Disclaimer: This question is not about PSQs, or "Problem Statement Questions". Rather it is about what users want Math.SE to be. I bring up PSQs to frame the issue.
Background: Recently there has been a great deal of discussion (and disagreement) about how to hand PSQs. The relevant threads are here and here. The main arguments in favor of removing/closing such questions were, as I interpret them:
Too many "bad" questions hurt the signal to noise ratio of the site and discourage the users who make valuable contributions from using it (as they get less out of it).
Other sites such as Physics.SE and Skeptics.SE have successfully implemented standards similar to those proposed.
These points are mostly reflected in Douglas Stones's answer to Qiaochu's PSQ question. The arguments against, mostly gleaned from comments to Qiaochu's question, seem to me as follows:
Homework questions are certainly "practical, answerable questions based on actual problems that you face" to quote the FAQ. They are occasionally quite interesting (one would hope!) and removing all PSQs is akin to throwing the baby out with the bathwater.
It is not necessarily cheating to ask PSQs, and even if it were, policing such things is not our job.
Many students, especially non-majors, are truly lost in the face of some exercises, and don't know how to describe where they're struggling.
Main Point: It seems to me that these different perspectives are a result of differing views on what Math.SE is. As I see it, there are three possibilities:
Math.SE is a place to do math (at all levels), asking questions as they arise and helping others with theirs when we can.
Math.SE is a place to learn and teach math, in addition to doing it.
Math.SE is a place for anything math and math-related.
This is not exhaustive, but I feel it roughly describes most users' feelings. The problem is that policies befitting each group are often harmful to the others. The broader the aim of Math.SE, the less effective it is at each of its goals, because the signal to noise ratio for each individual goal gets worse, and users have to spend more time dealing with elements of the site that don't interest them. Accordingly:
Question: What is Math.SE?
I don't expect consensus; anymore than I expect consensus in a parliamentary election. In any site of this size there are bound to be disagreements, and actions must be taken that will upset some users, as avoiding such actions would paralyze the site the same way requiring a consensus in parliament would paralyze a country. But I think this question needs answering before we can develop policies appropriate for the majority vision of the users of this site.
Some aspects of this have been discussed before. See here, here, and here.