I remember struggling greatly in algebra because it was like a boatload of recipes to remember. There was never much discussion on what these actions meant or why you do them.

There is this awful commercial in the states for an online tutoring project where the student asks "how do I find the area if a triangle?" The response is "well, Cindy, the formula for the area of a triangle is 1/2 b*h, so you take half the base and multiply by the height and that's how you find the area of a triangle."

Non of that is false, but all the poor girl in the commercial learned was yet another reasonless recipe.

I have to disagree here.

Recipe's are a very helpful fallback where you might be struggling with understanding the origins of material in a rigorous manner. You can inspect a triangle all you like, but at the end of the day it's much easier to simply remember the formula.

This, I am finding, is the only way I'm managing to actually understand complex analysis - take the formulas for the results, and remember how to apply them. It's revealing to me that what looks complex gets very simple in that manner (and also that I still get tripped up by elements of basic integration).

If I couldn't do this, then I'd be lost - and in a test simplification remembering how to work through the definition isn't possible (and isn't required thank god) because that's a path which leads to me spending 6 hours figuring out and trying to picture something in a way which makes sense.

The formula for quadratic equation solutions is not that hard to memorise. It is useful to remember since it makes it possible to get the solutions instantly, whether numerically or algebraically. It also exposes discriminant of the equation, another useful concept (to instantly determine the number real solutions). Of course, the derivation of these formulas should also be taught, but it is really inefficient to derive things from scratch every time.

>it is really inefficient to derive things from scratch every time.

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.

Isn't it the case that 99% of the population just needs to know it exists, and, in the very, very, very rare case they ever need it again, they would simply look it up?

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.

Kids forget. If you just introduce a concept, spend a day or even a week on it, then move on to a different concept, most kids (if not almost all) will forget it by the beginning of the next school year.

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 was unpleasent for me to memorise it. Could you elaborate on why it should be a goal to make students calculate solutions to a lot of quadratic equations?

It seems to me that training to derive a lot of stuff would enable them to solve more kinds of problems, would it not?

Quadratic equations are common. Very common. In physics, geometry, differential equations and so on. It is also the next step beyond linear equations. It is nice to be able to solve these quickly and to be able to tell their properties just by looking at the coefficients. The sum of roots, the product of roots. The axis of symmetry of the parabola. Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics.

>Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics.

Could someone have told us that in high school?

Yes, but this is actually false, so it's probably for the best.

Anything of degree 5 or higher is not guaranteed to have solutions solved by radicals (that is, a solution that can be expressed as some rational number to some exponent). For example, x^5 - x + 1 = 0 cannot factored into linear and quadratic polynomials in this way.* The proof for the insolvability is actually quite elegant.

Even so, factoring a polynomial from degree 3 or 4 into quadratics or linear terms is hard. The most general way I can think of is using the rational root theorem and plugging a few values in.

* - You can factor using ultraradicals (yes, it's a thing), but that is far above highschoolers or undergrads, even.

Just because the solutions are not guaranteed to have a closed form in radicals doesn't mean that my statement is false. In the field of real numbers any polynomial with a degree higher than two is reducible.

http://en.wikipedia.org/wiki/Irreducible_polynomial#Real_and...

I remember reading (with horror) the methods for factoring cubics and quartics and thinking, "Well, I guess that's what they did before they had Newton's method and computers." Ewwwww.

I think I've actually used the quadratic formula maybe twice outside of an academic math class in which the quiz question was designed to invoke the quadratic formula.

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.

1. I'm implemented the quadratic formula in software more than twice over the years, I think. In, you know, real programs I was paid money to write.

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..."

It's useful in "shoot a ball out of a cannon at 50 m/s at a 27' angle, how far does it go?" problems.