I've spent about equal amounts of time on programming and mathematics, but ... I'm confident that I can solve most typical programming problems, while even basic math problems are far more intimidating and error-prone ... I believe this asymmetry is due to the fact that one can "interact" with computer programs.

Quite true. This is one of the reasons there's so much interest in developing interactive proof assistants (HOL, Coq, Isabelle/Isar...) so that they can be used for "ordinary" mathematics. Not everyone likes both programming and math but for those who do, developing formalized mathematics on a proof assistant is a very engaging and even addictive experience.