
BILL BEYERS, Department of Mathematics and Statistics, Concordia University, Montreal, Quebec H4B 1R6, Canada 
Revealing the inner mathematician 
Teachers of university level mathematics often present the subject to their students in a way which is very different from the way they themselves understand it. The student is expected to see through the surface of the presentation (often rigorous, axiomatic) and somehow ``get it''. How you go about ``getting it'' and what it is precisely that you are supposed to ``get'' is usually unclear to the student.
Part of the problem is the logical structure of mathematics. When should rigor be introduced? In what context? And to which students? Bill Thurston wrote that ``what we [should be]doing [as mathematicians and teachers] is finding ways for people to understand and think about mathematics''. As teachers of mathematics, we must show the students how to think by revealing honestly how we think about mathematics.
These questions will be discussed in the context of various curricular reforms that we have undertaken at Concordia.