Problem solving is a kind of thinking similar to root-cause analysis. But you cannot be an effective problem solver by merely analyzing problems as they come along. You have to work in a constant state of active analysis, always investigating and discovering what connects everything together under the covers, behind the scenes. That means learning why things work, not just how. It means understanding how each piece of task and function relates to other tasks and functions. It requires fathoming big pictures and digging deep into minutiae and everything in between.
Then when a problem comes along, you already know where to start. You investigate from a position of deep and broad knowledge. With that knowledge, and a little luck, you instantly understand the problem, you know the solution or where to look for solutions. Even with truly wicked problems, you are still in the best possible position to find the best possible solution in the shortest amount of time.
Being a problem solver is an attitude, maybe even a personality type. It combines an active mind, a root-cause approach, a curious and exploratory disposition, a tenacious spirit, and a desire to learn more every day.
I read a lot of articles nowadays about students, especially in math, who memorize formulas but never learn why the formulas work, never understand the underlying math that is the foundation of the formula. My wife and my son have both taught math, and both have said it’s a pervasive deficiency. In my own brief foray into teaching, I found the same depressing absence of concentration and attention most of the time. The path of least resistance might get you through college, you might even memorize your way to a 4.0 GPA and get a good job, but it will never make you a problem solver.