# What should our goal be in answering questions?

If my search has failed me and this has been covered in some form, let me know and I'll delete this.

The post which is motivating my question is this post, where the OP had reduced a problem they are working on to a particular claim. I saw the post and decided to answer, more specifically I gave an answer detailing how to prove the claim, so that the OP could complete the problem using the method they started with. In the morning I came back to the post to find that another solution seems to be preferred by other users over mine, specifically a solution which completely forgets everything about OP's approach and details a much "higher level" approach to the problem which, it seems to me, is also using facts that the OP may not be familiar with (I am making an assumption based off the content of OP's post that they are just becoming familiar with the basics of modules).

There is no doubt that this solution is more "elegant", but to me this situation raises the question, what is our goal when we answer posts on this site? Is it to assist students in completing the solutions they've begun themselves, or is it to document the nicest possible solutions to a given problem? If it is the latter, why do we bother insisting that students detail what they have already attempted for the problem? Or is it something else?

• The general purpose of the site is Q&A answer the OP needs, and answers others might be able to use. For the former we need to know what restrictions in knowledge the OP has, what restrictions in usage the question permits if helping with ( but not fully doing) homework, etc. – user645636 Dec 19 '19 at 17:27
• The process of carefully parsing a user's Question to find the point where they ran into a difficulty is often intensive, and you are to be praised for doing so. At first glance you were able to validate the user's approach, and this helps with their learning style. Other Answers might also help in a different fashion, by showing a more elegant approach, and even this doesn't benefit the OP, it may well help others. In other words, it's all good. – hardmath Dec 20 '19 at 3:36
• There is nothing wrong per se in giving answers using tools at varying levels of sophistication. The axe I have to grind with the posters of those alternative answers is that they don't always check for earlier appearances of their "elegant" answer. This is unhealthy in my opinion. Duplication of material becomes a problem when done in larger scale. Somewhat related. – Jyrki Lahtonen Dec 20 '19 at 6:29
• To comment on the last line: "[...] why do we bother insisting that students detail what they have already attempted for the problem?" Showing one's attempt is only one of many ways to provide context. (And, the one that least interests me personally, unless the attempt has some useful or interesting nontrivial idea.) – Bart Michels Dec 20 '19 at 13:20
• @JyrkiLahtonen : I have myself been occasionally guilty on this front (duplication of content). It is best that this is brought to notice of OP so that the duplicate content be deleted. Recently I deleted two of my answers when I received a comment regarding this. – Paramanand Singh Dec 21 '19 at 6:58
• It is usually best to help the asker in the way he/she intends, but you are free to propose alternative approach. However one should preface such answer like "this is not exactly what you seek, but is an alternative approach which you may find useful". – Paramanand Singh Dec 21 '19 at 7:03
• All phenomena on the site (including voting behavior) should always be understood against the backdrop of what is happening on a macro level: all incentives are designed to maximize the site owner's metrics, which only partially coincides with whatever the community may want. (To see this; consider how the nature of questions and votes will look three months from now if new users are not allowed.) A lot of things can change because of the design of the system, rather than what users (us) actually try. – Herman Tulleken Dec 22 '19 at 15:13
• The question could be clearer, but it's not a stretch to guess that it is "What do I do from here?". You answered it, the other reply didn't. When I do this, I try to at least start my answer by acknowledging that it is a different approach, and most of the time it's also an approach that I feel will benefit the poster. You did well, accept that things aren't perfect and move on. – Git Gud Dec 27 '19 at 0:06

Regarding the voting on the cited question, I have noticed that "elegant" answers using more powerful tools tend to get more upvotes than more "elementary" answers (at least in analysis and in the related tags which I follow). I think that there are two large categories of upvotes, which account for a majority of the upvotes on these kinds of answers (and note that I am veering wildly into opinion here):

1. Aesthetics, in the sense that mathematicians often prefer to prove things at a higher level, in a more abstract manner, using fewer words. In a lot of ways, this makes sense—the whole reason to build up a body of theory is to allow for the use of powerful theorems which can make difficult results appear trivial. Hence some more advanced voters (PhD students, faculty in math departments, etc) may upvote more abstract arguments because they appeal to their sense of aesthetics.

