I recently read an article by a friend of mine, entitled "Why We Love Things That Won't End". It referred to TV and anime shows, book series', soap operas and the like. Ultimately, one theory was that we love them because while they're ongoing (and even afterwards), they foster creativity and community building.
|It's also freaking awesome to see how characters grow up over time.|
It got me thinking. Perhaps partly due to a joke my brother-in-law told me around the same time, about my "Taylor's Polynomials" math series. He said I'll have to keep it going forever, because the Taylor Series is infinite. Which raises a good point.
Math doesn't end.
But I don't mean that in the sense of integers continuing on forever. Nor do I mean it in the sense of "when will this period ever end". No, I mean it more in the sense of mathematics being all around us, and that there are always more discoveries to be made. (For that matter, Godel's Incompleteness Theorem even proved the existence of mathematical truths that cannot be proved.)
Now, the big news in discoveries for 2012 was whether Shinichi Mochizuki has found a proof for the ABC conjecture. But in May, there was also the matter of a teenager in Germany solving a 300 year old riddle by Newton. And who says discoveries have to involve theorems? A fair bit of statistical discussion also occurred recently, surrounding Nate Silver and his election predictions. We have the rover on Mars. And Canada got rid of the penny this year, so that's going to make a number of textbook questions irrelevant by 2015. There's always something going on out there.
Yet students think that, once they have the answer, "I'm done".
The system is partly to blame for this. We need something we can mark, be it a finished essay, a completed piece of artwork, or a solved math problem. Though society is also to blame. Mathematics is often met, not with wonder, but with fear, uncertainty, and the remark "well, I was never good at math". This results in a desire to have it be "done", as opposed to continue indefinitely. And of course, a solution being "the end" may ultimately have it's roots in the first QED (Quod Erat Demonstrandum, 'what was required to be proven').
Yet when the student perception of "I'm done" changes, it's wonderful.
Students form 'communities', both in and outside the class, in order to help each other out. (It's not enough that I've found an answer - you haven't.) They compare answers, finding new ways of reaching the same outcome. (It's not enough that I've found an answer - there was another path.) Sometimes they even start asking questions to extend into later problems. (Ideally less with "when am I ever going to use this" and more with "so what comes after degree mode".)
|Stop memorizing rules!|
The new question for an instructor then becomes, how can we change the "I'm done" perspective? More to the point, how can we do it without unnerving the student, while simultaneously keeping within the framework of all the standardized testing the government wants? (Even demands?) This, I dare say, is a work in progress. A work, perhaps, without end.
But that's not automatically a bad thing. Because it has created communities of mathematics educators, who are coming up with new, creative ideas. So here's to learning that never ends, and all the people that we meet along the way. Hope it was a good 2012!