What should be acceptable as "context"?

Question

As we all know, questions lacking context are strongly discouraged on this site. This includes mainly "homework questions" that look a bit like:

Prove that $\lim_{x\to0}x^2=0$ using $\epsilon$-$\delta$ definition of the limit

and contain nothing else in the question body. It is for this reason that the close reason of "lacking context and/or other details" exists, to ensure that we are not spammed with and overwhelmed by such questions which display no effort on the part of the OP.

I used to think this was quite clear: if the OP showed effort, even if it might have led to little to no progress, then it has context and should be allowed and not closed. But recently, I encountered this question about how deep the liquid in a half-full hemisphere should fill it up to. Seeing essentially no mathematical effort by the OP, I was immediately tempted to downvote and close the question as off-topic due to lack of context. This is especially since the question is something that could come up in any introductory course on calculus, just phrased differently. But at the same time, the OP did provide some sort of "context", albeit a non-mathematical one---that they were trying to measure the exact amount of vanilla extract to use in a cooking recipe. This makes the motivation clear in some sense, but the "context" provided isn't what one would usually expect, and certainly not one I would have considered prior to coming across this question.

So, my question is: How exactly should we define "context" in general, and in this particular case, should the question be regarded as lacking context?

Answer 1

For the specific question, the context seems completely clear. The question reads:

How deep is the liquid in a half-full hemisphere?

I have a baking recipe that calls for 1/2 tsp of vanilla extract, but I only have a 1 tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere.

My question is, to what depth (as a percentage of hemisphere radius) must I fill my teaspoon with vanilla such that it contains precisely 1/2 tsp of vanilla? Due to the shape, I obviously have to fill it more than halfway, but how much more?

(I nearly posted this in the Cooking forum, but I have a feeling the answer will involve more math knowledge than baking knowledge.)

The way the question is presented it seems clear that is an actual practical question (as opposed to a word-problem in an educational context).

This context is relevant for at least the following reasons:

• The main point is the actual answer, the methods are basically irrelevant.

• An approximate answer is sufficient. Indeed, an approximate numerical answer might be more helpful than an exact answer whose numerical value is not apparent.

Now, we can still strive to explain the answer and give an exact answer etc. But, still the context gives a clear indication what is expected from an answer: a ready to use number, and likely something that lends some credibility to the claim it is the correct answer, everything else is a bonus.

On the general problem.

The "work" and "effort" aspect is often over-stressed. Historically, it is to a large part a compromise-solution. What I, and at least some others, actually would want is that users asking explain the bigger picture in which the question arises and pinpoint a specific issue.

It was then claimed by others (and lets admit that at least for the sake of argument but there is at least some truth to it, I think) that this is not a realistic expectation for users below a relatively elevated level of mathematical or more broadly intelecttual sophistication.

Thus, came the compromise, they could at least explain what they were/are trying to do.

While users should put some effort into the question, via presenting it well and thinking first about it, it is not necessary to present attempts if the specific concern of the question is otherwise clear.

Contrary to some believes that is often not the case for what goes under PSQ. One could ask many questions related to:

Prove that $\lim_{x\to0}x^2=0$ using $\epsilon$-$\delta$ definition of the limit.

One of them being: What does a typical instructor expect and accept as satisfactory solution to this?

That often may be the intended question, but sometimes it might not be it, and in any case it should at least be made explicit. Frankly, personally, I might even accept that as context.

Answer 2

I often claim to have a de minimus requirement for context, in which even a slight indication of effort or interest in a problem suffices.

It is with the intent of offering an example that I feel narrowly misses this qualification that I post this recent Close review instance:

Title: Prove : every 5 integer numbers have 2 numbers that their sum or difference or multiplication divisible by 10

Body: "I think that I can prove that with pigenhole-principle[sic] , but I сan't do that."

I've left a Comment inviting the OP to add more in the way of context (and to use the body to give a self-contained problem statement). My thought about this example is that the OP knew this was an exercise to promote/reinforce learning the Pigeonhole Principle, and so included that phrase in lieu of a proper explanation of the the problem setup and goal, omitting all but the vague suggestion of an approach and "difficulty encountered".

I find this post unsatisfactory for Math.SE's goal of collecting excellent content to help students of mathematics at all levels, but given the luxury of waiting an hour or two (the post was three hours old when I came across it), I'd like to see the OP respond to my Comment. So I chose to "skip" in review, but will vote to place it on-hold (Close) if there is no response in the near future. [Absent that luxury I'd have voted to close it in its present form.]