If that was really true then teachers should encourge students who find an alternative but correct way to solve a problem. Most do not.
I'm not saying teaching math isn't important, I'm saying we are teaching it wrong. And as long as we continue to teach math illogically, I am going to continue to agree with students questioning why they need to learn this.
Except most equations you learn don't have common applications for most peoole. And you can easily look up the formula if you need it. Memorizing these things is only useful if you use it a lot, and if you use it a lot you will memorize them organically.
It teaches logic because it is a systematic system almost into perpetuity. However, there is some deviation when you get into advanced mathematics. Anything that is systematic will also accomplish the same thing. You can teach logic with a systematic approach to grammar, reading, definitely science, writing, music, or really just about anything if you set rules and always follow those rules. Of course once you've learned all the rules you then get to break them but that's where that advanced bit comes in to play.
Mathematics is unique in that the relationships it describes exist independently of our ability to apprehend them. Pythagoras' theorem was true long before we decided that the adjective precedes the noun, and it will be true long after there are no rational beings left to teach it.
No, it's about teaching the basic axioms that are necessary to understand the underpinnings of more complicated mathematic axioms. This, of course, does tend to improve logical thinking, but that's not the main point.
I'm not saying teaching math isn't important, I'm saying we are teaching it wrong. And as long as we continue to teach math illogically, I am going to continue to agree with students questioning why they need to learn this.