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> Advanced math that isn’t necessarily useful in your college career is absolutely essential to teach high quality thinking.

You've put into words something I wish I could tell everybody. The impact of achieving a level of rigorous logical thinking through math classes goes far beyond "Will I use it in my job," it makes a fundamental difference to the way you see the world and think through every decision in life.

What's so unfair is that, in my experience, whether a given student in average circumstances successfully attains this in either high school or college is, essentially, a crap shoot depending on the combination of teacher, student motivation and support, and any number of other environmental factors.

> calculus are not useful outside of STEM degrees or in “real life” is indeed what makes them useful since it allows them to remain essentially the only abstract thinking course high school students will be exposed to.

Nah, I would rather argue that discrete math is more valuable to students in both abstract thinking and in real life. In basic calculus, mostly you are just remembering formulas and equations like d/dx(e^x)=e^x. Not even touching the edge of abstract thinking.

> equations like d/dx(e^x)=e^x

That equation is extremely important though, it tells you that the rate of change for exponential functions is proportional to the current value. So in reverse, anything where the rate of change is proportional to the value is an exponential function and now you instantly know how fast those will grow.

So by knowing that we do get a lot of intuition for so many systems and processes. That is the power of math, understanding one thing improves your understanding for so many different otherwise unrelated systems.

So, play this for me.. Which specific profession I should be in for this to matter to me? Not that I should be curious about fundamentals, but something I can apply in a meaningful way?

Finance, economics, ecology, virology, electrical engineering, mechanical engineering, chemical engineering, software engineering, pretty much any field with a technical component (i.e one that uses numbers).

You could reasonably just look at 2^n for software, though in general usually continuous math is simpler then discrete math IMO. Doubling vs. e^x is kind of an exception to the rule, and if you do any software involving signal processing or simulations, you'll want to understand the continuous version.

Everyone has use for understanding and detecting exponential processes. And the most important and easiest to identify part of exponential processes is the derivative, their rate of change is proportional to their value.

It is like understanding basic dice outcomes, everyone should know that since it is so basic to understanding events that happens in the world, political discussions and advertisements and products.

> advanced math there is no place to hide flawed logical thinking

If that's what you think, your math isn't advanced enough :P (disclosure: neither is mine)

These days mathematical proofs tend to be dozens of pages long and require multiple experts days to check and validate.

I'm pretty sure programming fits your descriptions of nowhere to hide flawed thinking though.

> To the extent algebra and calculus are not useful outside of STEM degrees or in “real life” is indeed what makes them useful since it allows them to remain essentially the only abstract thinking course high school students will be exposed to.

Sure, but if "useless" things are desired, why don't we teach them comparative history of Hobbit society instead?

> so much like high school physics will not teach thinking but simply knowing how to apply the right formula at the right time...

Unfortunately, that seems to apply to a lot of high school math, including a lot of calculus classes. And, to shoot into my own ranks, college math and especially statistics classes are far from immune to this as well.