The problem with math is that the only strategy we’ve discovered (with any success) for making a concept easier to learn is to connect it with a concrete, real-world analogy. This works because people find it easier to learn something if they can connect it to something else they already know. The Achilles heel of this approach is that it discards the abstract form of the concept such that the full generality is lost on the student.

One example I see frequently around here is the much-maligned monad tutorial. Writers keep trying to connect it by analogy to concepts with which imperative programmers are already familiar. As a math student, I would much prefer to be given a basic statement of the definition of a monad, a list of properties, and maybe a theorem to prove that some structure is a monad instead of having to fall back to the definition. From there, I would just work through a few exercises and start to understand what it is and how it works. Most people don’t work this way, however. People seem to be extremely resistant to learning by experimentation. Instead, they prefer to be given all the answers.

I suppose this isn’t really a problem that’s unique to math. We’ve also discussed how a lot of computer users fail to truly understand how the software works and instead prefer to follow steps by rote in order to carry out one particular task.