[milesrout]

Procedural fluency is the basis for conceptual fluency. Maths is very layered: every layer builds on the next. You can't do algebra if your arithmetic is slow, as every algebra problem has half a dozen arithmetical subproblems. And you can't do calculus if your algebra is slow, as every calculus problem has half a dozen algebraix subproblems. And every differential equations problem requires you do to a bunch of calculus.

You can't understand a topic without doing lots of examples, so that you can feel out what is incident to the problem and what is inherent to the class of problems: when do we do integration by parts? When do we do integration by substitution? Etc.

So as a result you need to build procedural mastery at every layer before moving to the next and being able to build that conceptual understanding.

[Asraelite]

I strongly disagree. That's just how education is done as a matter of pragmatism.

In general it's very very difficult to impart conceptual knowledge to someone else because of the nature of human brains. We don't understand how we understand, so to speak, so we can't directly explain how to understand something to someone else.

We teach through examples because that is the best we can do. You're exposing someone to any idea over and over again until it clicks, but that click is what's important, not the examples. If you could somehow perfectly model how a student's brain works and know the exact combination of words to say and models to draw that would make it instantly click for them, then that's all that would be needed, no procedural mastery. But we don't know how to do that.

[Rendello]

> But we don't know how to do that.

That's because that's not how it works. From brains to LLMs [1], training creates a dense network of pathways that encode information and procedures. I lack many of the pathways to speak Swahili, and no mythical combination of words will change that. If I want to learn Swahili, I'll have to deliberately train myself to do so, building up those pathways incrementally. Then comes the click.

1. https://xkcd.com/2173/

[Asraelite]

How true what I said is depends on the domain. It is less applicable to subjects which involve simple facts with lots of rote memorization, and more applicable to subjects with difficult to grasp concepts.

You are comparing a subject near one of the spectrum (learning a language) to one that is probably as far as you can get to the other end of the spectrum (learning mathematics).