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I feel like calculus should probably start to be introduced in like maybe late-elementary-age? Certainly before high-school-age. I don't think everyone needs a dedicated 1-year course on it, but some of the concepts are certainly useful and understandable at that age. Regardless of whether its from compulsory education or some alternative education process.
Sorry for the rant. I long story short, I agree with you.
The quadratic formula.
When we learned to use it in algebra, it was just rote memorization that made little sense. We knew there was a proof for it, but we were told it was beyond our level and to just wait. When we finally touched on it again in Calculus, it was little more than a footnote. Since we had developed better tools for finding roots already, we did little more than note its existence and solve the problems more generally. I don't think we got around to the real proof of the quadratic formula until later with Linear Algebra. Most people aren't going to get that far. Most people don't have any need to. The quadratic formula is a bit of a chicken and egg problem. You need upper level math skills to prove it, but we learn it early in order to practice algebraic skills to get to that level.
I just wish that we'd have been taught some of those calculus fundamentals and ideas earlier. It would have been like a light at the end of the tunnel. Maybe we wouldn't be ready to rigorously work through limits and integrals before all that algebra practice, but even a child can understand acceleration and its relationship to changes in velocity. We have so many documentaries about special relativity, general relativity, and quantum mechanics. Almost no one watching these documentaries can do that math, but we don't worry about that. Our society could benefit from everyone having more general knowledge about the very broad strokes of calculus, differential equations, statistics, and combinatorics long before we worry about teaching the mechanics of those maths to them. Not everyone needs to know HOW to do them, but everyone can be taught to appreciate WHAT they do and WHY they are important and a part of every facet of our lives.
I remember trying to figure out if a specific infinite sequence converges or diverges because I was playing Sonic. I didn't have any algebra nor calculators that could handle precise calculations nor the terminology I refer to it as now, so I was just trying to guess by doing calculations by hand to see if it looked like it was plateauing to something or if it looks like it was gonna keep growing.
Not sure if its actually useful, but it was something I cared about regardless. Children should play with math more.