Function composition is very different from passing one function into another and the latter is definitely a concept that most Algebra 2 students would not be able to grok.

I knew I was making a strong assertion when I mentioned function composition. I'm grateful for your feedback, but perhaps you could elaborate? I don't see why they have to be "very different". I think you called me on a certain case, but an example like this demonstrates that they are not so "very different":

    def a(b):
        m = 0
        return b(m)

Or even:

    def a(b):
        m = b(1)
        return m+7

I don't see how these are not a form of function composition. I'm definitely no expert. But I'd really appreciate an elaboration so I can learn from any mistake I'm making.

Function composition is a single, special case of a function accepting functional arguments. You could define the compose operator as below:

    compose : (a -> b) -> (b -> c) -> (a -> c)
    a `compose` b = \x -> b(a(x))

However, there is a whole spectrum of additionally possible functions which can be built to accept functions as arguments. Here's a couple of example:

    partial3 : (Integer -> Integer -> Integer) -> (Integer -> Integer)
    partial3 f = f 3
    map : [a] -> (a -> b) -> [b]
    map [] f = []
    map (x:xs) f = (f x):(map xs f)

The first one takes a function and applies an argument, 3; the next one takes a list and a function and maps the list using the provided function. The idea of functions as arguments is a very powerful one, and from my understanding one with which people sometimes struggle.

One way to look at function composition is that it is a function to which you pass two functions and out comes a function:

  compose :: (b -> c) -> (a -> b) -> a -> c
  compose f g x = f (g x)

I'm not sure about the math curriculum in USA but in linear algebra you have functions that operate on functions.

Eliminating side effects is about a lot more than variable scope. One example would be reading from/writing to a file.

I think it's an accurate description. The OS code that reads a file involves changing variables outside function scope, eg pointer writes to memory mapped hardware. Simply forcing all variables to be final, all the way down, really does eliminate all side effects.