"Never ask a question you don't want the answer to." is one of my mother's better pieces of advice. Not always one of the easiest to follow. It comes to mind because I've been thinking off and on about Carl Zimmer's article An open letter to science students and science teachers. See the article for some examples of how not to ask questions, and the comments for how common it is to receive bad questions. Here, I'll take up the matter of how to ask good questions, at least as far as I've figured this out myself.
One step is to follow my mother's advice -- if you don't want the answer, don't ask the question. Another flavor of this mistake is to ask a question when you only want a particular answer -- say to ask whether something is true, and then being angry when the answer comes back the 'wrong' way. Not common in Zimmer's illustrations, but extremely common on blogs.
Since science is about learning more about how the universe works, a good question is one that helps you do that. "What about frogs?", for instance, is not a good question. Frogs are part of the universe, so learning more about frogs is a good thing. But that question doesn't take you anywhere. What about them? Are you interested in them as food? What they eat? Where they live? Whether they're poisonous? How to tell whether they're poisonous?
Those alternate questions of mine are narrower, but that's the point. The narrower focus lets the person you're answering know whether they've answered you -- which is a good thing. After the one question, you're ready to ask another and they're ready to answer it.
A second aspect which is violated often in the examples at Zimmer's is a failure to 'do your homework'. You don't have to know everything, getting towards that point is why you're asking questions. Where people go wrong is, for instance, to not read the article they say they're asking about. If the title is "How we know the moon is not made of green cheese", the question "Why do you think the moon is made of green cheese?" is a bad one.
Better is to ask why the moon looks like it's made of green cheese. The article answers how we know it isn't, but it didn't answer this question. (There is a greenish cast to the moon, at least to my eyes at certain times of the month.)
Even better is to follow up something that interested you from the article, like when the author mentions that the moon is made of a very dark rock (it is -- anorthosite, which is like what Hawaii and Iceland are made of). If the moon is made of a dark rock, why is a full moon so bright?
There are two good aspects to this question. First, it shows that you read deeper than just the title of the article. Second, it shows that you are trying to fit this piece of knowledge in with other things you know. Since scientists are constantly doing this themselves, they're happy to see you doing it too.
My son's 5th grade science class did a related good thing. Namely, after I had chatted about the kind of science I studied and asked the class for questions, first, they had questions, which was good. And second, they followed up each others' questions. It showed they were listening (always good) and the very good thing that they were bringing the earlier answers in to their minds and this was leading them to more questions.
It isn't for nothing that a common comment about a scientific result is that it raises more questions than it answers. Once you've gotten the hang of asking good questions, each answer breeds more questions.
Another thing I've seen in my visits to classes, including or especially middle school (grades 4-6, ages 9-12) is that if students do follow up questions, it's often only four or so questions before they get to something I can't answer, in fact that I know the answer is not currently known. Maybe they're the ones who will finally learn the answer. And, of course, open up a raft of new questions.
My question place posts are a place to practice.