Students fresh out of high school are often under the impression that mathematics is a discipline based entirely in recognizing the type of problem and applying an algorithm or cookbook method. These students think there are finitely many types of problems, and each type of problem corresponds to a method. From this perspective, "doing math" is about, in order,
- Identifying the type of the problem
- Remembering the method that corresponds to the problem
- Executing the method
However, one of the main goals of quality math courses is to carry students up to the next level of Bloom's Taxonomy, where they might comprehend instead of just remember what is going on. In fact, we hope they will comprehend strategies or facts that transcend specific "types" of remembered problems. I always hope to chip away at the idea of every problem being classifiable into memorized and cookbook categories.
I am interested in which topics in College Algebra or Precalculus best reward a student who ventures up a level in Bloom's Taxonomy in this way. These should be topics where a cookbook/remembering student will be able to solve the problem, but a comprehending student will be able to solve the problem more easily.