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.