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My kids are having trouble with determinants. I can sympathize; determinants involve fairly complicated and tedious calculation, and the motivation for handling determinants is not at all obvious. They get used later, of course! Not much point in teaching a technique that won’t be used. But, apart from some minor immediate applications that take more work than simply finding an answer by a combination of hand calculation and trial-and-error, determinants gain meaning only a month down the road, when you begin using diagonal matrices for Markov chains and Eigenvalues. And for teens with short attention spans and a shaky interest in the subject in the first place, a month may as well be a decade; by the time you get there, they’ve forgotten everything about determinants apart from hating them.

Just as bad, the need to push on to other topics before the term ends means there’s never enough time to demonstrate how much trouble determinants can save in the long run. They seem to be just another weary task, rather than a time-saver.

What determinants—well, the entirety of linear algebra, really—are for is computers. Many kinds of engineers do almost nothing but linear algebra in one guise or another, reducing complicated systems to “close enough” linear approximations, translating those into a huge matrix, and letting the mainframe crunch out the answer. Setting up the problem can take anywhere from hours to months; solving it can take microseconds (for answers that would take humans months to find) to days (for answers that humans couldn’t calculate accurately before the sun engulfs the earth). Problems get so big and complex that a whole branch of math exists simply to explore how to solve common linear programming problems more efficiently. But this, too, the students don’t get to see. There just isn’t time to show them, nor, for that matter, computers on which to demonstrate the principles in action.

Sharp students get an inkling of how these things may work, and how big the problems can get, on their own. (I recall getting excited about maximum and minimum values in calculus, and again in linear programming once I hit college. It seemed like this was the way the whole world worked, and we had been handed the key to perfecting the world: perform these calculations, and you can know how to make decisions for maximum effect, maximum return, minimum cost, minimum time, maximum anything you want.) But only inklings. And for the rest of the class, math quickly becomes a closed book.

I wish I could take one day a week, or even one day a month, simply to talk about what kind of things the techniques we study can really do, out beyond the timid and cartoonish world of exercises simple enough to do by hand. No, this won’t be on the test. No, there won’t even be any homework on it. Just sit back and listen. Give me your honest attention for fifty-five minutes, and I’ll try to explain why this seemingly pointless stuff is so cool. You don’t have to agree. You don’t have to care. You may never use this when you grow up. That’s okay. I just want you to grasp why other people will, and why this really is important, and why people with a passion for answers get excited about it. And I want you to realize that you can understand, loosely, how the answers are found and what the answers mean even if you can’t calculate them yourself today…or ever. You don’t need to master this right now. But you can “get” it, even if you can’t do it right now. And if you can get it, you will be a zillionty-billion times more likely to care when the subject affects your life, or when you need to vote on issues with a technical bent, or when you bump into a problem you can solve, with a little ingenuity.

But there’s never time. There just isn’t time. My kids need to pass their SATs, and my school needs them to pass its NCLB requirements and its Regents exams—even the kids who will never need to know this stuff in life or even in college. So I need to drill them in calculational methods, to make sure they can pass. And god, it’s so dreary. Even for the math geek standing in front of them, who knows what it’s all good for.

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