Never On Sunday

When teachers tell you, by way of encouragement, that mathematics is beautiful, don’t believe them. (The same applies to math geeks generally making this claim by way of apology for how they got that way.) Math can be beautiful; certain select portions are quite beautiful; but some very important math is ugly, brute force thinking, and cannot be improved upon.

Take the quadratic formula. Engineers and other math users often have cause to find which values of x will produce a result of zero for a given polynomial. When last I was taking math classes, my prof said there was no general algorithm for finding all such values for any arbitrary polynomial on any arbitrary set. But we can, at least, find the zero values for any quadratic polynomial on the real numbers—although it’s possible one must resort to complex numbers to find them.

The formula for calculating these zero values and the method by which it is derived are within the grasp of any reasonably on-the-ball adolescent, which is part of the reason everyone gets taught it. Unfortunately, a formula like

x=[-b +/- sqrt(b^2 – 4ac)]/2a

is just a mess, and eminently forgettable. I find myself forced to reproduce the formula just to make sure I’m getting it right on the rare occasion I need to use it. High school teachers, understandably enough, tend simply to barrel through the quadratic formula when they reach that chapter, showing how it can be done, why this is the proper answer, and telling their students to memorize it, then moving on to something a bit easier to swallow. Unfortunately, they miss an opportunity in so doing.

The quadratic formula has its uses. Everyone using math on a regular basis should know it, and be able to reproduce the thinking behind it. But there’s a bigger lesson to take away: that in real math, the answer isn’t generally obvious or tidy, and that even professional mathematicians thrash about looking for answers.

The critical step in producing the quadratic formula is the idea that we can construct a perfect square by adding the appropriate constant, and figuring out what that constant is. It turns out to be b^2/4a – c, and getting there without knowing the answer in advance requires some trial and error work. Too many textbooks, and teachers with them, simply pull b^2/4a – c out of the air and demonstrate why it works. Doing so gives the impression that students should realize what goes in the blank as easily as the teacher who quotes it, which reinforces the notion that math is opaque, arbitrary, and simply beyond the reach of ordinary mortals. Slowing down, making a few wrong guesses, showing how they get closer to the expression we want, and finally reaching it allows a teacher, like my freshman calc professor, to demonstrate that false starts are normal, and that mathematical thinking, including homework, should include some mistakes—as long as you catch them and go back and fix them properly. As Jim Munkres, my freshman calculus professor and one of the best teachers I was ever blessed to have, liked to say while performing a similar trial-and-error search for epsilon neighborhoods: “That’s how mathematicians look brilliant. They go back and erase all their wrong answers.”

It’s a lesson that stuck with me, far better than anything I learned about the law of cosines, or differential equations, or even painted networks. Ultimately, it’s a much more important lesson than any specific method for calculating any specific answer, and a useful one for people who won’t go on to use advanced math in daily life: we all have to stop and figure out the hard problems. Tinkering a bit before you get to the solution is normal. And a false start doesn’t mean you’re no good at math, nor that the effort of taking another stab at a solution isn’t worth it.