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If mathematicians and math teachers and textbooks in all cultures in all eras say that a certain theorem has been proved, then the reasonable layperson trusts them.

At the same time, a student who wants to understand mathematics works each day on trying to understand independently of such authorities why things in mathematics are as they are.

How widespread is knowledge of the point in the second paragraph above among those who have finished secondary-school mathematics? I.e. how many students know that pursuing independent understanding is the principal activity among those activities that constitute learning mathematics? Has anyone attempted to measure this objectively?

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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. // Added to clarify: I moved the comments to chat as they seemed centered around old versions of the question and as such obsolete. At the same time I did not want to 'completely' delete them as there seem to have been some lose ends that somebody may or may not want to pursue further. $\endgroup$ – quid Apr 29 '17 at 23:44
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    $\begingroup$ Related meta post: matheducators.meta.stackexchange.com/q/538/77 $\endgroup$ – Joel Reyes Noche Apr 30 '17 at 3:17
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    $\begingroup$ I would think this would not have been studied, because, although I do agree with the premise that striving for understanding first-hand rather than merely believing authorities' claims is a good thing, it is not clear to me that there is any sort of consensus about this in the first place, and many people would claim the opposite, etc. Indeed, it is my impression that very many people think math is about "following rules", even when the rules are inexplicable. $\endgroup$ – paul garrett May 1 '17 at 12:25
  • $\begingroup$ Googling on keywords like university student attitudes mathematics turns up a large amount of information, but it mostly seems to be about things like fear of mathematics, i.e., these are studies of the type of student population that has an inability to carry out mathematical algorithms. $\endgroup$ – Ben Crowell May 1 '17 at 15:33
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    $\begingroup$ Everybody please try to focus on the question at hand. If you want to discuss the subject, there is a chat-room already. One cannot move comments more than once; if the comment thread continues to grow there will be a comment-purge. $\endgroup$ – quid May 4 '17 at 18:26

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