Is the calculus the language of mathematics? Well no, but a language, and can everyone speak it at some level?
At my age learning new things, even old things that I learned much earlier and have forgotten, is not easy. It’s hard. But somehow of all the things that make me know that I’m still alive learning new things is best.
People of my own generation are now dying, and at increasing rates. I’m not, or rather I’m still alive, while more and more of my Harvard classmates and school colleagues, including John Updike, Ted Kennedy, and Ted Sizer, all of whom died last year, are not.
The latest new thing for me is the calculus. My first exposure to this was the elementary calculus course that I took my third year at Harvard, in 1952-53, being told, even then, that it would look good on my application to Medical School. During that year, and at best, I did learn how to take a derivative and an integral, but in doing so I was only applying to problems rules I had memorized.
But at that long ago time I had no more understanding of the calculus, than those who memorize the multiplication table or techniques for adding fractions and taking percentages understand the real number system on which they are based.
Today the calculus is one of the things that are keeping me mentally active. Much earlier it was language learning, always in order to be able to read the authors I much appreciated, if not loved, in their own languages. In particular I learned Russian and Italian through the stories of Chekhov and Pirandello.
Chess too, almost throughout my lifetime, was always out there challenging me, always something I wanted to become better at, but early on I realized that this was not to be. And it wasn’t age that stopped me from making the desired progress, rather something like the carrying capacity of my brain cells which was not up to carrying more than a move or two in advance of the board.
Let me say as a parenthesis just one thing about our ability to learn, because it’s important to understand that our ability to learn, most often and most unfortunately referred to as IQ or intelligence, is what most separates us, first in our families, then in our school environments, and finally in life.
Those who somehow reach Yale and Harvard, as the present eight, soon to be nine, with the confirmation of Elena Kagan, members of the Supreme Court, by their own oft repeated demonstrated ability to learn whatever it was in their path, have thereby separated themselves as by an impenetrable wall from most of their less capable of learning fellows.
It’s true in a very general sense that everyone can learn, and therefore that everyone ought to be provided with learning opportunities, as in school. It’s not true, however, that everyone can learn everything, and in particular that everyone should attend college, for the challenge of the latter, if, as it should be, substantial, will be too much for too many.
For it’s just not true in any number of specific instances that everyone can learn as much as everyone else. Just as some make more money, some will inevitably learn a lot more than others. Equality of results we don’t have.
Again this, our learning capacity, more than anything else is what separates us. And we ought not to fight it. It’s the way things are. We ought not to be trying to narrow if not eliminate such things as achievement gaps among various populations. For we can’t. The gaps are inevitable and unavoidable.
Instead of trying to paper them over we ought to cease giving them undue importance, as in the constant reference to school drop out rates or the widely varying achievement levels between different school populations. What one can learn, not what one cannot learn, ought to be getting all of our attention.
For me at the moment what’s getting my attention is the calculus. Something I can learn, and, in apparent contradiction to what I’ve just said, something I now believe everyone can learn, at least at some meaningful level. This is because of the richness of the subject, there being enough there for everyone, there being countless avenues of approach.
That there is enough there for everyone (also true of chess, by the way) brilliant inner city math teachers from Jaime Escalante on have shown.
Although the differential and integral calculus were only discovered, almost simultaneously in the 17th. century by Newton and Leibnitz, the underlying questions and concepts behind the calculus, all those ideas that the calculus would eventually explain, have been with us at least since the time of the Greeks in the 5th century BC.
Men had for long struggled with such things as the so-called paradoxes of Zeno, with measuring the instantaneous speed of a moving object, with computing the area under a curve. The calculus made the solution to these and many other problems ridiculously easy.
“With calculus,” and I quote, “the mathematical description of the physical universe became possible for the first time and modern science was born. Throughout the 18th and 19th centuries calculus was used to describe an enormous variety of physical phenomena from the motion of planets to electromagnetic radiation. Today calculus remains at the heart of the way we think of the physical world (science) and our methods for manipulating it (technology).”
This must have been what attracted me to the calculus. “It’s at the heart of the way we think of the physical world and our methods of manipulating it.” For me the calculus is also the culmination of all of school mathematics up until that point, that point being the calculus class. And stopping before that point seems to me now as being absurd.
Most of school mathematics, in particular algebra, geometry, and trigonometry, seem to me to have little if any real world application outside of the calculus. Yet it is these pre-calc courses, that for most students make up the whole of their middle and high school mathematical experience. Stopping before they reach the prize? Absurd.
For most students their experience of school mathematics is much like their experience of a foreign language. For just as most language students never get to the point of being able to use the language they have spent so much class time learning, most math students never get to the point of “speaking” the language of mathematics, which is not algebra, geometry, or trigonometry, but the calculus. (Perhaps an exception to this was Euclidean geometry which in ancient Greece may have been the language of mathematics, just as calculus is now.)
Now it seems to me that no less than the goal of learning a foreign language is to read and speak that language the goal of all those math classes is to eventually to be able to use the calculus. Unfortunately most language students never reach fluency in the foreign language, and most math students never make it to the calculus.
In fact, we go on pretending to teach, and the students go on pretending to learn.
The calculus is such a beautiful and powerful tool that it should be introduced if not taught early on, as was done in some “whole” mathematics programs, as in the now discontinued (helas!) SMP Math out of England. Once again, it can be understood at some meaningful level by everyone.
Exponential, logarithmic and trig functions, limits, sequences and series, polynomials and their graphs, polar coordinates and the unit circle, and all other such concepts, all the stuff of pre-calculus math, become, at least for me, most interesting and most alive when embodied in the calculus, the calculus enabling them, as it were, to blossom.
It does overwhelm me, all that I never learned before and am trying to learn now. Will I one day be able to speak this language with those who are fluent, and be listened to and understood? Probably not, but I am on a good road, and there’s so much out there to see and experience along the way. That’s more than enough.
