At almost any undergraduate institution mathematics instruction is by nature a combination of inspirational discussion and routine exercises. Up until recently I considered inspiration and routine to be mutually exclusive. However during my three years at Pacific I have become convinced that it is in fact a false dichotomy: one can inspire and teach necessary routines at the same time. This simple realization now serves as the foundation of my teaching philosophy. I believe that even the most "boring" mathematical problem can be turned into a discovery project if pursued properly.
I will illustrate my approach using the following standard exercise from Multivariate Calculus: Find the set of points in space which are equidistant from two given points A and B. The set in question is a plane perpendicular to AB passing through its midpoint. In principle, one can arrive at this conclusion with little effort but then the answer is not very revealing and often leaves the student with no sense of accomplishment. In order to bring the class to deeper understanding of the mathematics involved, I suggest that the problem is first "projected" into two dimensions. In the plane the set is a straight line perpendicular to the segment AB, passing through its midpoint. The image of a straight line invariably provokes a conditioned response of finding the point-slope equation. Yet the point-slope description does not apply in three dimensions! Projecting the problem into lower dimensions instead of solving the problem directly leads students to deeper appreciation of the distance formula and a [somewhat startling] realization that the familiar concept of slope needs to be replaced with a more general concept of a vector. Thus by looking at the problem from different angles students accumulate important mathematical insights faster than it usually happens when they follow examples done on the board.
Once the answer for particular points A and B is found (in any problem), I ask the students what would happen if the points were allowed to vary. It soon becomes apparent that only the numeric coefficients in the equation vary while the equation itself remains linear. This leads to an early discovery of the general equation of the plane in three dimensions which the students would otherwise see much later in the course. As a final challenge, I sometimes ask the class to describe the points that are twice as close to B than they are to A. The answer in this case turns out to be a sphere and here again the students have the opportunity to exercise their creative thinking by projecting the problem into two dimensions and following the intuition they developed earlier on. Most importantly, rather than doing the bare minimum the students learn to investigate mathematical problems, as well as interpret and articulate their findings. Getting the right answers is only a part of the mathematical learning: the true mathematical skill is the ability to ask the right questions.