> Thus a multifacetted image of mathematics as a coherent subject, all of whose many aspects are well connected, is important for a successful teaching of mathematics to students with diverse (possible) motivations.
Somewhat useless personal anecdote to follow on this quote that I also liked:
Prior to my PhD studies I had heard of math and experienced it largely as computation. Arithmetic, matrix multiplication, integration by parts etc etc. This is, to my mind, the most terribly boring part of mathematics.
It's not certain, but I suspect that providing at least a few alternate characterizations of mathematics to students stuck doing computations for years and years will almost certainly help some of them find their way to regions of the subject that they find interesting.
I had a somewhat similar experience. No PhD but I had always hated math as a kid learning nothing but, as you say, rote computations in school.
By chance I stumbled on Frege's philosophy of mathematics/investigations into the foundations of mathematics in college and suddenly math was actually really interesting. I find proofs, set theory, algebras, and other mathematical domains closely related to logic way more fun than rote computations. I recall thinking I could have really fell in love with math and maybe even excelled at it if my schooling had ever given me so much as a hint that this stuff was also part of mathematics and it wasn't mere repetition of the same old computations, and that all those formulas build on each other and actually have very interesting justifications.
Somewhat useless personal anecdote to follow on this quote that I also liked:
Prior to my PhD studies I had heard of math and experienced it largely as computation. Arithmetic, matrix multiplication, integration by parts etc etc. This is, to my mind, the most terribly boring part of mathematics.
It's not certain, but I suspect that providing at least a few alternate characterizations of mathematics to students stuck doing computations for years and years will almost certainly help some of them find their way to regions of the subject that they find interesting.