A long calculation is like playing a computer game.
This is actually somewhat true, for the sort of computer games where you realize that you've gone down the wrong path and have to backtrack. In both cases you're trying to find your way through a maze of twisty little passages, all alike, from the beginning of the maze (the beginning of the game, or the state of having no knowledge) to the end of the maze (the end of the game, or the result of the computation). There are in both cases a small number of ways to succeed (to successfully reach the end of the game or the result of the computation), but usually more than one. (In the case of computation, "how many ways" there are to do a particular computation depends very strongly on what sorts of computations you consider to be "the same".) But there are a much larger number of ways to fail; the problem is to find a path to "success" in some reasonable amount of time.
Neither breadth-first search (where one does all the possible one-step computations, then all the possible two-step computations, and so on) nor depth-first search (where one follows a computation to its logical end, and only then tries something else) are really practical in most cases; it feels like a lot of learning "how to do mathematics" is learning when to give up on one particular approach to a problem and try something completely different. And a large part of that is to get rid of the qualifier "all alike" in my title. If equations look "all alike", one has no way of knowing what to do; mathematical education is as much about developing a facility for knowing what sort of mathematical expressions are amenable to what sort of tools as it is about learning how to use those tools.