Excel in Math

From elementary school through calculus, students are taught (implicitly, perhaps) that the path to success is through memorizing solution strategies. This is done by viewing many examples, extracting solution strategies from those examples, noticing similarities between the current problem and seen examples to decide which solution strategy corresponds to this problem, and modifying the solution strategy sufficiently to solve the problem. When done most efficiently, this process requires little-to-no thought.

Sooner or later, students—especially those who excelled in math thus far—are thrown into upper-division mathematics and are often baffled by their sudden incompetence. Perhaps they properly identify the cause and consciously make the necessary changes to how they approach mathematics, perhaps they become increasingly frustrated but stick with it and eventually figure it out subconsciously, or perhaps they conclude that math wasn’t really for them after all and abandon their passion to study something different.

However, I believe that all students can successfully make this transition if they know what changes they need to make and are motivated enough to implement those changes. 

I claim there are two such changes. Students must (1) learn effective problem solving and apply those skills to solve unfamiliar problems and (2) learn what a proof is and what it is not. While there are many other important skills students must learn and implement to be a successful math student (such as learning how to understand definitions and theorems, how to study effectively, test taking strategies), I believe those two are the most often overlooked and thus play the biggest part in aiding students through this transition.