(Here I am focusing on math outside of the math taught in pure mathematics programs.)
If anything the focus on math being a tool should be even bigger than it is today if you ask me. Most if not all of the math you get taught in school is math that was invented to solve a problem, it was not created as art. Introducing this historical context would probably help students in understanding why learning math is useful instead of just teaching math without this context as is usually done today.
The beauty, in theorems and their proofs, comes in my opinion from understanding the nitty-gritty details. Without understanding the logical steps it's hard, at least for me, to understand where the beauty lies. I cannot appreciate mathematical ingenuity without understanding it. This is a bit of an abstract question, but perhaps you could elaborate on how you see this beauty without understanding?
Not OP, but I agree with OP, assuming by "people" they mean "people already inclined towards math." Self-evidently, IMO, most students need numeracy much more than they need pure math.
In school, I struggled with memorization & attention to detail. For the life of me, I couldn't quite finish an algebra-heavy problem without writing 3*4=15 or outright skipping a step in a conic rotation or whatever. I'd get dinged every... single... time, and got relatively bad grades.
Shortly later, when I arrived back at higher level math through cs & stats, I could ruin the curve in my applied classes. I'd relied so much on mathematical intuition early on to compensate for a lack of memorization ("what are the steps again? Whatever - I'll just think it through now") that I was quite good at putting pieces together in odd ways to solve data science problems.
Not to mention - with proper context, as you say, all that linear algebra that so annoyed me all the sudden seemed intuitive. "OHHHH so THAT'S why this exists"
I was first driven out of math by a relentless focus on math-as-a-tool, and was able to return at a level where it became more about creating new logic. I'm convinced that I would have thrived in an earlier class that focused on elegance & logic - what makes a good proof? - and I'd have stuck with it.
While I can relate to your complaint that some steps were a bit too simple to include... The point of all those steps was precisely to show that you hadn't memorized the results. At the risk of overly simplifying high school maths (it's hard if it's new) I would say that in high school the math is quite basic. You could pretty much remember the first step, not get most of it in between, and still produce the right answer because you remember the answer format. It's just not feasible to have a individual discussion with every student to see if they really understood the material. Requiring students to include the steps in between is a way to prevent memorization, although maybe not the best way for each student.
It's good that the same thinking served you well later on though!
Replace arithmetic with “math appreciation”? No thanks.
I’m still disappointed by the way music and art are taught at the elementary and middle school levels. Singing, playing an instrument, drawing are all things which can effectively be taught and are so much more useful than useless watered down “art” and “music” I had to take in grade and middle school.
Based on the username and its recent birth, you created this account just to paste this comment. What does it mean? (I think that you are commenting sarcastically on the way math is taught, but that's just a guess.)
If anything the focus on math being a tool should be even bigger than it is today if you ask me. Most if not all of the math you get taught in school is math that was invented to solve a problem, it was not created as art. Introducing this historical context would probably help students in understanding why learning math is useful instead of just teaching math without this context as is usually done today. The beauty, in theorems and their proofs, comes in my opinion from understanding the nitty-gritty details. Without understanding the logical steps it's hard, at least for me, to understand where the beauty lies. I cannot appreciate mathematical ingenuity without understanding it. This is a bit of an abstract question, but perhaps you could elaborate on how you see this beauty without understanding?