A list is a monad

I mean if someone is learning chess for the first time, then yes you should start with the rules rather than jumping right into waxing philosophic about positional strategy to show off how smart you are.

I think the point is that a monad is a useful concept _purely_ because of what it _allows_ you to do, and _not_ because of anything syntactical. Those rules that you're obviating there, the commutative squares, are precisely what then lets us have powerful intuitions about these objects. The type signatures matter a lot less. If, for example, you don't have functoriality (which is false for `std::vector`, for instance, since `std::vector<bool>` is special-cased) you lose the ability to reason powerfully about abstract algorithms.

Thus, explaining the syntax and where the type variables go is explaining the least relevant thing about monads to their power and importance. It's certainly easy to showcase both the syntax and the list and maybe monads, that's part of the "monad tutorial fallacy". Gaining intuition for how to think about monads _in general_ is a lot harder and requires practice. Like, yes, list and maybe are "containers", but is `(->) t` a container? Is `IO`? How do these compose, if at all? What is this about "effect" semantics, "I thought monads were just burritos/containers"? etc. These are the hard, both conceptually and pedagogically, questions. Yes you need to know the syntax to use it in any given programming language, but knowing what scabbard your knife fits in doesn't give you the skills of how to use knife :)

Yeah I am aware that most teachers, like another commenter said, do teach math just by spitting out the rules and then people memorize them and sure who am I to argue against that. It's not how I learned math and most people I know who are well versed in math including those with PhDs (and working as a quant I am fortunate enough to work with many) also don't really learn things that way either...

When I taught my daughter chess at the age of 5, I did not teach her by laying out the rules and in general when I teach people anything, I don't start by emphasizing rules. Rules are often the least interesting part of any subject and I find rules are much easier to learn once a sufficient amount of motivation has been given first, and then we get to a point where we use rules to crystalize our intuition and express rigorously what we've come to learn. The rules are the end result of learning, not the starting point.

With chess, I taught my daughter by simply playing a simple game where I had a queen, and she had one king and two pawns, and her goal was to get one pawn across the board and my goal was to stop her.

The king and pawns are arranged so that with perfect play she is guaranteed a win, and eventually she learned the pattern that allows her to always win. I then switched it up so she gets the queen and I get the king/pawns but I arrange the king/pawns so that I will always lose if she plays perfectly.

After this, I added more pawns for white and a bishop for black. The motivation for adding these pieces is that to play perfectly and always win as white, you need to learn how pawns define the physical structure of the board, making walls and allowing for control over certain parts of the board, but this also comes at the cost of introducing clutter and making it hard for the king to maneuver.

After these principles are learned, then I introduce the knight, because the knight can jump over walls and isn't as burdened by clutter. I don't just put a knight in the game out of nowhere and make her memorize how it moves... by the time the knight is introduced, it feels natural to want a piece that has better maneuverability over the extra pawns.

And so on so forth... each piece is introduced with some kind of motivation for why it exists. It's not just there for the sake of existing because of some rule that you just need to memorize. It exists because it adds some kind of flavor and richness to the game and you end up absorbing the rules rather than just memorizing them.

Now I'm dwelling a bit on chess here... but the principle applies to programming as well. I don't think if you ever just listed the rules you did as a means of teaching someone about monads anyone would ever come to appreciate why these rules matter or why they should waste any time learning them and in fact I am fairly confident that this kind approach to teaching people functional programming is why it's taken so long for mainstream languages to appreciate these features and adopt them.

it's great you should write an article about this!