[talyian]

Learning math isn't "really, really hard", but it is unforgiving in a particular kind of way: you're basically required to internalize a concept fully before going on to later concepts. (This is coming from being mostly self-taught up through calculus-level math)

Students often don't fully grasp one concept before the teacher moves onto the next topic. Once this happens, the student is missing a crucial building block and tends to just get stuck with all the later things that build on it. This is a common experience in schools and I suspect is the origin of the "math is hard" trope.

It can be hard for a student to self-evaluate whether he/she has full mastery of a topic. This is exacerbated by situations where a struggling student will go to office hours or work with classmates and "solve" a problem, without having acquired the skills to solve slightly different permutations of that same problem on their own.

In programming, we don't have this problem. You can write HelloWorld without understanding console IO, terminal protocols, window buffers. You can write a webapp without knowing the OSI stack. Software is built upon abstractions.

[andrewla]

This rant got longer than I thought it would. tl;dr I think that "you just have to have the dedication to keep learning more even when the foundation is weak."

I'm not an educator so I can only speak from my experience and the experiences of people that I know.

I've found the opposite around foundational concepts -- very often you simply learn how to use them mechanically with no real understanding, and then move on. Only much later do you get the epiphany that lets you unlock the understanding.

I found this first with trigonometric functions; I had "SOH-CAH-TOA" memorized, and the double-angle and a+b things memorized, and could solve basic problems about triangles. It wasn't until much later on (in calculus) that I broke through and really understood what the hell was going on.

Limits I learned early in calculus, and found them incomprehensible -- I could mechanically complete epsilon-delta proofs and intuitively apply them to things like single-point singularities or rational function evaluation, and even follow the sin(x)/x proof, but it wasn't until much later (after differential calculus, and maybe even later than that) that it just clicked and I had trouble remembering what I found difficult about the concept.

Same with quotient groups in abstract algebra; it wasn't until field theory and Galois theory that I realized why my textbook used the Z/nZ notation instead of just saying Z_n like everyone else did.

Learning about asymptotes in rational functions was something that my high school did in pre-calc; that was a concept that I was perfectly capable of mechanically executing without any understanding. There are a million examples of this kind of thing for me.

And so on -- through differential geometry and topology and algebraic topology, complexity theory (formal computer science notation is made even worse if you've actually used computer programming languages, because it seems like an alien language -- it's always stuff like "take the machine S(sigma, alpha, q, r_alpha, delta) over the language M_gamma" -- what? In what sense is this a machine?). I can even feel echoes of this in terms of learning addition and multiplication vs. understanding what multiplication is -- why is multiplication associative is an intuition that came late to me, after I was able to see the geometric analog of multiplication.

For a long time my thought was that this could be fixed by introducing clarifying concepts earlier -- I feel like the concept of a limit as you go to infinity is easier to understand than local ones (something like, if the limit is infinity, a game, "hey, I say infinity, so you say, 'bet it's not bigger than 1,000!', and I say 'if I plug in 10, then the function evaluates to 2,000', and you say 'bet it's not bigger than a million'" -- if I can prove that I will always win this game, then I've proven the limit).

But I think now that the progression from rote and memorization to deep understanding is something that can't be hurried -- as a student, you just have to have the dedication to keep learning more even when the foundation is weak.