Most people can't do this or have a very hard time learning much easier things.
I find that the crowd that talks about "pointless learning/testing/education" are often either ones that struggled mightily and were never really that smart, or they are so smart that they are above it.
As for the “most people can’t derive the quadratic formula,” you might be right about my blind spots, but I think it’s equally likely that I’m right and have the necessary point of view to see that most people can’t do it because it’s taught and tested poorly. Both explanations would equally explain it being easier for me to derive the formula than memorize it.
> to see that most people can’t do it because it’s taught and tested poorly.
Or they just don't want to learn it. Some people don't like math. Or maybe people are lazy - they like it but can't overcome procrastination to really learn it. Or they are more focused on something else, like I was as an adolescent (computers.) There are certainly people who got sick, or went on vacation during that week of school, etc.
My point is, we always like to blame things that aren't actionable, i.e. "the system." It was just "taught poorly." There are certainly cases of that being the truth, but if you look at the time constraints and all other details, it's hard to just blame the system. How do you actually fix the system?
Yeah, given all the constraints, I agree completely. My complaint is that there is a giant emphasis on testing in really scalable manners that take people and dialogue out of it. So which constraint is the system (the way things are taught and tested) most sensitive to? I expect that it's teacher to student ratios, which my thesaurus says is a synonym for money, simultaneously the easiest and hardest constraint to change :-(
It's much more important to know that there is a quadratic formula, or more fundamentally that every quadratic equation has 0-2 roots (exactly 2 if dealing with complex numbers) and what the different cases loom like (does the parabola touch or intersect the x-axis?). It's typically more important to be able to solve a quadratic equation through guessing, factoring or completing the square. Because these things teach you something about how mathematics works, regurgitating some formula serves no purpose, you can just ask Wolfram|Alpha instead. And even if you can't complete the square etc., I'd much rather people understood the conceptual side of it instead of remembering formulas.
The importance has always been "how its applied." Tons of people failed classes despite knowing the formula. It has always been about the basic application.
I don't think many math tests consisted of "write down the quadratic formula" and that's it.
But nowadays, you can just feed the equation to Wolfram|Alpha (or Sage, or Mathematica, ...), so there is no point in blindly memorising formulas.
And I don't think I've ever had a formula sheet in one of my maths exams in school...
Every few years I need the quadratic formula for something, and I just derive the thing instead of remembering or looking it up. I’ve essentially traded some memorization for some understanding. We’re surely going to do it wrong, so I’d err on the side of too much understanding and too little memorizing. If you have a real feel for how a thing works, it’ll stick with you longer than the date of such and such battle.