> 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.