Morris Orzech - Linear algebra and rigour--mixed messages as an opportunity

I teach linear algebra to students just out of high school. My class consists of prospective majors in mathematics and theoretical physics, and other honours students who like mathematics and want strong preparation for its use and study. Exposure to mathematical rigour is a fitting ingredient for this course, but most of us would recognize the need to balance it against constraints such as time, student preparedness and aesthetics. What do students make of this ``balancing act''? Most apprehend it as a mixed message that engenders confusion about what is expected of them. Making our expectations explicit (``Here is a list of proofs you should know.'') is sensible, but by itself misses an opportunity to confront students with practical and intellectual aspects of proof strategy and proof culture. Finding problems and activities to exploit this opportunity is challenging, but my experience indicates it can also be a good learning experience for student and teacher.