A different class

I've always found it compelling to think that our ancestors from thousands of years ago were no less clever, no less smart, than we are today, and that they merely had a bit less experience, had had a few fewer millennia in which to sort things out by trial and error and intentional experimentation, than have we. Given several dozen more centuries in which to try their hands at various critical and computational maneuvers, certainly they'd have come to many of the same conclusions as we have by now. (You must admit that we've been given a distinct advantage by the astute application of printing technology and modern methods of data storage, data recovery, and data transmission.)

One day at some point during my third year of undergraduate study at the University of Denver I was idly toying with some polygons that I'd circumscribed with a unit circle and I noticed it wasn't hard to recover an inductive formula for the lengths of the polygonal segments that made of a circumscribed 2ⁿ-gon a circumscribed 2ⁿ⁺¹-gon instead. With a little basic trigonometry (it turns out that the Law of Cosines works best) you can arrive at an iterated radical formula for the number π.

I was flabbergasted, thrilled by my discovery, and the next day I told my adviser, excitedly, about what I'd found.

His response was something along the lines of "oh, Euler's formula!" I'd recovered a formula first noticed by the great Swiss mathematician Leonhard Euler (the 300th anniversary of whose birth was recently celebrated in the math community), akin to an even earlier formula, the first successful arbitrary approximation of π, due to the French mathematician and astronomer François Viète.

If you're going to get scooped by someone, Euler, one of the most prolific mathematicians in history, is not a bad one by whom to be scooped. Still, that discovery that your discovery is not a discovery at all, or at least not a new one, can be unsettling. Certainly it's happened to us all, and it happens more frequently when you make it your business to ask tough questions. How often do even the biggest names in math research get one-upped by slightly cleverer colleagues?

Asking tough questions is the job of the mathematician, so it's imperative that young math-minded minds get used to tackling tough questions in a controlled environment, one in which the answers are already known to be known, and in which tough but tractable questions can be set up for what they are: challenges and tests of skill, yes, but not traps meant to lure the student into a sense of hubristic invention.

Put another way, if you know from the get-go that the discovery you're about to make is not a new one you can take your attention from the statement of the theorem on the page in front of you and place it where it really belongs, on the path you're about to trace out that will lead you to the theorem at its end. That same path, you'll know as you walk along it, is the same as or similar to the one taken by hundreds of highly intelligent human beings who came before you...but like they did before, you'll make your way along the path yourself, and the fact that the land at which you'll find yourself at the end has already been mapped out and explored doesn't make that land any less beautiful or wondrous.

Discovery is like that.

While running this morning I thought of a discovery activity I can use in MATH 280 this coming fall when it comes time to rap about equivalence relations, a topic that proofs dauntingly difficult to a large number of students.

I'll gather several dozen small objects of various kinds and bring them to class in a big ol' bag and empty the bag onto the classroom floor.

"Sort 'em out," will be the order of the day.

"How?" I can imagine students asking.

"You tell me." They'll pick through the pile of stuff scattered before them, and after a bit of trial and error patterns will emerge: the Tonka truck matches up with the lemon-shaped lemon juice bottle (for obvious reasons), and by the same logic the wingnut and the nickel get tossed in the same subpile, and the magnolia leaf meets up with the mango. Without realizing it, the students have constructed an equivalence relation, creating classes whose elements exhibit demonstrably reflexive, symmetric, and transitive properties.

"Can you do it another way?" The next iteration takes a bit more thought, and perhaps now inorganic objects are grouped together while once-living things share a different class. Or perhaps size proves to be the most distinguishing characteristic. Somehow a new partition emerges, and another equivalence relation is born.

A similar exercise may well work to demonstrate order relations. Confronted with a disorderly mess of objects, can the students impose some kind of order on them? What properties does this "order" satisfy? What properties does it not satisfy? Does the order need to be a total one?

Surely the students, without formal knowledge of the definition of the phrase equivalence relation will be able to build several such relations of their own, and having done so will be far likelier to recognize such relations when they encounter them in more mathematical contexts. Moreover, they'll have a greater appreciation for the technical definition of equivalence relations when it's given to them.

That's the power of discovery: you're much likelier to remember and understand something you discovered yourself than something someone else discovered for you and merely told you about.

Why in the hell don't we teach like this more often?

I know an answer to that question already (and my colleagues and students should feel free to supply many more in the comments section): because it's difficult to do so. Setting the stage for incipient discovery is far more difficult than describing what discovery looks like.

I admit that, though I hope that my classes set students up for discovery more often than those of less ambitious instructors, I make use of discovery-based pedagogical methods more rarely than I should. I'm trying, my friends, I'm trying to address that. I hope to devote a good deal of time this summer both to my own discovery (during the hours I spend with my REU students and the other students with whom I'll be doing original research) and to developing means by which I can facilitate others' discoveries on their own.

What discoveries, new and old, await us? I'm tremendously excited to set out on this summer's journey.

I am not Euler, and you are not me. Yet we're all human, we're all clever and intelligent, we're all naturally inquisitive, and we're all equally capable of discovery should we put ourselves in positions from which discovery is easily possible. In this regard no one of us is in a different class.