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Is there any research out there on how an instructor's philosophical beliefs about mathematics might affect some aspect of his or her impact as a teacher? My intended meaning of 'impact' is broad, covering not just how well students perform in test questions on a particular topic, but possibly including how likely they are to go on in math, lead successful lives, believe in ghosts, et c.

One could certainly imagine different philosophies coloring how we explain mathematics. A platonist might describe a proof as explaining and confirming why a given theorem is true, while a formalist might describe a proof as what makes the theorem true in the first place. A realist might explain the derivative as giving the velocity of a particle from the position function, while an operationalist would explain the derivative as an invented concept that, in our observations of the world, relates these two measurements. (Actually, the operationalist might treat velocity itself as an invented concept.) These differences may seem minor on each instance, but they can add up.

I am looking only for references to actual research on this question, if there is any. Please keep your own speculations on these topics to yourselves. I understand that effect sizes may simply be too small for any research to be possible.

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  • $\begingroup$ One statistic you might be able to find would be a comparison between active research mathematicians and those who are primarily in teaching positions. Do students of research mathematicians tend more to go into research? How would you fairly filter for bias on school rank etc.... good luck. I will delete this comment if it is distracting. Let me know. $\endgroup$ – James S. Cook Sep 8 '15 at 2:05
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My apologies for my small sample size... but, still, 40+ years of teaching almost gets me to statistical significance? :)

Yes, early in my career-arc, I was essentially conformist, and would aver that the course objectives officially espoused were truly what they were. Students did not "like" this, but could easily endorse it. (Yikes...) This is not about philosophy of mathematics, though, but about the social deconstruction of the university experience...

At the next stage, I still did believe that the traditional+orthodox description of "mathematics", and how one learns it, and the course sequences, and so on, were essentially accurate, if belabored. It took several years for this to be beaten out of me. :) That is, essentially none of my advanced undergrad or graduate students responded in the "appropriate" manner. That is, the philosophy to which I "adhered" did not sell, at all.

Next, try to explain the post-Wallis experimental/inductive reality of research in mathematics... Well, turns out, this is repellent to many of the people who've alread self-selected for "math major" (based on the traditional-orthodox pop idea of what it is...)

So, yes, philosophical, and socio-ideological, attitudes of the instructor have an impact on the students. Perversely, often kindness is mistaken for weakness, etc., because of popular social "constructs", ...

So, we don't quite get to any genuine philosophy of mathematics. It is hard enough to convince most students, especially undergrads, that mathematics could possibly be anything more than "a school subject".

And, then, the usual scenario in which lower-division math is merely a filter/weeder, rather than a genuine educational (much less scholarly/intellectual!!!) endeavor, subliminally confirms the apparent non-sensical-ness of trying to pretend that math is anything more than "another brick in the wall".

(Yes, I have tried to push back, against this sort of thing, but both colleagues and the students themselves have such powerful prior expectations that it's hard to accomplish much...)

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