I know I don’t have a right to say this, that calculus is the language of mathematics. I’m not a mathematician, nor am I even good at math. Whatever it is that I’m calling mathematics, I do like, and have at various points in my life, all in the times since my own schooling, not during it, been fascinated by mathematics, math problems, and especially the calculus, which I’m now calling the language of mathematics because it now seems to me to include most everything else.
My own experience, no more than glimpses of the beauty of math through a dark glass as sit were, does makes me think that everyone could profit from some experience of the calculus, no less than of Shakespeare’s language, Franz Schubert’s Piano Trio in E-Flat (Notturno), the geological history of the earth, Charles Darwin’s theory of evolution, and any number of other of man’s achievements.
But this will never happen, as in my own case, until the content of a liberal education, math, music, art, science and the humanities, is no longer considered the substance of the school curriculum, but the on-going substance of our lives. for at the present time we have most of all we have impoverished the experience of the liberal arts by turning them over to the schools, and allowing popular culture to replace them as the principal content of our thinking and feeling lives.
As a civilization we lost the true path when we confined, at least for the young, their learning to their time in school. It wasn’t always that way. For most of man’s 100,000 plus years on earth learning was always the fixed and not dislodgeable accompaniment of life itself and never over except at death.
The calculus if beautiful. But it’s the rare kid in school who is able to appreciate this beauty. It takes time, as in my own case a lot of time. In school kids are rarely influenced, let along impressed or “knocked over” by the very best that has been thought and said (and done).
And if we insist too much on their grasping the beauties and truths of our lives, of some of what we have grasped only from many years of life experiences after school, we probably risk turning them even further away, insuring their falling even more tightly into the grip of the more comfortable icons of popular culture.
What to do? What to teach young children? I’m not going to write about this, other than to say that schooling should be most of all about acquiring skills, once with the bow and arrow, but now with any number of the countless instruments at our disposal, including pen and brush, measuring and mapping devices, computer programs, string and wind instruments, not to mention reading, writing, and arithmetic.
Knowledge will only come with time and experience. In any case what they “learn” in school they will forget unless they go on to use it in their lives. Whereas this is less true of skills, and if the skills taught have been well chosen for the individual learner it will be these skills that they will take with them no matter what they go on to do.
The tragedy of our public schooling, at least for a good number of kids, perhaps more than half if the critics are right, is not that the kids have acquired little knowledge and understanding of the liberal arts, but that they have dropped out of school entirely, or may have even graduated, but in possession of few or no skills.
But I’m getting away from my subject, which is calculus as the language of mathematics. Just today two experiences seemed to support my position. Earlier this morning I was watching a u-tube video of David Jerison’s Single Variable Calculus class on the Web via MITOpenCourseware.
His subject was implicit differentiation and inverses, but the language he used was mostly the language of trigonometry. He even said at one point that the “trig identities” etc. had to be learned and memorized, one had to achieve fluency in their use, as (he didn’t say it but I said it to myself) with acquiring vocabulary while learning a foreign language. So much of the earlier school mathematics seems to me now only to be a preparation for the calculus later.
The other experience was my discovery of Steven Strogatz’s 2009 Book, The Calculus of Friendship. Strogatz is the author of a recent series written for the New York Times, He says it best
Steven Strogatz is a professor of applied mathematics at Cornell University. In 2007 he received the Communications Award, a lifetime achievement award for the communication of mathematics to the general public. He previously taught at the Massachusetts Institute of Technology, where he received the E.M. Baker Award, an institute-wide teaching prize selected solely by students. “Chaos,” his series of 24 lectures on chaos theory, was filmed and produced in 2008 by The Teaching Company. He is the author, most recently, of “The Calculus of Friendship,” the story of his 30-year correspondence with his high school calculus teacher. In this series, which appears every Monday, he takes readers from the basics of math to the baffling.