Well yeah, it would be really inefficient to derive the formula and then plug numbers into it, but you don't have to do that, you can just teach people to complete the square instead. It's pretty much as fast as using the formula (since it's the same operations), but it builds on your existing equation manipulation skills and so you understand every single step. And if you ever need to actually know the equation for whatever reason, you can derive it easily.
I'd rather spend our children's time letting them know that these formulas exist, making sure they can perform them as needed, and then have them move onto something new. No need to drill / memorize trivial (as opposed to fundamental and foundation) stuff that is 2 seconds a way with google.
The worst kind of math education was that endless dribble associated with memorizing useless patterns that could easily be looked up in a book. At one point I had (painfully, oh my god, so painfully) memorized about 20 different patterns so I could chunk up integrals into products of u(x)v(x)only to spit them back on exam day, and never again look back on any kind of calculus. That was not a pleasant day week in high school. Would have been much better spent learning Geometry, Prob/Stats, Discrete Math, Linear Algebra, or any other host of mathematically oriented topics.
Personally, it think the time for manipulation of symbols has passed. Math should be only about concepts and hardware or software should do he manipulation. But then you run into systemic issues like "how do I test all students". The factory-styled learning environment needs to die.
It seems to me that training to derive a lot of stuff would enable them to solve more kinds of problems, would it not?
Could someone have told us that in high school?
I'm a computer scientist, one of the more mathematical occupations.
I've never been in a life-or-death situation where I thought, "aha, thank God I memorized that formula in high school!". Also, I don't actually remember it anymore, other than "negative b plus or minus the square root of b squared minus four-a-c, all over two-a".
Well huh. I guess I do remember it, but apparently at a verbal-auditory level rather than visual-symbolic. As usual for me.
Still: I've never had occasion to use that recitation I just made.
2. Have you ever been in a life-or-death situation where anything you learned in class in high school was useful to you? "Well, I would have died, but then I remembered that Julius Caesar's last three words were not actually 'Et tu, Brute!' according to Shakespeare..."
