Monad documentation hate

Monads are a useful construct in Haskell. Even though it's a pure-functional language, monads allow you to "cheat" a little, or at least pretend to, and carry state through computations (or do other stuff). This is pretty important, since a truly pure functional language wouldn't be able to do useful things like generating output.

Unfortunately, just looking at the documentation that exists online about monads, I wouldn't be entirely convinced that they're not just an elaborate joke cooked up by functional programming partisans - the equivalent of a linguistic snipe hunt. Let's look at some examples.

First, the Haskell wiki article on monads, since that comes up pretty high in search results. This is pretty close to coming straight from the source; I bet they have some great, readable documentation, right?

This is the introductory image they use. The caption reads: "Representation of the Pythagorean monad". Thanks, Haskell wiki! Since you haven't yet explained what a monad is, or why there would be a Pythagorean one, this diagram means nothing at all to me. Actually, it's worse than that - having put some serious effort into learning what monads are, and actually having a basic understanding of what they do, this diagram still makes no sense to me. Awesome!

Or, let's look at this tutorial called "A Gentle Introduction to Haskell". It's the first result when I search for "haskell using monads", so it must be pretty useful, right? Unfortunately, it begins by diving straight into the "mathematical laws" that "govern" monads, whatever that means. Actually, I'll just quote it:

Mathematically, monads are governed by set of laws that should hold for the monadic operations. This idea of laws is not unique to monads: Haskell includes other operations that are governed, at least informally, by laws. For example, x /= y and not (x == y) ought to be the same for any type of values being compared.

Which is great and all, but doesn't explain a whole lot to somebody who doesn't yet know what monads are. This is not how you write introductory documentation, guys! To borrow an example from the quote, you don't introduce the == operator by saying that it follows a mathematical law. You say what it actually does first!

So what is a monad, actually? As far as my (admittedly extremely rough) understanding goes, it's the functional equivalent of a wrapper class in OO languages. You can use it to wrap up some boilerplate code that would otherwise be really irritating, in a more or less standard way. Haskell also provides some syntactic sugar in the form of "do" blocks, which you can use for some monads to make your code look like imperative code.

See, now that wasn't so hard, was it?