Other useless school trivia: the quadratic formula | Hacker News Reader

Other useless school trivia: the quadratic formula

lemire.me

10 pointsplatz8 comments

As an aspiring mathematician and current math grad student, I am often confronted by students facing existential questions about life and the role mathematics plays in it. [For that matter, I also face these questions, just pitched a little differently]

I think I can summarize the existential angst into a single question: does everyone need to learn mathematics?

I waver on how I feel about this. On the one hand, the answer is clearly no. Not everyone needs to learn mathematics.

Recently, there was a post on HN that prompted discussion about whether or not going to college was necessary to achieve "the American Dream." One of the higher comments also said no. But not going to college limits the paths one can take forward. This is parallel to my current thinking about general mathematics education.

Being innumerate is bad for personal development. Numeracy is required for many professions while innumeracy is targeted and taken advantage of by news makers and, for lack of a better word, propagandists.

That aside, this doesn't actually answer the question of whether or not everyone should learn math. I suggest that there are reasons for everyone to become numerate. Becoming numerate in the American education system is thought of as a byproduct of the mathematical system. Learning the quadratic formula by rote does not contribute to numeracy. It seems plausible that having the mental faculty to understand its derivation does.

But I would also argue that becoming good at programming, physics, or chemistry (among others) also develops good numeracy. [I have thought about how "critical thinking skills," to use a buzzword, relates to basic numeracy. There are many studies, and the Math Wars is a battlefield]. In physics and chemistry, the quadratic formula might arise in the process of understanding something else. It doesn't bother me if a student sees a quadratic, knows there exists a formula to compute its roots, and then proceeds to look up/use this black box formula -- if the quadratic arose from good reasoning about some physical phenomenon (like finding orbits, or kinematics, or even making plans in games like KSP). To this, the author would reply that such a student is capable of understanding and perhaps even rederiving the quadratic formula on his or her own.

I suppose I could summarize my view by saying that I have conflicting thoughts on the merits for or against rote memorization in early mathematics... or for that matter calculus (which I'm currently teaching).

It's really easy to memorize if your teacher presents it to the tune of pop goes the weasel... Whether it's worthwhile to memorize or not, use a computer or not, the valuable conceptual struggle teachers need to focus in on is in what the answer means, and when to apply it, not in deriving the formula, though that can be a useful struggle if you doubt the power of proof or the rules of algebraic manipulation.

The reason there's little point in memorizing these formulas is that it's easier to use a computer. If you don't understand it anyways, why not just plug it into a computer? How is it plugging it into a paper formula any better?

Solving it by hand gives an exact answer, which in some contexts, might be preferable. Forman Acton, in his classic book, Numerical Methods That Work, uses the quadratic formula as an example to illustrate how a computer can give a wildly incorrect solution.

Computer algebra systems can symbolicly solve quadratics precisly (think Mathematica).

Right. Anything that a human can solve by following simple mechanical steps can be easily done by a computer. (See Turing's definition of computability.) Many people now are just following simple mechanical steps without understanding. They may as well use Mathematica or Wolfram|Alpha for such simple problems http://www.wolframalpha.com/input/?i=12x%5E2%2B2x-30%3D0 They should focus on learning how to convert real problems into a format Wolfram|Alpha can understand.

Knowing the formula is important even if you are going to let a computer deal with actually computing roots from coefficients you give it, because you can look at the formula and get insight into how changes to the coefficients affect the roots.

In real life applications, we are often not simply presented with coefficients and required to accept the consequences. The coefficients are often the parameters we get to control in order to achieve a desired outcome.

I disagree with the author. I've found that rote memorization (especially in quantitative endeavors) gives one a foundation from which creativity can spring.

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