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This question: How to Garner Mathematical Intuition is (from what I understand) about how to "teach intuition". The question does seem to be off topic because it opens up to opinion based answers.

But the question actually does good in that it cites an academic journal article on the topic. The article seems to outline the "issue" that on the one hand for mathematicians intuition is a key tool, but on the other, intuition might not be taught (see my answer to the question for more on my view on this).

So the question, here, is whether or not this question is on topic. It seems like a completely valid mathematics education question that would fit well with a non-existent RUME (research in undergraduate mathematics education) tag. But on the other hand, I do agree that the answers will come close to being opinion based.

So maybe more specific: Are questions related to RUME on topic?

One might say that the question is off topic because it asks for opinions about what can be done about something that an article has pointed out is a "problem". And so one could say that RUME questions are on topic if they just ask objectively about what has been done in research. But, in my opinion, those two would be difficult to separate since for example here, some might argue that there isn't a problem...

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    $\begingroup$ Relevant blog post: Real Questions Have Answers. $\endgroup$
    – dtldarek
    Commented Aug 4, 2013 at 20:25
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    $\begingroup$ This seems a good question to me, and on topic for this site. If there are worries about objective answers, make it CW. $\endgroup$
    – Andrés E. Caicedo
    Commented Aug 4, 2013 at 22:38
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    $\begingroup$ I would welcome such questions. $\endgroup$
    – Potato
    Commented Aug 4, 2013 at 23:36
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    $\begingroup$ Are you so sure the cited question is a RUME question? It starts out very promisingly, by citing a RUME article. But it doesn't ask about how to teach intuition. It asks research mathematicians to speak up about acquiring mathematical intuition. To my mind, as soon as you direct a question to research mathematicians, you are in danger of losing the thread to math education. (Maybe it sounds like I'm being ironic; I'm definitely not.) $\endgroup$
    – Pete L. Clark
    Commented Aug 5, 2013 at 0:16
  • $\begingroup$ @PeteL.Clark: I have only skimmed the article that was cited in the question. I probably should have read all of it since I gave an answer. It sounded like the article was making a push for trying to actively teach intuition in some way. But as you say, it is probably more correct to say that the article asks research mathematicians to speak up. The question posted here on Math.SE. was about how we can teach intuition. I have some familiarly with RUME and to me that sounds like a RUME question. In my answer I was trying to say that maybe it isn't something we should teach directly. $\endgroup$
    – Thomas
    Commented Aug 5, 2013 at 0:45
  • $\begingroup$ @PeteL.Clark: (cont.) And maybe that is because it can't be taught directly, but is something that is acquired indirectly. Maybe I actually disagree with the article. I will read the whole article now. $\endgroup$
    – Thomas
    Commented Aug 5, 2013 at 0:47
  • $\begingroup$ I realized that it might have sounded like I am more familiar with RUME than I actually am. I have very little experience or knowledge... $\endgroup$
    – Thomas
    Commented Aug 5, 2013 at 0:58
  • $\begingroup$ @Thomas: Don't worry, I haven't read the article either! (And everything I know about RUME comes from watching colleagues: ME is very big at my university.) $\endgroup$
    – Pete L. Clark
    Commented Aug 5, 2013 at 2:11
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    $\begingroup$ @Thomas: I just read the article. Unfortunately my first reaction was that I didn't like it very much, and I was even a bit disappointed with it as an exemplar of ME research. (For instance at one point the author writes "In my research, I found" by which she means "When I talked to mathematicians, they said...". The idea that chatting with a bunch of mathematicians and recording their answers as quotes could constitute an important part of ME research is a bit distressing to me.) Still, there is some food for thought there. $\endgroup$
    – Pete L. Clark
    Commented Aug 5, 2013 at 2:39
  • $\begingroup$ @PeteL.Clark: Ok, so I finally read the whole thing. From what I can see, this article fits in some type of pedagogy language. The author talks about models that she works with and says that the article is a continuation of previous work. One might would probably need to understand this to get what she is writing. That said, I don't understand what she is writing. At the end, she for example says: $\endgroup$
    – Thomas
    Commented Aug 5, 2013 at 16:45
  • $\begingroup$ (cont.) "I would like to encourage mathematicians, indeed anyone who has responsibility for the learning of mathematics, to open mathematical activity to include the subjectivity of intuitions, to model their own intuitive processes, to create the conditions in which learners are encouraged to value and explore their own and their colleagues' intuitions and the means that they use to gather them..." My question would now be: ok, so how do we do that? Nothing in the article seems (to me) to touch on how one might actually do this. And there is nothing about if this can be done. $\endgroup$
    – Thomas
    Commented Aug 5, 2013 at 16:46
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    $\begingroup$ (cont.) She does say that some mathematicians are saying that they get intuition through experience and knowledge. Ok, so I guess we just have to keep the actual math to gain the experience so that we can develop the intuition. Also, this article is clearly an opinion based article. But the author doesn't do much in terms of dealing with possible objections. Isn't that a requirement in an argumentative essay/article? (And this is again the reason for asking whether the above question is even on topic. I do personally like the discussion and the topic). $\endgroup$
    – Thomas
    Commented Aug 5, 2013 at 16:49

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First, let me address the titular question. If research in math ed. questions are not appropriate for MSE, then they need to be addressed either at MO or some other site. I think that such questions would be better received at MSE than MO, and I don't want to start occasionally browsing yet another math QA site, so for this reason alone I suggest that these questions be welcomed here.

Finally, specific questions involving research in math ed. need not solicit opinions. Particular math ed. questions that are subjective and/or argumentative may be closed as such, same as any other question.

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Let's distinguish two types of question

(1) "What are some research papers on how to teach intuition in mathematics?" GOOD QUESTION

(2) "In your experience, how can we teach intuition in mathematics?" BAD QUESTION

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