A third group became animated when I mentioned the philosophical problems. A fourth group seemed moved only by applications. Moreover, there was not much overlap among the groups. The mathematics should connect to what interests the students. Moreover, I realized, it CAN connect.
Every mathematical idea has a history.
Many mathematical ideas are associated with a philosophical problem. And, of course, most mathematical ideas have applications. I was especially attuned to these ideas because my colleagues and I had been discussing ways to keep good mathematics students from leaving the calculus sequence due to lack of interest in, well, calculus. Those discussions, in which we shared experiences like mine, led to our developing a new curriculum for mathematics and science majors based on connections. The first-year sequence has replaced the first year of calculus taken by students majoring in mathematics and natural sciences.