My biggest mistake as a SWE (now in my 30s) was not learning higher level mathematics and allowing what knowledge I did possess to wither on the vine.
[edit] and I’m dreading my kids getting past elementary school math because they’re gonna be like “why the hell am I spending months of my life on quadratic equations?” and I’m not gonna have an answer, because IDK why we did that either. At least I have answers for calculus, even if they’re not much good (“so you can do physics stuff”, “right, but will I ever need to do physics stuff?”, “uhhh… unless you really want to, no.”)
That's the same answer as the quadratic formula, but makes a lot more sense to me! Of course I've cooked the numbers so that you don't wind up with surds in the answer, but those are just complications in bookkeeping, not in concept.
And good luck taking on calculus without being super solid in the mathematical tools you use against quadratics -- factoring, completing squares, manipulation of binomials, pairing up like terms, etc.
For example, the way to solve a quadratic is to reduce it to a form one knows how to solve (via competing the square).
The specifics of the mathematics are not the prize, the methods of thinking are.
I don’t think the methods of thinking per se are why we teach math, though. Might be part of it, but if that’s all, I think we could do a lot better for a lot less effort for all concerned. I think it’s because the math itself is useful. If in fact the point were to teach methods of thinking, I doubt we’d teach it as we do math—why would we, when it generates such resistance and loathing from so many students?
Edit: I can recommend this book for a self-guided study
https://archive.org/details/zeldovich-higher-mathematics-for...
The author was a Soviet nuclear physicist (who participated in the creation of the H-bomb), so his main point isn't rigor. It can be a nice change of perspective from standard American texts.