Re-examination of Standard Upper-Division Courses

Upper division courses in college are where math majors learn real mathematics. For the first time they get to examine the foundations of algebra, geometry and analysis, come face-to-face with the deductive nature of mathematics on a consistent basis and, most importantly, learn to do serious theorem-proving. For reasons not unlike these, most mathematicians enjoy teaching these courses more than others. While teaching graduate courses may be professionally more satisfying, it also involves more work, and the teaching of lower division courses — calculus and elementary discrete mathematics — is a strenuous exercise in the suppression of one's basic mathematical impulses. By contrast, the teaching of upper division courses involves no more than doing elementary mathematics the usual way: abstract definitions can be offered without apology and theorems are proved as a matter of principle. This is something we can all do on automatic pilot.

But have we been on automatic pilot for too long?

Mathematicians approach these courses as a training ground for future mathematicians. Even a casual perusal of the existing textbooks would readily confirm this fact. We look at upper division courses as the first steps of a journey of ten thousand miles: in order to give students a firm foundation for future research, we feed them technicality after technicality. If they do not fully grasp some of the things they are taught, they will when they get to graduate school or, if necessary, a few years after they start their research. Then they will put everything together.