I did a degree in applied math. You’d think this would be “math you’ll use,” but the fact is that despite my program having a CS concentration, most of the stuff I did was not really applicable in practice.
However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?). Obviously in the former case I wind up doing actual proofs, and in the latter I make strong arguments based on logical consequences of established or presumed facts, or find flaws or gaps in arguments that are being considered.
I really wish I’d spent a lot more time on proofwriting than say, vector calculus.
Of course you may want specific math to solve real problems, and that’s a real need too! Not to diminish your point at all, just advocating for proofs to be seen in a practical light.
Proofs are indeed very practical. For example, I believe Leslie Lamport mentioned somewhere that he only came up with the final version of Paxos once he tried to prove it, and noticed that some condition he assumed wasn't necessary at all.
The reasoning behind having geometry be the standard high school sophomore math class is that that’s the age where kids would be ready to do proofs. Except that curriculum designers seem to have forgotten this and except in honors classes, most sophomores don’t get taught proofs in geometry and instead get a set of inert rules about shapes that they have no use for.
Geometry uses a deduction-only basket of proof techniques that don't prepare students for proofs done afterwards. I would like to see it replaced by elementary number theory which naturally uses a lot of induction/recursion and has some uses for proof by contradiction.
Geometry has really all that is needed for proofs:
* Axioms
* Substitution
* Modus Ponens
* Universal Quantification
Induction or proof by contradiction are just special cases of this.
But yeah, geometry for introducing proofs is difficult, because it is so easy to confuse visual intuition with proof. At the very least, you need a capable teacher who knows the difference. But nobody expects children to understand it all from the get go. A healthy struggle to disentangle intuition and proof, and then to entangle them again later on once you know the difference, that's the path to understanding mathematics.
I got a math degree, but I definitely failed at understanding that difference when learning geometry in high school. My teacher was good, but not good enough in separating that, and with classroom sizes getting larger by the decade, I think it's not the best approach to require that kind of tip-toeing. I'm not sure what would be better though, I've thought about maybe combinatorics could be a good replacement, but I also don't have the brain of a teenager anymore and I don't teach them either, so I don't know how far of a reach that would be.
The thing about geometry is that it does not take long before you've taught those four things, and then you start teaching stuff that is specific to plane geometry.
There are worse things to learn than plane geometry. It's actually a good thing to have a fixed topic to really learn those 4 things. Because once you really understood those 4 things, you are done, and you know everything about proofs in general there is to know.
Applied-focused courses (like modeling, optimization, etc.) are very broad, and I think that's where some application gets lost. For me, learning proofs within an applied-subject-domain (like partial differential equations) helped me a lot, because it was a different "flavor" of proofs, where you could understand that applied domain much much deeper.
However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?). Obviously in the former case I wind up doing actual proofs, and in the latter I make strong arguments based on logical consequences of established or presumed facts, or find flaws or gaps in arguments that are being considered.
I really wish I’d spent a lot more time on proofwriting than say, vector calculus.
Of course you may want specific math to solve real problems, and that’s a real need too! Not to diminish your point at all, just advocating for proofs to be seen in a practical light.