And now, a brief break from non-blogging:
Today, I’d like to start with a specific example and move on to a general point. The specific example is a way to approximate the squares of numbers and then refine those approximations to get exact answers, and the general point concerns the place such techniques should have in mathematics education.
My last calculator broke years ago, so when I have to do a spot of ciphering, I have to work the answer out in my head or push a pencil. (If the calculation involves more numbers than can fit on the back of an envelope, then it’s probably a data-analysis job which is being done on a computer anyway.) Every once in a while, the numbers teach you a lesson, in their own sneaky way.
It’s easy to square a smallish multiple of 10. We all learned our times tables, so squaring a number from 1 to 9 is a doddle, and the two factors of 10 just shift the decimal point over twice. Thus, 502 is 2500, no thinking needed.
Now, what if we want to square an integer which is near 50? We have a trick for this, a stunt which first yields an answer “close enough for government work,” and upon refinement gives the exact value. (I use the “close enough for government” line advisedly, as this was a trick Richard Feynman learned from Hans Bethe while they were calculating the explosive power of the first atomic bomb, at Los Alamos.) To get your first approximation, find the difference between your number and 50, and add that many hundreds to 2500. The correction, if you need it, is to add the difference squared. Thus, 482 is roughly 2300 and exactly 2304, while 532 is roughly 2800 and exactly 2809.
I wouldn’t advise teaching this as “the way to multiply,” first because its applicability is limited and second because it’s, well, arcane. What a goofy sequence of steps! Surely, if we’re drilling our children on an algorithm, it should be one which works on any numbers you give it. The situation changes, though, after you’ve seen a little algebra, and you realize where this trick comes from. It’s just squaring a binomial:
[tex](a + b)^2 = a^2 + 2ab + b^2,[/tex]
which when [tex]a = 50[/tex] becomes
[tex](50 + b)^2 = 2500 + 100b + b^2.[/tex]
And once you’ve seen it like this, you can generalize it and cultivate your own tricks from this seed.
We were taught to multiply binomials in eighth grade, if my memory does not fail me completely. Our lessons involved multiplying a pageful of binomials with the good ol’ FOIL method, squaring another pageful, multiplying a few dozen more for review. . . eventually moving on to the exciting operation of factoring a trinomial!
I do not remember middle-school math with any particular fondness. And, let’s face it, I was a nerd (he says as he reaches for the diet soda bottle perched precariously atop the DVD box sets of Firefly and The Prisoner). I don’t know the best way to fold stuff like the squaring numbers near fifty trick into the curriculum, but I suspect that items of this sort might help at least some fraction of the students stay awake. If we were shown how the algebraic formulas could be helpful in doing arithmetic, then the lesson would have had more purpose and been more memorable, which I think would be beneficial. Budding scientists need to learn about taking successive approximations, too. And if we stick such lessons in a coursebook somewhere, then maybe we can boost the number of teachers at all levels who are wise in these ways, and who are thus capable of teaching them at other grade levels when the circumstances require it.
(Dealing with a complicated machine requires flexibility, and you can’t be flexible when you only know how to do one thing.)
Education is not a straight-line race. I see it more as a great big scavenger hunt (in a dark forest belurked with goblins, naturally). There are treasures which everybody needs to bring back, like how to do long division, and there is a path through the forest where the terrain has been pounded flat and much of the undergrowth trimmed back. The path works for a lot of people — maybe a majority, maybe just a plurality — but some have no luck following it, and others can scamper over the rocks to reach many of the prizes faster.