by Brendan Ames
I try to help my students develop their mathematical intuition and reasoning skills rather than simply teaching them how to make calculations. My lectures tend to follow a somewhat strict formula:
- Introduce a problem
- Discuss/investigate methods of solving the problem
- Arrive at a method for solving the problem
- Apply this method to a few illustrative examples
In a lower level class, Step 2 most likely takes the form of a simple example that can be extrapolated to explain some universal behavior, but it usually involves a more formal proof in senior courses. In either case, I am able to explain where the formula for making a calculation comes from (and hopefully solidify my students’ understanding of the formula) while also illustrating how it can be used.