2. A cargo cult mentality, in the sense that there are a lot of undergraduates who use this site who are impressed by the use of powerful tools to solve simple problems. For example, a user recently asked about a YouTube video which made the following claim

First, suppose that our initial chunk is part of a parabola, or if you like a cubic, or any polynomial. If I then tell you that my mystery function is a polynomial, there's always going to be exactly one polynomial that continues our initial chunk. In other words, a polynomial is completely determined by any part of it.

One answer was relatively elementary: if $$f$$ and $$g$$ are polynomials which agree on an infinite number of points (such as an interval), then $$f-g$$ is a polynomial, which is zero on an infinite number of points; but the number of zeros of a polynomial cannot exceed the degree of that polynomial, unless that polynomial is the zero polynomial. Therefore $$f-g$$ must be the zero polynomial, so $$f = g$$.

Another answer invoked the Identity Theorem from complex analysis. A commenter to that answer noted that this was like swatting a fly with a nuclear bomb (and I can't disagree). Yet, as I recall, this answer had more upvotes. I suspect that many of these upvotes came from folk who may have been introduced to the Identity Theorem via that answer. They were not familiar with it before, but upvoted out of a sense of awe or worship for the use of heavy tools.

On the other hand, answers which use more "elementary" techniques have disadvantages which often prevent them from being upvoted. They tend to be more involved and technical, as one is restricted to using less powerful tools. Because of this, they are often harder to read and/or tedious to follow.

For what it is worth, I tend to prefer more elementary answers to elementary questions.

A last comment: I don't think that we insist that students provide their attempts. Rather, we insist that askers (who may or may not be students) provide context for their questions. The goal of providing context is to ensure that terminology is being used consistently, to help set the level of an answer, and to help make a question more searchable for future askers.

One form of context is an attempt at a problem—in principle, such context helps determine where an asker is really stuck, and also weeds out a lot of cheating attempts. In practice, I think that this form of context tends to clutter questions and make them unclear. If I were king of the world, I would not regard "an attempt" as context, and would ask for more useful information (what book is the problem from? what theorems can you invoke? what level of answer are you expecting? etc).

• Whether or nor '"elegant" answers using more powerful tools tend to get more upvotes than more "elementary" answers' is highly context dependent. If the tags attract more readers at higher level then that may be true. But if not the opposite may be true (I know that for a fact because I often try to expose higher-level ideas in elementary number theory and algebra questions and the opposite often occurs, e.g. trivial FGITW answers often get many more votes than deeper answers). This effect is exacerbated with wider elementary exposure (e.g. drive-by voting from the network Hot List). – Bill Dubuque Dec 19 '19 at 21:23
• Do you have a link to that post using the Identity Theorem? A quick search didn't work. – Bill Dubuque Dec 19 '19 at 21:37
• @BillDubuque I would have put it into the body of the answer if I could find the post again. I couldn't find it, and am relying on my memory. Regarding the content of your first comment, I cannot disagree---this is likely very context dependent; I should say that my observation comes largely from analysis related tags. – Xander Henderson Dec 19 '19 at 21:55
• @BillDubuque I've found the post, and edited my answer to include details. Note the post in question was---briefly---on the HNQ. – Xander Henderson Dec 19 '19 at 22:40
• Great! The theorem name was not in the answer, which is why my search failed. I added it. – Bill Dubuque Dec 19 '19 at 23:49
• What a fantastic answer! Thank you for your insight into this. – Teresa Lisbon Dec 24 '19 at 13:24

If a question can be answered in multiple ways it increases the value of the question since it can expose you to different styles of solving a problem. This is typically reflected in the upvotes to both the questions and the answers.

Anecdotally, the community values variety, brevity, and readability. Several of my most popular answers were second or third answers, one sentence, and in natural language. This is particularly true in questions about definitions or motivating examples.

Paradoxically, several highly valued answers were indirect and didn't answer the question fully. In these cases I gave them a general overview of how to solve the problem rather than the explicit computation. Helping a student understand the material is often more valued than the explicitness of the answer.

When computations are involved or a question requires a longer answer readability becomes an important. Answering someone else's question is an inherently social activity so presenting them in a way they can understand is important. Organized, well-formatted and easily followed answers have more value.