7:02
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I sometimes write answers to questions using techniques that are likely too advanced for OP. If I'm aware I'm doing it, I try to add a note at the beginning, something like:

This is an advanced answer.

or,

This is a more abstract approach.

or, probably best,

If you know the Banach Fixed Point theorem, ...

An answer is a written communication. It should anticipate to whom it is communicating. If you start with algebraic manipulations, without any preparation for the fact that you are about to make an appeal to some high-power theorem, it can be an irritating experience for the reader. It will feel like an appeal to a higher power - "Then, a miracle occurred." If you think OP, or anybody with a similar question, is unlikely to understand this answer, you are wasting that person's time by not warning him in advance.

As a general rule, I want readers to know what kind of answer they are getting before they start.

That said, any time there is a question asking to prove there are infinitely many primes of the form $4k+1$ or $6k+1$ or $30k+1$, some smart-@$$ writes in an answer that the result is a direct result of Dirichlet, which is hardly useful.

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So my question is, does MSE etiquette have a mechanism for this? Is there a way to determine the appropriate amount of "technology" for a problem?

It is requested from users asking question to provide context for their problem. Their background is such context. It is alright to ask for this background. Once determined, one can make a reasonable judgment.

Is there a way to express to another answer that you think their answer is inappropriate for the OP and their needs? Perhaps should there be?

A way to do this is a comment. Note though that not everyone will react positively to such remarks. In extreme cases you can consider casting a down-vote too (if you are convinced the answer is not useful in a strong sense.) It should however be kept in mind that absent any context what you think is appropriate level is your opinion and the answerer might just have a different one.

In my opinion to ask the questioner to make clear what type of answer they are looking for is the way to go.

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Thanks to OP for bringing up this topic. +1 for the same. I believe that an answer given on MSE has two purposes:

1) Help the OP get the answer to his question. This means that it is always a good idea to guess the level of OP (either from context of the question or by asking OP directly). If OP is not satisfied with an answer (just because it is at a different level) and faces similar instances multiple times probably he will be less inclined to use MSE. MSE thus loses one user (and probably a host of potential nice questions). Hence there should be some effort from users to provide an answer at a level suitable for OP and this kind of behavior should be encouraged.

2) Help others apart from OP get answer to the question posted. If an answer is on a different level compared to what is expected by OP then it may not be so useful to OP at that moment (and OP can indicate this by not voting/accepting it, or by a comment) but at the same time it may be useful to others who can comprehend the answer. This way it adds to the knowledge base of MSE and hence this behavior should not be discouraged by users of MSE.

I think MSE's mechanism of voting/accepting/commenting takes care of such answers in most appropriate manner. At least I never felt that answers at an advanced level are a problem for me. I can ignore them at that moment and perhaps return to them when my level has increased somewhat.

On the other hand I have seen that most of the answers which use advanced tools and techniques often get more votes (because of their small size) and answers which are based on simple ideas and elementary techniques are kind of treated as just another answer. I don't want to complain here, but I kind of follow "simpler is better" policy especially when dealing with maths and apply it during voting/giving an answer.

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    $\begingroup$ I happen to agree about the "simpler is better" philosophy, particularly as some of those questions can have some elementary but nice answers. I'm far more interested in a process that will give me a better notion of the solution to an equation, rather than just being told, "So if you take some point, and you iterate it a whole bunch, then you'll eventually get kinda there." At least give me a sense of how good my approximations will be when. $\endgroup$ – AJY Jul 4 '15 at 7:02
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    $\begingroup$ @AJY: I am glad that you like "simpler is better". It is only because some people have followed this policy that I have acquired some knowledge of maths without taking advanced courses in maths after my school years. When you are writing on web, you should put an effort to express your ideas in simplest possible manner to reach a wider audience. $\endgroup$ – Paramanand Singh Jul 4 '15 at 7:09
  • $\begingroup$ My experience is opposite yours: generally more elementary answers tend to get voted much higher than deeper answers (probably because they are comprehensible to a wider segment of the community, and they are quicker to compose, so generally have higher view count). $\endgroup$ – Bill Dubuque Jul 5 '15 at 2:21
  • $\begingroup$ @BillDubuque: Since there are so many questions on so many topics it is possible that our experiences can be different in this regard. $\endgroup$ – Paramanand Singh Jul 5 '15 at 4:34
  • $\begingroup$ Actually, I think the answers in the middle get most votes. :-) For a run-of-the-mill answer to get many votes: it should be short, simple, and contain a "nice" idea. (And this relative to the standards of the main math.se population.) $\endgroup$ – quid Jul 5 '15 at 11:01
  • $\begingroup$ @ParamanandSingh Possibly. Your tags seem to imply that most of your activity here is in calculus and related areas. But most of my activity is in other fields (number theory and abstract algebra). I would not be surprised if calculus is a bit different in that regard. $\endgroup$ – Bill Dubuque Jul 5 '15 at 12:38
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When answering questions, I think it goes beyond considering the complexity level of the answer or even alternative approaches to getting the same answer.

For example when I see answers that show how to fit a curve to some data and the resulting answer gives fitted curve goes through the cloud of points appropriately and adequately which answers the question, I also will consider if the residuals are what should be expected given the particular curve fitting method and what might be done with the estimates of the coefficients. Even if the fit has a good appearance, if the objective is to compare coefficients of different datasets, careful consideration of the residuals is imperative to believing one has appropriate standard errors for the coefficients. That kind of assessment for curve fitting seems to be rare in these forums. (And I'm likely to hear a lot of pushback about that.)

My point is that one shouldn't always just answer the question (at whatever level of complexity or approach) but also consider what might be done with that answer - even if it is just asking "Have you considered these consequences?" That's where I see the biggest advantage to all of the experience available in these forums.

To get to the original question: I don't think that there's a particular etiquette. It has to vary by the individual's expertise, experience, and time available. And when one oversteps with an answer, this crowd is not shy in stating opinions.

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