[MrVitaliy]

I think it's a weird way to learn math, and I learned it this way in school. Most of these courses just teach information memorization and recall. sin(x)^2 + cos(x)^2 = 1, etc.

I would start with something like Elementary Analysis: The Theory of Calculus, and work from there. You'll eventually arrive at the same place -- Calculus but from a much stronger mathematical foundation.

[ammasant]

You learned using the AOPS books? Don't be fooled by the titles, these books exclusively use a proof-based approach to construct a pretty wide foundation around these topics.

[blagie]

AoPS are among my favorite math books, but they're definitely not proof-based or particularly rigorous in terms of formalism.

They do focus on complex problem solving, which is equally important. The key value-add of AoPS are interesting, often beautiful examples and problems.

However, they don't do proofs or formalism much. They don't do applications or show what math is useful for. And they completely, totally, and universally screw up units (you'll have problems trying to equate a length with an area and similar; that's true of their classes as well, and RSM is similar).

I don't think there's a one-stop-shop for math, though, which does everything right. AoPS is at the peak of their particular game (which is right in the name: problem-solving).

That's best complemented by:

- Something which does data, applications, visualizations, and storytelling well.

- Something which does early exposure / surface learning well

- Something which is more formal and rigorous in terms of proofs and derivations

- Something which touches on a broad set of interesting topics (graph theory, oddball parts of geometry, etc.)

- In 2024, I would add something which does computational mathematics well

Nothing I know of does all those well in a one-stop-shop.

[dcw303]

I have not found that to be the case, the books I have read have gone into deep foundational detail to build up knowledge. Perhaps you're referring to Vol 1 & 2 of "The Art Of Problem Solving"? I haven't read them but from what I know they are a distillation of core concepts for students looking to do competitive maths.

It's confusing because that title is also the name of the publisher / website of the series of the books I'm reading.